=2 [10, 38]. Thus, the magnetization (4) becomes modified to

eq ¼ cN G !

Onsager symmetry relations then require [12, 39] that additional forces involv-

Now, using the latest modified magnetization (9), the magnetization part in (1) can be eliminated, similar to previous discussion in Section 2.2. We thus have the following ferrohydrodynamic equation of motion (including elongational flow

transversal, and oblique magnetic fields [6, 9].

Pattern Formation and Stability in Magnetic Colloids

4. Agglomeration and elongational flow effects

change the obtained states at least qualitatively.

, and the velocity u

þλ2S H! .

4.2 Modified internal magnetic field

!

! � H !

magnetic field H

S ¼ ∂iuj þ ∂jui

where G ¼Ω

effects) [12]:

20

the 2-CCF basic state.

4.1 Introduction and motivation

$$\left(\partial\_t + \overrightarrow{u} \cdot \nabla\right)\overrightarrow{u} + \nabla p\_M - \nabla^2 \overrightarrow{u} = -c\_N^2 \left[\overrightarrow{H} \,\nabla \cdot \overrightarrow{G} + \overrightarrow{H} \times \nabla \times \overrightarrow{F}\right]/2\tag{12}$$

where pM is the dynamic pressure incorporating all magnetic terms which can be written as gradients, i.e., ∇ H ! � H � � ! and ∇ Heq ! � H ! Þ� M h� i ! . See also (5) for comparison without considered elongational flow.

Assuming that the internal magnetic field is equal to the external imposed magnetic field (it is known as a leading-order approximation [24], see also following section 5) which is sufficiently good for a 'first' numerical investigation for the effect of elongational flow. Therefore, (10) can be simplified:

$$\begin{aligned} \left(\partial\_t + \overrightarrow{\boldsymbol{u}} \cdot \nabla\right) \overrightarrow{\boldsymbol{u}} + \boldsymbol{p}\_M - \nabla^2 \ \overrightarrow{\boldsymbol{u}} &= \mathfrak{s}\_N^2 \left\{ \nabla^2 \ \overrightarrow{\boldsymbol{u}} \ -2\lambda\_2 \nabla \cdot \mathscr{S} \ \overrightarrow{\boldsymbol{H}} \ -\boldsymbol{H} \\ &\times \left[ 2\nabla \times \left( \overrightarrow{\boldsymbol{\Omega}} \times \overrightarrow{\boldsymbol{H}} \right) - \overrightarrow{\boldsymbol{H}} \ \times \nabla^2 \ \overrightarrow{\boldsymbol{u}} \ + \lambda\_2 \nabla \times \mathscr{S} \overrightarrow{\boldsymbol{H}} \right] \end{aligned} \tag{13}$$

In this approach, the magnetic field and all the magnetic properties of the ferrofluid could be influenced by the velocity field, the transport coefficient λ2, and the magnetic field Niklas parameters sx and sz as described before (7). However, it is worth to mention that in this approach, the material dependent transport coefficient λ2 is a parameter [12], which can be scaled by the term describing the

elongational flow and is independent of H ! . Thus, it only appears in combination with the symmetric component of the velocity field tensor. However, real ferrofluids

show H ! dependence in the microscopic aspect of the dynamics of ferrofluids with chains or nonspherical particles [25].

## 4.3 Linear stability behavior and analysis

Before we come to the influence of λ2 onto the fully nonlinear Eqs. (11), it is worth to have a look at the linear marginal stability threshold of the different flow structures. Therefore, in the following, we first investigate the linear growth behavior of either stationary axisymmetric m ¼ 0 (TVF) and nonaxisymmetric m ¼ 1 (SPI) oscillatory vortex flow of a ferrofluid in the TCS that is placed in homogeneous axial magnetic fields H ! ext ¼ Hz e ! z (Figure 1). In the literature [4, 5, 7, 40–42], one can find many theoretical works that analyze (via linear stability analysis) the influence of rotational symmetric magnetic fields, i.e., axial, azimuthal, and radial ones on the flow of a ferrofluid in the TCS. However, for a symmetry breaking (e.g., transverse), magnetic field such linear stability analysis is not known for the corresponding basic state at such a field configuration. For a detailed description regarding the linearization around the CCF and solution of the linearized NSE, we refer to the Appendix in [43].

### 4.3.1 Marginal stability thresholds

In order to discuss how the transport coefficient λ2 influences the growth of m ¼ 0 and m ¼ 1 vortex perturbations, we determine the stability of the CCF basic state against these perturbations, considering a magnetic field H ! ext ¼ Hz e ! z, applied in pure axial direction.

Figure 9 presents the bifurcation thresholds in the Re2 � Re1 diagram for either axisymmetric, m ¼ 0, and nonaxisymmetric, m ¼ 1, vortex flow in magnetic fields at (a,b) Hext ¼ 67:7 kA=m and (c) Hext ¼ 135:4 kA=m for two different axial

respectively. Ignoring the effect at small Hext, all thresholds increase linearly with Hext. The slopes are given by ∂Re1,stabð Þ H =∂H and vary slightly either with λ2 and k. For outer cylinder at rest, Re2 ¼ 0, (Figure 10(I)(a)) the slopes of the m ¼ 0 and m ¼ 1 curves are comparable, they differ significantly for counter-rotating cylinders, Re2 ¼ �100 (Figure 10(I)(b)). For the latter, this results in intersections of the m ¼ 0 and m ¼ 1 stability boundaries. Such intersections have been observed

In general, modifications of the Hext field dependence of the bifurcation thresholds due to λ2 tend to increase, when the cylinders are stronger counter-rotating (Figure 10(I)). Re2 becomes more negative, and thus, the strain increases. On the other side, they also depend on the axial wavenumber k (see also Section 4.3.4). Considering fixed k, the modifications due to λ2 increase with increasing magnetic field Hext while remaining qualitatively the same. However, the general tendency visible in Figure 11, considering elongational effect, is to shift the stability boundaries downward in Re1. Concrete amounts depend on the parameters k and Re2. Note that this relative small destabilizing elongational effect is always overcompensated by the significant stronger general stabilization effect of the magnetic field itself.

Instead of looking at the growth rates—the real part of the eigenvalue of these solutions (previous Sections), one can also consider the marginal spiral frequency ω—the imaginary part of the marginal eigenvalue at the bifurcation threshold of m ¼ 1 (SPI) vortex flow. Figure 10(II) shows the frequency, corresponding to the m ¼ 1 bifurcation thresholds presented in Figure 10(I). Main effect is the increase of the SPI frequencies with the field Hext, accordingly to the increase of onset Reynolds numbers with Hext (Figure 10(I)). Thus, the threshold frequencies are

larger for Re2 ¼ �100 than for Re2 ¼ 0. In general, the effect induced by

elongational flow λ2 6¼ 0 at Re2 ¼ 0 it can be slightly decreased.

elongational flow is minor, significant smaller, and only secondary. Analog to the slightly down shift in the onsets (Figure 10(I)) also the SPI frequencies are slightly reduced (Figure 10(II)). Strong counter-rotating cylinders reduce in general the frequency for finite λ2 (Figure 10(II)), while for larger wavenumbers, a finite

(I) Variation of the (a) m ¼ 0 (TVF) and (b) m ¼ 1 (SPI) bifurcation thresholds and (II) corresponding SPI frequencies ω with magnetic field Hext for different λ2 as indicated. The top row refers to Re2 ¼ 0 and the bottom one to Re2 ¼ �100. The wavenumbers are (1) k ¼ 2:8274 and (2) k ¼ 3:1415, respectively.

also in experiments and also in full nonlinear simulations [6] with λ2 ¼ 0.

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

DOI: http://dx.doi.org/10.5772/intechopen.80301

4.3.3 Frequencies of helical SPI states

Figure 10.

23

#### Figure 9.

Stability thresholds of the CCF basic state against vortex growth of (a) m ¼ 0 (TVF) and (b) m ¼ 1 (SPI) in the Re2 � Re1 diagram of control parameters. Thin solid and dashed lines with symbols show the results for λ2 ¼ 0 and λ2 ¼ 0:8 for a magnetic field Hext ¼ 67:7kA=m. The axial wavenumber is k ¼ 2:8274. (c) Same curves, with wavenumber k ¼ 3:4558 and magnetic field is Hext ¼ 135:4kA=m. The lowest thick curve in (a) and (b) refers in each figure to the threshold in absence of a magnetic field Hext ¼ 0. Insets show corresponding SPI frequencies at the bifurcation thresholds.

wavenumbers. Note that the representative axial wavenumber chosen in (a) k ¼ 2:8274 is as detected in experiments [10]; (b) k ¼ 3:4558. Insets in Figure 9(b,c) show corresponding SPI frequencies at the bifurcation thresholds. As already discussed in Section 3.3.4, the CCF basic state is stabilized against the two types of vortex perturbations in any magnetic field (the lowest thick solid curve in each plot in Figure 9 refers to the case without any magnetic field H = 0 (c.f. Figure 8(a) for nonlinear calculation)). Consider elongational flow (λ2 ¼6 0), the basic state becomes destabilized in comparison with the case λ2 ¼ 0. Regarding Figure 9(a,b), both bifurcation thresholds for m ¼ 0 and m ¼ 1 vortices are shifted to lower values of Re1. Thus, for the axial wavenumber of k ¼ 2:8274, the λ2 term reduces the general stabilization effect caused by an applied magnetic field. For parameters as presented in Figure 9(a,b), this holds for the whole range of presented Re2, even when the changes of the stability boundaries are quite weak, in particular for corotating cylinders. It is important to mention that it never over-compensate the field induced stabilization. Figure 9(c) illustrates in general a strong parameter dependence of the elongational flow effect. While at Re2 < 22 for TVF and Re2 < 4 for SPI, increasing λ2 also has destabilizing effect, and for corresponding larger Re2, the effect is just opposite. Increasing λ2 even enforce the stabilization due to the applied magnetic field.

It is worth to mention that, in contrast to the stabilization of the CCF state by a magnetic field, the λ2-induced modifications of the linear growth of vortices depend significantly on their axial wavenumber k as will be discussed below.

#### 4.3.2 Magnetic field dependence of thresholds

Figure 10(I) illustrates the variation of the bifurcation thresholds m ¼ 0 (TVF) and for m ¼ 1 (SPI) with the magnetic field H ! ext ¼ Hz e ! z, incorporating the elongational flow effect. The top row of figures refers to Re2 ¼ 0, and the bottom one to Re2 ¼ �100 and wavenumbers are k ¼ 2:8274 (left) and k ¼ 3:1415 (right),

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

respectively. Ignoring the effect at small Hext, all thresholds increase linearly with Hext. The slopes are given by ∂Re1,stabð Þ H =∂H and vary slightly either with λ2 and k. For outer cylinder at rest, Re2 ¼ 0, (Figure 10(I)(a)) the slopes of the m ¼ 0 and m ¼ 1 curves are comparable, they differ significantly for counter-rotating cylinders, Re2 ¼ �100 (Figure 10(I)(b)). For the latter, this results in intersections of the m ¼ 0 and m ¼ 1 stability boundaries. Such intersections have been observed also in experiments and also in full nonlinear simulations [6] with λ2 ¼ 0.

In general, modifications of the Hext field dependence of the bifurcation thresholds due to λ2 tend to increase, when the cylinders are stronger counter-rotating (Figure 10(I)). Re2 becomes more negative, and thus, the strain increases. On the other side, they also depend on the axial wavenumber k (see also Section 4.3.4). Considering fixed k, the modifications due to λ2 increase with increasing magnetic field Hext while remaining qualitatively the same. However, the general tendency visible in Figure 11, considering elongational effect, is to shift the stability boundaries downward in Re1. Concrete amounts depend on the parameters k and Re2. Note that this relative small destabilizing elongational effect is always overcompensated by the significant stronger general stabilization effect of the magnetic field itself.

### 4.3.3 Frequencies of helical SPI states

Instead of looking at the growth rates—the real part of the eigenvalue of these solutions (previous Sections), one can also consider the marginal spiral frequency ω—the imaginary part of the marginal eigenvalue at the bifurcation threshold of m ¼ 1 (SPI) vortex flow. Figure 10(II) shows the frequency, corresponding to the m ¼ 1 bifurcation thresholds presented in Figure 10(I). Main effect is the increase of the SPI frequencies with the field Hext, accordingly to the increase of onset Reynolds numbers with Hext (Figure 10(I)). Thus, the threshold frequencies are larger for Re2 ¼ �100 than for Re2 ¼ 0. In general, the effect induced by elongational flow is minor, significant smaller, and only secondary. Analog to the slightly down shift in the onsets (Figure 10(I)) also the SPI frequencies are slightly reduced (Figure 10(II)). Strong counter-rotating cylinders reduce in general the frequency for finite λ2 (Figure 10(II)), while for larger wavenumbers, a finite elongational flow λ2 6¼ 0 at Re2 ¼ 0 it can be slightly decreased.

#### Figure 10.

(I) Variation of the (a) m ¼ 0 (TVF) and (b) m ¼ 1 (SPI) bifurcation thresholds and (II) corresponding SPI frequencies ω with magnetic field Hext for different λ2 as indicated. The top row refers to Re2 ¼ 0 and the bottom one to Re2 ¼ �100. The wavenumbers are (1) k ¼ 2:8274 and (2) k ¼ 3:1415, respectively.

wavenumbers. Note that the representative axial wavenumber chosen in (a)

significantly on their axial wavenumber k as will be discussed below.

Figure 10(I) illustrates the variation of the bifurcation thresholds m ¼ 0 (TVF)

elongational flow effect. The top row of figures refers to Re2 ¼ 0, and the bottom one to Re2 ¼ �100 and wavenumbers are k ¼ 2:8274 (left) and k ¼ 3:1415 (right),

!

ext ¼ Hz e !

z, incorporating the

4.3.2 Magnetic field dependence of thresholds

Figure 9.

22

SPI frequencies at the bifurcation thresholds.

Pattern Formation and Stability in Magnetic Colloids

and for m ¼ 1 (SPI) with the magnetic field H

k ¼ 2:8274 is as detected in experiments [10]; (b) k ¼ 3:4558. Insets in Figure 9(b,c) show corresponding SPI frequencies at the bifurcation thresholds. As already discussed in Section 3.3.4, the CCF basic state is stabilized against the two types of vortex perturbations in any magnetic field (the lowest thick solid curve in each plot in Figure 9 refers to the case without any magnetic field H = 0 (c.f. Figure 8(a) for nonlinear calculation)). Consider elongational flow (λ2 ¼6 0), the basic state becomes destabilized in comparison with the case λ2 ¼ 0. Regarding Figure 9(a,b), both bifurcation thresholds for m ¼ 0 and m ¼ 1 vortices are shifted to lower values of Re1. Thus, for the axial wavenumber of k ¼ 2:8274, the λ2 term reduces the general stabilization effect caused by an applied magnetic field. For parameters as presented in Figure 9(a,b), this holds for the whole range of presented Re2, even when the changes of the stability boundaries are quite weak, in particular for corotating cylinders. It is important to mention that it never over-compensate the field induced stabilization. Figure 9(c) illustrates in general a strong parameter dependence of the elongational flow effect. While at Re2 < 22 for TVF and Re2 < 4 for SPI, increasing λ2 also has destabilizing effect, and for corresponding larger Re2, the effect is just opposite. Increasing λ2 even enforce the stabilization due to the applied magnetic field. It is worth to mention that, in contrast to the stabilization of the CCF state by a magnetic field, the λ2-induced modifications of the linear growth of vortices depend

Stability thresholds of the CCF basic state against vortex growth of (a) m ¼ 0 (TVF) and (b) m ¼ 1 (SPI) in the Re2 � Re1 diagram of control parameters. Thin solid and dashed lines with symbols show the results for λ2 ¼ 0 and λ2 ¼ 0:8 for a magnetic field Hext ¼ 67:7kA=m. The axial wavenumber is k ¼ 2:8274. (c) Same curves, with wavenumber k ¼ 3:4558 and magnetic field is Hext ¼ 135:4kA=m. The lowest thick curve in (a) and (b) refers in each figure to the threshold in absence of a magnetic field Hext ¼ 0. Insets show corresponding

It is worth to mention that the crossover values of kcross for m ¼ 0 and m ¼ 1 perturbations also depend sensitive on other the control parameters, e.g., Re2. Based

Similar to the stability thresholds, also the spiral frequencies (Figure 11(2)) increase with increasing λ2 when k is large and decrease for smaller k as a result of switching on the elongational effect. Thereby, the crossing wavenumber, kcross, separating the increase and decrease in ω is slightly smaller than the one for crossing the marginal thresholds (Figure 11(a1,b1)). Following the speculation by Odenbach and Müller [10] on the physical origin of the elongational effect, one can give a reasonable explanation for such an observed wavenumber dependence of the λ2 effect on the growth rates of vortex flow. In [10], the authors argue that the microscopic origin of a finite λ2 might be attributed to the finite asphericity of the colloids, i.e., the presence of (short) particle chains in the ferrofluid. Small axial wavenumber k will allow these chains more easily to remain aligned in the direction of the magnetic field in vortex flows instead in a flow with large axial wavenumber and large azimuthal vorticity. Larger wavenumber means smaller wavelength and therefore more higher probability to unalign in the given field. Thus, since the magneto-rotational dissipation in the latter flow is larger than in the former one, it requires larger centrifugal forces, i.e., larger values of Re1, to drive the growth of a large k flow. However, it is important to state that this argumentation also holds some weakness: In the presence of chains, one could expect also for small k vortices an increase of the threshold Reynolds number Re1. On the contrary, incorporating the elongational term ð Þ λ2 6¼ 0 in the magnetization balance changes the momentum balance of the vortex flow such that the growth of vortices is enhanced for k < kcross (left in Figure 10) and reduced for k>kcross (right in Figure 10) relative to the

on the observations in Figure 11, it is fair to assume that, depending on the

Following, we will consider the full nonlinear ferrohydrodynamical equation of motions incorporating elongational flow effects (11) in order to investigate the influence of a finite transport coefficient λ2 at different applied

the situation λ2 ¼ 0 (Figure 10(c) and Figure 11). Considering the effect of

netic field, i.e., comparing λ2 ¼ 0 and any value λ2 ¼6 0.

elongational flow, the critical bifurcating point of primary centrifugal instability will be shifted either up or down. As detected in the linear analysis, the potential destabilizing effect of λ2 never overcomes the general stabilizing effect of the external applied mag-

The general modification in the bifurcation threshold of primary instabilities ((w)TVF) and ((w)SPI) due to finite values λ2 6¼ 0 remains qualitative as detected and described before in the linear stability analysis in pure axial field. However, the absolute values are slightly different. A direct comparison for axial field (Figures 9-11) results in variation about one to two percent between the calculated onsets using either the linear analysis or full nonlinear equations [6]. While the different field orientations result in different magnitudes of stabilization of the basic states CCF and 2-CCF, respectively, as discussed in Section 3.3.4, a finite value λ2 does not depend on different field orientations. The variation effect with λ2 is almost identical for all considered field configuration. However, in general, the modifications are slightly larger under oblique magnetic fields. Depending on other system parameters, e.g., wavenumber k, cylinder rotation speeds Re1,Re2, the effect is either stabilizing or destabilizing with respect to

wavenumber k, the γ-point (Section 3.3.4) will change.

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

DOI: http://dx.doi.org/10.5772/intechopen.80301

reference case of λ2 ¼ 0.

4.4 Nonlinear analysis

magnetic fields.

25

4.4.1 Bifurcation thresholds

#### Figure 11.

(1) Stability boundaries of the CCF basic state against the growth of m ¼ 0 (TVF) (blue) and m ¼ 1 (SPI) vortex flow (orange) in the k � Re1 diagram for λ2 ¼ 0 (solid lines) and λ2 ¼ 0:8 dashed lines. (2) corresponding marginal spiral frequencies ω at the m ¼ 1 thresholds. Further parameters: (a) Re2 ¼ 0 and (b) Re2 ¼ �100. The (axial) magnetic field is constant with Hext ¼ 80kA=m. Results from linearized equations [43].

A general observation is that all modifications of the spiral frequencies due to finite λ2 6¼ 0 are most pronounced for the smaller wavenumbers (here k ¼ 2:8274 in Figure 10) in counter-rotating cylinders. The field dependence is virtually not much modified due to variations in λ2, and the growth rate of the imaginary part of the marginal eigenvalue with increasing λ2 is quite similar to that of its real part.

## 4.3.4 Axial wavenumber dependence

Following, we will first discuss the situation for a (moderately) larger wavenumber, taking k ¼ 3:4558 as one specific, representative example. For larger value k (Figure 9(c)), the situation becomes more complicate. Incorporating the elongational flow scaled by the parameter λ2 increases and decreases the growth rate, with respect to λ2 ¼ 0, and thus shifts the stability boundaries, up or down, depending on Re2.

Figure 11 illustrates the relation between the wavenumber dependence of the stability boundaries and of the spiral frequencies. This is of particular interest in experiments, since the wavenumbers of some vortex structures show a strong variation when changing the magnetic field [9, 44]. Reindl et al. [9] also observed hysteresis between vortex flows with different k when increasing and decreasing the Reynolds numbers Re1. The bifurcation thresholds (Figure 11, top row Re2 ¼ 0 and the bottom row Re2 ¼ �100; the (axial) magnetic field is fixed at Hext ¼ 80 kA=m) for m ¼ 0 and m ¼ 1 vortex flow structures in the k � Re1 diagram change when the elongational term ð Þ λ2 ¼6 0 is considered in the linearized ferrofluid equations [43]. Right column of Figure 11(2) presents the wavenumber dependence of the corresponding marginal spiral frequencies at the onsets for m ¼ 1 SPI.

Typically, the thresholds for λ2 6¼ 0 (in Figure 11 the case of λ2 ¼ 0, lines with squares) are shifted downward in Re1 at smaller wavenumbers k and upward at larger k. Resulting in a fixed magnetic field, the elongational effect destabilizes (stabilizes) the CCF state against growth of vortices with small (large) wavenumbers, either for m ¼ 0 or m ¼ 1 perturbations. Consequently, the stability onsets cross for different values of λ2 (λ2 ¼ 0, λ2 ¼ 0:8 in Figure 11) at a specific value of kcross the wavenumber.

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

It is worth to mention that the crossover values of kcross for m ¼ 0 and m ¼ 1 perturbations also depend sensitive on other the control parameters, e.g., Re2. Based on the observations in Figure 11, it is fair to assume that, depending on the wavenumber k, the γ-point (Section 3.3.4) will change.

Similar to the stability thresholds, also the spiral frequencies (Figure 11(2)) increase with increasing λ2 when k is large and decrease for smaller k as a result of switching on the elongational effect. Thereby, the crossing wavenumber, kcross, separating the increase and decrease in ω is slightly smaller than the one for crossing the marginal thresholds (Figure 11(a1,b1)). Following the speculation by Odenbach and Müller [10] on the physical origin of the elongational effect, one can give a reasonable explanation for such an observed wavenumber dependence of the λ2 effect on the growth rates of vortex flow. In [10], the authors argue that the microscopic origin of a finite λ2 might be attributed to the finite asphericity of the colloids, i.e., the presence of (short) particle chains in the ferrofluid. Small axial wavenumber k will allow these chains more easily to remain aligned in the direction of the magnetic field in vortex flows instead in a flow with large axial wavenumber and large azimuthal vorticity. Larger wavenumber means smaller wavelength and therefore more higher probability to unalign in the given field. Thus, since the magneto-rotational dissipation in the latter flow is larger than in the former one, it requires larger centrifugal forces, i.e., larger values of Re1, to drive the growth of a large k flow. However, it is important to state that this argumentation also holds some weakness: In the presence of chains, one could expect also for small k vortices an increase of the threshold Reynolds number Re1. On the contrary, incorporating the elongational term ð Þ λ2 6¼ 0 in the magnetization balance changes the momentum balance of the vortex flow such that the growth of vortices is enhanced for k < kcross (left in Figure 10) and reduced for k>kcross (right in Figure 10) relative to the reference case of λ2 ¼ 0.

## 4.4 Nonlinear analysis

A general observation is that all modifications of the spiral frequencies due to finite λ2 6¼ 0 are most pronounced for the smaller wavenumbers (here k ¼ 2:8274 in Figure 10) in counter-rotating cylinders. The field dependence is virtually not much modified due to variations in λ2, and the growth rate of the imaginary part of the marginal eigenvalue with increasing λ2 is quite similar to that of its real part.

(1) Stability boundaries of the CCF basic state against the growth of m ¼ 0 (TVF) (blue) and m ¼ 1 (SPI) vortex flow (orange) in the k � Re1 diagram for λ2 ¼ 0 (solid lines) and λ2 ¼ 0:8 dashed lines. (2) corresponding marginal spiral frequencies ω at the m ¼ 1 thresholds. Further parameters: (a) Re2 ¼ 0 and (b) Re2 ¼ �100. The (axial) magnetic field is constant with Hext ¼ 80kA=m. Results from linearized equations [43].

Following, we will first discuss the situation for a (moderately) larger wavenumber,

(Figure 9(c)), the situation becomes more complicate. Incorporating the elongational flow scaled by the parameter λ2 increases and decreases the growth rate, with respect to

Typically, the thresholds for λ2 6¼ 0 (in Figure 11 the case of λ2 ¼ 0, lines with squares) are shifted downward in Re1 at smaller wavenumbers k and upward at larger k. Resulting in a fixed magnetic field, the elongational effect destabilizes

wavenumbers, either for m ¼ 0 or m ¼ 1 perturbations. Consequently, the stability onsets cross for different values of λ2 (λ2 ¼ 0, λ2 ¼ 0:8 in Figure 11) at a specific

(stabilizes) the CCF state against growth of vortices with small (large)

taking k ¼ 3:4558 as one specific, representative example. For larger value k

λ2 ¼ 0, and thus shifts the stability boundaries, up or down, depending on Re2. Figure 11 illustrates the relation between the wavenumber dependence of the stability boundaries and of the spiral frequencies. This is of particular interest in experiments, since the wavenumbers of some vortex structures show a strong variation when changing the magnetic field [9, 44]. Reindl et al. [9] also observed hysteresis between vortex flows with different k when increasing and decreasing the Reynolds numbers Re1. The bifurcation thresholds (Figure 11, top row Re2 ¼ 0 and the bottom row Re2 ¼ �100; the (axial) magnetic field is fixed at Hext ¼ 80 kA=m) for m ¼ 0 and m ¼ 1 vortex flow structures in the k � Re1 diagram change when the elongational term ð Þ λ2 ¼6 0 is considered in the linearized ferrofluid equations [43]. Right column of Figure 11(2) presents the wavenumber dependence of the corresponding marginal spiral frequencies at the onsets for m ¼ 1 SPI.

4.3.4 Axial wavenumber dependence

Pattern Formation and Stability in Magnetic Colloids

Figure 11.

value of kcross the wavenumber.

24

### 4.4.1 Bifurcation thresholds

Following, we will consider the full nonlinear ferrohydrodynamical equation of motions incorporating elongational flow effects (11) in order to investigate the influence of a finite transport coefficient λ2 at different applied magnetic fields.

The general modification in the bifurcation threshold of primary instabilities ((w)TVF) and ((w)SPI) due to finite values λ2 6¼ 0 remains qualitative as detected and described before in the linear stability analysis in pure axial field. However, the absolute values are slightly different. A direct comparison for axial field (Figures 9-11) results in variation about one to two percent between the calculated onsets using either the linear analysis or full nonlinear equations [6]. While the different field orientations result in different magnitudes of stabilization of the basic states CCF and 2-CCF, respectively, as discussed in Section 3.3.4, a finite value λ2 does not depend on different field orientations. The variation effect with λ2 is almost identical for all considered field configuration. However, in general, the modifications are slightly larger under oblique magnetic fields. Depending on other system parameters, e.g., wavenumber k, cylinder rotation speeds Re1,Re2, the effect is either stabilizing or destabilizing with respect to the situation λ2 ¼ 0 (Figure 10(c) and Figure 11). Considering the effect of elongational flow, the critical bifurcating point of primary centrifugal instability will be shifted either up or down. As detected in the linear analysis, the potential destabilizing effect of λ2 never overcomes the general stabilizing effect of the external applied magnetic field, i.e., comparing λ2 ¼ 0 and any value λ2 ¼6 0.

## 4.4.2 Flow structure modifications

Aside the modification of the primary stability thresholds, there are further flow structural changes/modifications which can be observed for finite value λ2. Figure 11 shows, for different field configurations T: ð Þ sx ¼ 0:1; sz ¼ 0 , A: ð Þ sx ¼ 0; sz ¼ 0:1 , O: ð Þ sx ¼ 0:1; sz ¼ 0:1 , how the maxima of vorticity η vary for either the full flow structure Δη≔η λð Þ� 2 6¼ 0 η λð Þ 2 ¼ 0 (Figure 12(a)), and reduced 2-fold subspace (Figure 12(b)) varies with λ2 for the basic state configuration ð Þ Re1 ¼ 60; Re2 ¼ 0 . Remember, that only for sx ¼ 0, one has the classical CCF, and for any sx 6¼ 0, a modified basic state is present [6, 9, 37] (i.e., symmetry breaking effect of transversal field). Although for a given configuration (A, T, O), the absolute maxima of these values are significantly different, and their relative differences Δ scale is linear with λ2 starting at zero. Lines in Figure 12 are calculated by linear regression fitting to the numerical data points. The effect of λ2 is decoupled from the magnetic field configuration and with this the considered flow structure.

Either for classical "pure" CCF or field modified 2-CCF, the variation in maxð Þ Δη grows linear with λ2 (Figure 12(a)) (note the significant smaller amplitude in the m ¼ 2 subspace). Further confirmation for this is the fact that an analog linear growth can be also found when considering other parameters, at which a supercritical flow state exists [37]. This holds for toroidally closed structures (w)TVF, as well as for helical flow states (w)SPI. Only the absolute values in maxð Þ Δη are significant larger.

It is important to mention that the slopes in Figure 12 (for O sð Þ x ¼ 0:1; sz ¼ 0:1 ) not only depend on the applied field orientation (O) but also depend on the strength of the applied magnetic field. With increasing the magnetic field strength sx, sz, the slopes become steeper. This scenario is illustrated, exemplary for 2-CCF basic state in transversal field configuration in Figure 13.

Moreover, similar characteristics hold for the different primary bifurcating structures. Figure 14 elucidates this effect for the three different field configurations A, T, and O as indicated (see also Figures 4 and 6 in [37] for another example considering a transversal magnetic field in finite system configuration). For a given field configuration, the slopes increase with λ2, when increasing the magnetic field strength sx. In general, slopes are smallest for transversal and largest for oblique fields, respectively. The variation in the slopes is continuous with increasing the field strength, but not linear; for larger field strength, the variation in the slopes becomes smaller.

4.4.3 Wavenumber modification and selection

(they differ less than 2 percent).

(strength as indicated). Lines are linear fits.

wavenumbers k.

27

Figure 13.

Figure 14.

As seen in the former discussion (section 4.3.2), an elongational flow can have strong effect on the axial wavenumber k. Consider the full equations (and periodic boundary conditions), one obtains similar results as presented in Figure 10 for linear analysis of the marginal stability thresholds for TVF and SPI. As described before, the agreement between the results of linear and full equations is very close

maxð Þ¼ η η λð Þ� 2 η λð Þ 2 ¼ 0 for (a) transversal (T), (b) axial (A) and (c) oblique (O) magnetic field

However, in order to see such wavenumber modification from another perspec-

Starting with a classical TVF as initial flow state, with n ¼ 22 vortices, which corresponds to an axial wavenumber k ¼ 3:41 in the bulk (illustrated on the most left border in Figure 15), we consider a situation with an applied pure transversal magnetic field which is continuously increased for two values, either ignoring any elongational flow effect, λ2 ¼ 0 (Figure 14(a)), or considering such effect λ2 ¼ 0:2 (Figure 15(b)) (note 0.2 is quite small). In any case, either for λ2 ¼ 0 and for λ2 ¼ 0:2, by increasing the magnetic-field strength, sx, the number of vortices is reduced, which leads to an increase in the axial wavelength (i.e. decrease in axial

tive, it makes sense to study a finite system with sufficient large aspect ratio enclosed by fixed end plates (assumed to be stationary) on top and bottom. As in a finite system, the axial wavenumber k is natural selected by the system itself (depending on the various system parameters), it is possible to see the direct influence, considering a finite value λ2 on to the flow structures and their axial

As Figure 11 for basic state 2-CCF at Re1 ¼ 60, Re2 ¼ 0. Variation with λ2 of vorticity maxima maxð Þ¼ η η λð Þ� 2 η λð Þ 2 ¼ 0 for transversal field (T) at different field strength sx as indicated.

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

DOI: http://dx.doi.org/10.5772/intechopen.80301

Supercritical (w)TVF solution at Re1 ¼ 100, Re2 ¼ 0. Variation of vorticity maxima

#### Figure 12.

Variation of vorticity maxima with λ2 for (a) maxð Þ¼ η η λð Þ� 2 η λð Þ 2 ¼ 0 at full solution and (b) maxð Þ¼ ηð Þ m ¼ 2 η λð Þ� 2 η λð Þ 2 ¼ 0 at only 2-fold symmetric subspace (empty for A) for basic flow (CCF and 2-CCF, respectively) at Re1 ¼ 60, Re2 ¼ 0 and different field orientations: Axial (A) sx ¼ 0:0, sz ¼ 0:6; transverse (T) sx ¼ 0:6, sz ¼ 0:0 and oblique (O) sx ¼ 0:6, sz ¼ 0:6. Lines are linear fits.

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

#### Figure 13.

4.4.2 Flow structure modifications

Pattern Formation and Stability in Magnetic Colloids

in transversal field configuration in Figure 13.

larger.

becomes smaller.

Figure 12.

26

Aside the modification of the primary stability thresholds, there are further flow

It is important to mention that the slopes in Figure 12 (for O sð Þ x ¼ 0:1; sz ¼ 0:1 ) not only depend on the applied field orientation (O) but also depend on the strength of the applied magnetic field. With increasing the magnetic field strength sx, sz, the slopes become steeper. This scenario is illustrated, exemplary for 2-CCF basic state

Moreover, similar characteristics hold for the different primary bifurcating structures. Figure 14 elucidates this effect for the three different field configurations A, T, and O as indicated (see also Figures 4 and 6 in [37] for another example considering a transversal magnetic field in finite system configuration). For a given field configuration, the slopes increase with λ2, when increasing the magnetic field strength sx. In general, slopes are smallest for transversal and largest for oblique fields, respectively. The variation in the slopes is continuous with increasing the field strength, but not linear; for larger field strength, the variation in the slopes

Variation of vorticity maxima with λ2 for (a) maxð Þ¼ η η λð Þ� 2 η λð Þ 2 ¼ 0 at full solution and (b)

transverse (T) sx ¼ 0:6, sz ¼ 0:0 and oblique (O) sx ¼ 0:6, sz ¼ 0:6. Lines are linear fits.

maxð Þ¼ ηð Þ m ¼ 2 η λð Þ� 2 η λð Þ 2 ¼ 0 at only 2-fold symmetric subspace (empty for A) for basic flow (CCF and 2-CCF, respectively) at Re1 ¼ 60, Re2 ¼ 0 and different field orientations: Axial (A) sx ¼ 0:0, sz ¼ 0:6;

structural changes/modifications which can be observed for finite value λ2. Figure 11 shows, for different field configurations T: ð Þ sx ¼ 0:1; sz ¼ 0 , A: ð Þ sx ¼ 0; sz ¼ 0:1 , O: ð Þ sx ¼ 0:1; sz ¼ 0:1 , how the maxima of vorticity η vary for either the full flow structure Δη≔η λð Þ� 2 6¼ 0 η λð Þ 2 ¼ 0 (Figure 12(a)), and reduced 2-fold subspace (Figure 12(b)) varies with λ2 for the basic state configuration ð Þ Re1 ¼ 60; Re2 ¼ 0 . Remember, that only for sx ¼ 0, one has the classical CCF, and for any sx 6¼ 0, a modified basic state is present [6, 9, 37] (i.e., symmetry breaking effect of transversal field). Although for a given configuration (A, T, O), the absolute maxima of these values are significantly different, and their relative differences Δ scale is linear with λ2 starting at zero. Lines in Figure 12 are calculated by linear regression fitting to the numerical data points. The effect of λ2 is decoupled from the magnetic field configuration and with this the considered flow structure. Either for classical "pure" CCF or field modified 2-CCF, the variation in maxð Þ Δη grows linear with λ2 (Figure 12(a)) (note the significant smaller amplitude in the m ¼ 2 subspace). Further confirmation for this is the fact that an analog linear growth can be also found when considering other parameters, at which a supercritical flow state exists [37]. This holds for toroidally closed structures (w)TVF, as well as for helical flow states (w)SPI. Only the absolute values in maxð Þ Δη are significant

As Figure 11 for basic state 2-CCF at Re1 ¼ 60, Re2 ¼ 0. Variation with λ2 of vorticity maxima maxð Þ¼ η η λð Þ� 2 η λð Þ 2 ¼ 0 for transversal field (T) at different field strength sx as indicated.

#### Figure 14.

Supercritical (w)TVF solution at Re1 ¼ 100, Re2 ¼ 0. Variation of vorticity maxima maxð Þ¼ η η λð Þ� 2 η λð Þ 2 ¼ 0 for (a) transversal (T), (b) axial (A) and (c) oblique (O) magnetic field (strength as indicated). Lines are linear fits.

### 4.4.3 Wavenumber modification and selection

As seen in the former discussion (section 4.3.2), an elongational flow can have strong effect on the axial wavenumber k. Consider the full equations (and periodic boundary conditions), one obtains similar results as presented in Figure 10 for linear analysis of the marginal stability thresholds for TVF and SPI. As described before, the agreement between the results of linear and full equations is very close (they differ less than 2 percent).

However, in order to see such wavenumber modification from another perspective, it makes sense to study a finite system with sufficient large aspect ratio enclosed by fixed end plates (assumed to be stationary) on top and bottom. As in a finite system, the axial wavenumber k is natural selected by the system itself (depending on the various system parameters), it is possible to see the direct influence, considering a finite value λ2 on to the flow structures and their axial wavenumbers k.

Starting with a classical TVF as initial flow state, with n ¼ 22 vortices, which corresponds to an axial wavenumber k ¼ 3:41 in the bulk (illustrated on the most left border in Figure 15), we consider a situation with an applied pure transversal magnetic field which is continuously increased for two values, either ignoring any elongational flow effect, λ2 ¼ 0 (Figure 14(a)), or considering such effect λ2 ¼ 0:2 (Figure 15(b)) (note 0.2 is quite small). In any case, either for λ2 ¼ 0 and for λ2 ¼ 0:2, by increasing the magnetic-field strength, sx, the number of vortices is reduced, which leads to an increase in the axial wavelength (i.e. decrease in axial

wavenumber k) due to elimination of vortices in the bulk. Therefore, the corresponding axial wavenumber k in the bulk becomes reduced. Remember that the initial state is a classical, stationary TVF, which becomes modified to stationary wTVF as soon as sx 6¼ 0. For the field generated wTVF, the wavy-like modulation is not visible in Figure 15. However qualitative, the same scenario also holds for either axial and oblique magnetic fields, with changes in the corresponding parameters sx, sz at which the wavenumber k changes, i.e., vortices disappear.

form chains of various lengths and tend to agglomerate in general. One possibility in order to make a more realistic approach for ferrofluids and to come close to such a configuration in a real ferrofluids is to consider the effect of elongational flow. Such a flow becomes incorporated by the symmetric part of the velocity gradient field tensor in the nondimensional ferrohydrodynamic equations of motion (11) and scales by a material dependent transport coefficient λ2. By such an assumption, the latter is given by a pure scalar number, describing the strength/influence of all

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

DOI: http://dx.doi.org/10.5772/intechopen.80301

Consider either linear stability analysis or fully nonlinear equations, one finds that incorporating such elongational flow can have either stabilization or destabilization effect (i.e., modification in the supercritical primary bifurcation thresholds) with respect to the reference situation in absence of such flow ð Þ λ2 ¼ 0 . The modifications due to finite value λ2 are qualitative similar and in the same range for different considered magnetic fields, slightly largest in case of an oblique field compared to pure axial or transversal ones. Typically, the modifications due to λ2 6¼ 0 are larger for helical, nonaxisymmetric, m ¼ 1, (w)SPIs than for toroidally closed, axisymmetric, m ¼ 0, (w)TVF. Thus, the regim of primary stable bifurcating (w)TVF increases. This observation is similar and goes in hand with the previously described general stabilization of any applied magnetic field (Section 3). There we saw that the stability thresholds of helical flow states are shifted further than those of toroidal closed flow structures. However, it is important and fundamental to state that the elongational effect λ2 ¼6 0 is never big enough to over compensate the general, field-induced stabilization. Only the magnitude/amplitude in shift can be reduced. Further, it is worth to mention that linear stability analysis is limited to axial magnetic fields; as for fields with finite transversal component, no basic state can be derived, simulations of full nonlinear equations confirm similar effects. The modifications due to λ2 are sensitive to various different system parameters (e.g., Reynolds numbers, radius ratio, etc.), in particular sensitive to the axial wavenumber k. Here, smaller k are strongly effected than larger ones. Thus, considering finite size TCS, a considered elongational flow results in modifications of flow structures in the sense of changing number of vortices in the bulk, which coincides with a different wavenumber selection. Thus, depending on the particular wavenumber, the presence of any elongational effect λ2 ¼6 0 can be either stabilizing or destabilizing for the basic state. In finite system, this results in different selection and/or modification of number of vortex pairs within the annulus,

Regarding the flow structural properties, a finite transport coefficient, λ2 6¼ 0, does not change the qualitative shape of the flow pattern. Only quantitative differences can be observed. Essentially, all structural properties remain as generated/ induced due to the external applied magnetic field. Thus for instance, the azimuthal mode-2 symmetry originated in the symmetry-breaking transverse magnetic field is preserved; only the strength of different flow quantities is modified (in general they increase) according to finite λ2. Mainly, the flow states become linearly enforced, e.g., speaking in quantity of the azimuthal vorticity, which holds for both the basic flow (CCF, 2-CCF) and the primary instabilities the supercritical centrifugal unsta-

Aside the properties and behavior of the magnetic particles itself and their

interaction with the applied magnetic field (see last sections), the correct

5. Modified field dependence - internal magnetization

former mentioned properties.

depending on the parameter.

ble flows ((w)TVF, (w)SPI).

5.1 Introduction and motivation

29

In general, the exact field strength sx for vortex elimination and/or the number of destroyed vortices depends on λ2. In the example of Figure 15, the initial flow has 22 vortices with an axial wavenumber k ¼ 3:41. Neglecting elongational effects λ2 ¼ 0 (Figure 15(a)) until the parameter sx approximately reaches the value about 0.7, the flow remains stable, and the number of vortices in the flow can be reduced to n ¼ 20 with k ¼ 3:04. Further increasing sx to about 0.85, the number of vortices can be more reduced to n ¼ 16 with k ¼ 2:58. Hereafter, no further changes in the number of vortices n appear, before reaching the boundary threshold, sx≈0:9. Consider elongational flow λ2 ¼ 0:2 (Figure 15(b)), the behavior of the vortex state is similar to the case neglecting such effects. The flow remains stable before the parameter sx reaches about 0.75. At the same time, the number of vortices is also reduced in analog manner to n ¼ 20 with k ¼ 3:04. Hereafter, as before, the number of vortices n remains unchanged until the parameter sx increases to the boundary threshold, sx≈0:95. Consider values λ2>0:7, the flow structure (for given parameters) is not affected by the magnetic-field strength. A flow with n ¼ 22 vortices corresponding to axial wavenumber k ¼ 3:41 in the bulk is remaining in the bulk.

An interesting phenomenon, which also has been reported in experimental studies [9] as in numerical simulations, is the fact that one or two vortex pairs can be eliminated. However so far, there is no physical explanation for this observation/ phenomenon. Analog to the presented scenario in Figure 15 for pure transversal magnetic field, one also finds similar reduction scenarios for pure axial and oblique magnetic fields. As to expect, the absolute values of field strength, sx, sz, at which the elimination of vortices happen is different, depending on the applied field.

### 4.5 Resume

Real ferrofluids consist of a suspension of particles with finite size and mainly almost ellipsoid shape, as well as there are particle-particle interactions that tend to

#### Figure 15.

Contours of the radial velocity component u at mid-gap with variation of field strength sx at Re1 ¼ 88:35, Re2 ¼ 0 for (a) λ2 ¼ 0 and (b) λ2 ¼ 0:2. Left border in each plot gives the initial state, a classical TVF with n = 22 vortices corresponding to an axial wavenumber k = 3.41 in the bulk. Note that for sx ¼ 0, only wavy-like modified flow states exist. Red (yellow) [dark gray (light gray)] corresponds to positive (negative) values. The max (min) level is �9.98.

## Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

wavenumber k) due to elimination of vortices in the bulk. Therefore, the

sx, sz at which the wavenumber k changes, i.e., vortices disappear.

Pattern Formation and Stability in Magnetic Colloids

axial wavenumber k ¼ 3:41 in the bulk is remaining in the bulk.

4.5 Resume

Figure 15.

28

(negative) values. The max (min) level is �9.98.

An interesting phenomenon, which also has been reported in experimental studies [9] as in numerical simulations, is the fact that one or two vortex pairs can be eliminated. However so far, there is no physical explanation for this observation/ phenomenon. Analog to the presented scenario in Figure 15 for pure transversal magnetic field, one also finds similar reduction scenarios for pure axial and oblique magnetic fields. As to expect, the absolute values of field strength, sx, sz, at which the elimination of vortices happen is different, depending on the applied field.

Real ferrofluids consist of a suspension of particles with finite size and mainly almost ellipsoid shape, as well as there are particle-particle interactions that tend to

Contours of the radial velocity component u at mid-gap with variation of field strength sx at

Re1 ¼ 88:35, Re2 ¼ 0 for (a) λ2 ¼ 0 and (b) λ2 ¼ 0:2. Left border in each plot gives the initial state, a classical TVF with n = 22 vortices corresponding to an axial wavenumber k = 3.41 in the bulk. Note that for sx ¼ 0, only wavy-like modified flow states exist. Red (yellow) [dark gray (light gray)] corresponds to positive

corresponding axial wavenumber k in the bulk becomes reduced. Remember that the initial state is a classical, stationary TVF, which becomes modified to stationary wTVF as soon as sx 6¼ 0. For the field generated wTVF, the wavy-like modulation is not visible in Figure 15. However qualitative, the same scenario also holds for either axial and oblique magnetic fields, with changes in the corresponding parameters

In general, the exact field strength sx for vortex elimination and/or the number of destroyed vortices depends on λ2. In the example of Figure 15, the initial flow has 22 vortices with an axial wavenumber k ¼ 3:41. Neglecting elongational effects λ2 ¼ 0 (Figure 15(a)) until the parameter sx approximately reaches the value about 0.7, the flow remains stable, and the number of vortices in the flow can be reduced to n ¼ 20 with k ¼ 3:04. Further increasing sx to about 0.85, the number of vortices can be more reduced to n ¼ 16 with k ¼ 2:58. Hereafter, no further changes in the number of vortices n appear, before reaching the boundary threshold, sx≈0:9. Consider elongational flow λ2 ¼ 0:2 (Figure 15(b)), the behavior of the vortex state is similar to the case neglecting such effects. The flow remains stable before the parameter sx reaches about 0.75. At the same time, the number of vortices is also reduced in analog manner to n ¼ 20 with k ¼ 3:04. Hereafter, as before, the number of vortices n remains unchanged until the parameter sx increases to the boundary threshold, sx≈0:95. Consider values λ2>0:7, the flow structure (for given parameters) is not affected by the magnetic-field strength. A flow with n ¼ 22 vortices corresponding to form chains of various lengths and tend to agglomerate in general. One possibility in order to make a more realistic approach for ferrofluids and to come close to such a configuration in a real ferrofluids is to consider the effect of elongational flow. Such a flow becomes incorporated by the symmetric part of the velocity gradient field tensor in the nondimensional ferrohydrodynamic equations of motion (11) and scales by a material dependent transport coefficient λ2. By such an assumption, the latter is given by a pure scalar number, describing the strength/influence of all former mentioned properties.

Consider either linear stability analysis or fully nonlinear equations, one finds that incorporating such elongational flow can have either stabilization or destabilization effect (i.e., modification in the supercritical primary bifurcation thresholds) with respect to the reference situation in absence of such flow ð Þ λ2 ¼ 0 . The modifications due to finite value λ2 are qualitative similar and in the same range for different considered magnetic fields, slightly largest in case of an oblique field compared to pure axial or transversal ones. Typically, the modifications due to λ2 6¼ 0 are larger for helical, nonaxisymmetric, m ¼ 1, (w)SPIs than for toroidally closed, axisymmetric, m ¼ 0, (w)TVF. Thus, the regim of primary stable bifurcating (w)TVF increases. This observation is similar and goes in hand with the previously described general stabilization of any applied magnetic field (Section 3). There we saw that the stability thresholds of helical flow states are shifted further than those of toroidal closed flow structures. However, it is important and fundamental to state that the elongational effect λ2 ¼6 0 is never big enough to over compensate the general, field-induced stabilization. Only the magnitude/amplitude in shift can be reduced. Further, it is worth to mention that linear stability analysis is limited to axial magnetic fields; as for fields with finite transversal component, no basic state can be derived, simulations of full nonlinear equations confirm similar effects. The modifications due to λ2 are sensitive to various different system parameters (e.g., Reynolds numbers, radius ratio, etc.), in particular sensitive to the axial wavenumber k. Here, smaller k are strongly effected than larger ones. Thus, considering finite size TCS, a considered elongational flow results in modifications of flow structures in the sense of changing number of vortices in the bulk, which coincides with a different wavenumber selection. Thus, depending on the particular wavenumber, the presence of any elongational effect λ2 ¼6 0 can be either stabilizing or destabilizing for the basic state. In finite system, this results in different selection and/or modification of number of vortex pairs within the annulus, depending on the parameter.

Regarding the flow structural properties, a finite transport coefficient, λ2 6¼ 0, does not change the qualitative shape of the flow pattern. Only quantitative differences can be observed. Essentially, all structural properties remain as generated/ induced due to the external applied magnetic field. Thus for instance, the azimuthal mode-2 symmetry originated in the symmetry-breaking transverse magnetic field is preserved; only the strength of different flow quantities is modified (in general they increase) according to finite λ2. Mainly, the flow states become linearly enforced, e.g., speaking in quantity of the azimuthal vorticity, which holds for both the basic flow (CCF, 2-CCF) and the primary instabilities the supercritical centrifugal unstable flows ((w)TVF, (w)SPI).

## 5. Modified field dependence - internal magnetization

## 5.1 Introduction and motivation

Aside the properties and behavior of the magnetic particles itself and their interaction with the applied magnetic field (see last sections), the correct

assumption of the applied magnetic field is another crucially point in describing magnetic flows and their dynamics. The frequently used, most common assumption is to consider the internal magnetic field within the magnetic fluid (e.g., a ferrofluid) to be equal to the external applied field. Following, we will see that this is only a leading-order approximation. By accounting for the ferrofluid's magnetic susceptibility, a uniform externally imposed magnetic field is modified by the presence of the magnetic fluid (ferrofluid) within the annulus. The modification to the magnetic field appears in an radial dependence 1=r2 and their magnitude scales with the susceptibility χ. To be concrete, such modification to the imposed magnetic field can be substantial, also for ferrofluids typically used in laboratory experiments. These have significant consequences on the structural properties and stability of the basic states, as well as on the supercritical (primary) bifurcating solutions.

## 5.2 Modified internal magnetic field

As already seen in the previous sections, for solving the equation of motion (1), we have to consider the magnetic field within the annular gap between the cylinders. In the last sections, we used the simplest approach with the assumption to take the magnetic field to be identical to the applied external field [4, 6]. However, this simple approach is only a leading-order approximation, and depending on the magnetic susceptibility of the ferrofluid, the magnetic field in the gap is modified.

Assuming infinitely long cylinders, the magnetic boundary conditions are

$$
\overrightarrow{H} = \overrightarrow{H}\_{\text{ext}} - \mathbf{M}\_r \overrightarrow{e}\_r \tag{14}
$$

H !

motion:

H ! ¼ 2

¼ � 2HT

ext=K � � r1=r

where K ¼ ð Þ 2 þ χ

2 � � ½ � χ cos 2ð Þþ θ ΩcN sin 2ð Þθ ex

n

DOI: http://dx.doi.org/10.5772/intechopen.80301

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

field for a ferrofluid between two cylinders.

Hx 1 � ζ=r

2 � � cosð Þθ er

unchanged/untouched from such considered radial field dependence.

ð Þ 2 þ ζ K

5.3 Flow structural modifications

bifurcating solutions ((w)TVF and (w)SPI).

5.3.1 Primary instabilities

31

the dimensionless scalar parameter

! þ½ � χ sin 2ð Þþ θ ΩcN cos 2ð Þθ ey

2 � χ2η2. A special case of (15) when ΩcN ¼ 0. This corre-

sponds to an equilibrium magnetization and has been already observed in an earlier work by Odenbach and Müller [38]. They considered the limit of an equilibrium magnetization and used a similar derivation of the modification of the magnetic

Based on (15), one can use the following magnetic field in the equation of

ζ ¼ χ 2 þ χ r 2

characterizes the strength of the radial field dependence ζ. The ferrohydrodynamical equation of motion (6) and the Niklas parameters sx and sz (7) remain

As we learned in previous sections, a magnetic field with a finite transverse component modifies either basic state or primary bifurcating flow structures in a wavy-like manner. More details of these states have been reported in numerical simulations [6], for which a homogeneous internal magnetic field equal to the external applied one has been supposed, as well as in real experimental studies [9]. The question now is how does the former derived radial field dependence ζ affects the stability of the basic states (CCF and 2-CCF) and the characteristics of the

We begin our discussion by considering a purely axial applied magnetic field (A), with fixed field strength sz ¼ 0:6, which leaves the flows, in particular its structural properties unaffected (TVF and SPI). Figure 16 shows the influence of the radial field parameter, ζ, (17) (exemplary chosen (a) ζ ¼ 0, (b) ζ ¼ 0:4 and (c) ζ ¼ 0:8), on the bifurcation thresholds for TVF and SPI in an axial magnetic field. As in absence of any field dependence, ζ ¼ 0 (Figure 16(a)), the bifurcations continue being supercritical. Main effect on the bifurcation curves is that with increasing ζ, the thresholds for the onset of both TVF and SPI are shifted to higher Re1 (Figure 16(b,c)). Thereby, the threshold for the toroidally closed TVF being shifted significantly further (to larger values Re1). Note the different scaling on the abscissa in Figure 16(c). Aside from the enhanced stabilization of the CCF basic state on top of the field introduced stabilization itself, one observes an important consequence to the relative shifts in the two bifurcation curves. Consequently, the bifurcation curve of TVF is shifted more than the one of SPI, the co-dimension-two point, γ, is also shifted to more positive Re2 (c.f. Figure 15(a,b)). Interestingly, this is just the opposite way of the shift with increasing field strength sz in a pure axial magnetic field (the same holds for sx in transversal field), which we saw in Figure 7

! �ζ=r

! h

2 sin ð Þθ eθ

! i

þ Hzez

1 (19)

!g,

(17)

(18)

at r ¼ r1 and r ¼ r2, where H ! ext is the homogeneous external applied magnetic field in the absence of the ferrofluid-filled annulus and Mr is the radial component of the magnetization M ! .

The following ansatz satisfies a solenoidal field, ∇� H ! ¼ 0:

$$\begin{aligned} \overrightarrow{H} &= \overrightarrow{H}\_{\text{ext}} + \left[ \left( a\_1 - b\_1 / r^2 \right) \cos \left( \theta \right) + \left( a\_2 - b\_2 / r^2 \right) \sin \left( \theta \right) \right] \overrightarrow{e}\_r \\ &+ \left[ \left( a\_2 + b\_2 / r^2 \right) \cos \left( \theta \right) - \left( a\_1 - b\_1 / r^2 \right) \sin \left( \theta \right) \right] \overrightarrow{e}\_\theta \end{aligned} \tag{15}$$

and then the boundary conditions defining the external field are.

To calculate the four unknown constants ð Þ a1, a2, b1, b2 from the boundary conditions (12), we need the radial component of the magnetization which results from substituting (13) into the magnetization (3):

$$\begin{split} \mathbf{M}\_{r} &= \left[ \chi \mathbf{H}\_{\text{ext}}^{T} + \chi (a\_{1} - b\_{1}/r^{2}) - c\_{N} \boldsymbol{\Omega} (a\_{2} + b\_{2}/r^{2}) \right] \cos \left( \theta \right) \\ &+ \left[ c\_{N} \boldsymbol{\Omega} \mathbf{H}\_{\text{ext}}^{T} + \chi (a\_{2} - b\_{2}/r^{2}) + c\_{N} \boldsymbol{\Omega} (a\_{1} + b\_{1}/r^{2}) \right] \sin \left( \theta \right), \end{split} \tag{16}$$

where HT ext stands for the transverse component of Hext ! . Using the continuity of the magnetic field condition on the cylinders, the coefficients can be calculated. Substituting them into (13), finally the resulting magnetic field is given by

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

$$\overrightarrow{H} = -\left(2\mathbf{H}\_{\text{ext}}^T / \mathbf{K}\right) \left(r\_1/r^2\right) \left\{ \left[\chi\cos\left(2\Theta\right) + \Omega c\_N \sin\left(2\Theta\right)\right] \overrightarrow{e}\_x^+ + \left[\chi\sin\left(2\Theta\right) + \Omega c\_N \cos\left(2\Theta\right)\right] \overrightarrow{e}\_y^+\right\},\tag{17}$$

where K ¼ ð Þ 2 þ χ 2 � χ2η2. A special case of (15) when ΩcN ¼ 0. This corresponds to an equilibrium magnetization and has been already observed in an earlier work by Odenbach and Müller [38]. They considered the limit of an equilibrium magnetization and used a similar derivation of the modification of the magnetic field for a ferrofluid between two cylinders.

Based on (15), one can use the following magnetic field in the equation of motion:

$$\overrightarrow{H} = 2\frac{(2+\zeta)}{K}H\_{\text{x}}\left[\left(1-\zeta/r^{2}\right)\cos\left(\theta\right)\overrightarrow{e\_{r}} - \zeta/r^{2}\sin\left(\theta\right)\overrightarrow{e\_{\theta}}\right] + H\_{\text{z}}\overrightarrow{e\_{z}}\tag{18}$$

the dimensionless scalar parameter

$$
\zeta = \frac{\chi}{2+\chi} r\_1^2 \tag{19}
$$

characterizes the strength of the radial field dependence ζ. The ferrohydrodynamical equation of motion (6) and the Niklas parameters sx and sz (7) remain unchanged/untouched from such considered radial field dependence.

## 5.3 Flow structural modifications

## 5.3.1 Primary instabilities

assumption of the applied magnetic field is another crucially point in describing magnetic flows and their dynamics. The frequently used, most common assumption

ferrofluid) to be equal to the external applied field. Following, we will see that this is only a leading-order approximation. By accounting for the ferrofluid's magnetic susceptibility, a uniform externally imposed magnetic field is modified by the presence of the magnetic fluid (ferrofluid) within the annulus. The modification to the magnetic field appears in an radial dependence 1=r2 and their magnitude scales with the susceptibility χ. To be concrete, such modification to the imposed magnetic field can be substantial, also for ferrofluids typically used in laboratory experiments. These have significant consequences on the structural properties and stability of the basic states, as well as on the supercritical (primary) bifurcating

As already seen in the previous sections, for solving the equation of motion (1), we have to consider the magnetic field within the annular gap between the cylinders. In the last sections, we used the simplest approach with the assumption to take the magnetic field to be identical to the applied external field [4, 6]. However, this simple approach is only a leading-order approximation, and depending on the magnetic susceptibility of the ferrofluid, the magnetic field in the gap is modified. Assuming infinitely long cylinders, the magnetic boundary conditions are

> ext � Mr e !

field in the absence of the ferrofluid-filled annulus and Mr is the radial component

2 cosð Þþ θ a2 � b2=r

2 cosð Þ� θ a1 � b1=r

cosð Þþ θ a2 � b2=r2

cosð Þ� θ a1 � b1=r2

ext þ χ a1 � b1=r2 ð Þ� cNΩ a2 þ b2=r2 ð Þ cosð Þθ

the magnetic field condition on the cylinders, the coefficients can be calculated. Substituting them into (13), finally the resulting magnetic field is given by

ext þ χ a2 � b2=r2 ð Þþ cNΩ a1 þ b1=r2 ð Þ sin ð Þθ ,

To calculate the four unknown constants ð Þ a1, a2, b1, b2 from the boundary conditions (12), we need the radial component of the magnetization which results from

and then the boundary conditions defining the external field are.

1

2

ext stands for the transverse component of Hext

! and H !

2 sin ð Þθ e

2 sin ð Þθ eθ

sin ð Þθ er

ð Þ¼ r ¼ r2 Hext

sin ð Þθ eθ

!

r (14)

! r

2 cosð Þþθ

!.

. Using the continuity of

! (15)

! þ a2 þ b2=r2

1

(16)

ext is the homogeneous external applied magnetic

! ¼ 0:

1

2

!

þ a1 � b1=r2

H ! ¼ H !

!

The following ansatz satisfies a solenoidal field, ∇� H

ext þ a1 � b1=r

ext þ a1 � b1=r2

! þ a2 þ b2=r2

þ a2 þ b2=r

is to consider the internal magnetic field within the magnetic fluid (e.g., a

solutions.

5.2 Modified internal magnetic field

Pattern Formation and Stability in Magnetic Colloids

at r ¼ r1 and r ¼ r2, where H

! .

of the magnetization M

ð Þ¼ r ¼ r1 H

cosð Þ� θ a1 � b1=r2

!

Mr ¼ χHT

1 sin ð Þ� θ eθ

substituting (13) into the magnetization (3):

þ cNΩHT

H !

a2 � b2=r2 2 sin ð Þ� θ er

where HT

30

H ! ¼ H !

As we learned in previous sections, a magnetic field with a finite transverse component modifies either basic state or primary bifurcating flow structures in a wavy-like manner. More details of these states have been reported in numerical simulations [6], for which a homogeneous internal magnetic field equal to the external applied one has been supposed, as well as in real experimental studies [9]. The question now is how does the former derived radial field dependence ζ affects the stability of the basic states (CCF and 2-CCF) and the characteristics of the bifurcating solutions ((w)TVF and (w)SPI).

We begin our discussion by considering a purely axial applied magnetic field (A), with fixed field strength sz ¼ 0:6, which leaves the flows, in particular its structural properties unaffected (TVF and SPI). Figure 16 shows the influence of the radial field parameter, ζ, (17) (exemplary chosen (a) ζ ¼ 0, (b) ζ ¼ 0:4 and (c) ζ ¼ 0:8), on the bifurcation thresholds for TVF and SPI in an axial magnetic field. As in absence of any field dependence, ζ ¼ 0 (Figure 16(a)), the bifurcations continue being supercritical. Main effect on the bifurcation curves is that with increasing ζ, the thresholds for the onset of both TVF and SPI are shifted to higher Re1 (Figure 16(b,c)). Thereby, the threshold for the toroidally closed TVF being shifted significantly further (to larger values Re1). Note the different scaling on the abscissa in Figure 16(c). Aside from the enhanced stabilization of the CCF basic state on top of the field introduced stabilization itself, one observes an important consequence to the relative shifts in the two bifurcation curves. Consequently, the bifurcation curve of TVF is shifted more than the one of SPI, the co-dimension-two point, γ, is also shifted to more positive Re2 (c.f. Figure 15(a,b)). Interestingly, this is just the opposite way of the shift with increasing field strength sz in a pure axial magnetic field (the same holds for sx in transversal field), which we saw in Figure 7

#### Figure 16.

Bifurcation curves for the onset of TVF (blue) and SPI (orange) in Re2 � Re1 diagram for different ζ-values (a) ζ ¼ 0, (b) ζ ¼ 0:4 and (c) ζ ¼ 0:8 with an applied axial magnetic field (A) with sz ¼ 0:6. ((a) is a replot of Figure 7(c) for comparison). Note the different scale on the ordinate in (c). Simulation results of full nonlinear equations (11) consider a short periodic TCS with axial wavenumber k ¼ 3:927 (same hold for Figures 15 and 16).

favors the helical instability. Close to the onset, the variation of the maxima of all

Bifurcation curves for the onset of wTVF (blue) and wSPI (orange) in Re2 � Re1 diagram for different ζvalues (a) ζ ¼ 0, (b) ζ ¼ 0:4, and (c) ζ ¼ 0:8 with an applied oblique magnetic field (O) with sx ¼ 0:6 ¼ s � z. ((a) is a replot of Figure 7(d) for comparison). Note different scaling in (c).

Figure 19 illustrates the flow structural properties of wTVF and wSPI in either a transverse magnetic field T (a,b) and an oblique magnetic field O (c,d), both with assumed radial field dependence ζ ¼ 0:8. Note, the flow structures of TVF and SPI under a pure axial applied magnetic field are not shown, as they virtually do not differ from the ones in absence of a magnetic field. Only the absolute values (e.g. in η) are different. Comparing these flow structures to the ones for ζ ¼ 0 cases as earlier presented in Figure 6 (Section 3.3.3), one sees the modulations in all structures to be enhanced/enforced. In general, the waviness (axial variation in azimuthal direction) of

(See also Figures 2 and 6 for comparison to flow states without radial field dependence. The radial field dependence is considered to be ζ ¼ 0:8. Top row: Azimuthal vorticity isosurfaces η over two axial wavelengths with pure transversal applied magnetic field (T): (a) wTVF at Re2 ¼ 0, Re1 ¼ �130, (b) wSPI at Re2 ¼ �150, Re1 ¼ 190; and with oblique applied magnetic field (O): (c) wTVF at Re2 ¼ 0, Re1 ¼ �150, and (f) wSPI at Re2 ¼ �150, Re1 ¼ �200 Red (yellow) isosurfaces correspond to positive (negative) values as indicated. The bottom row shows the mode amplitudes j j um,n of the radial velocity field u corresponding to the

flow components u, v, w is almost linear.

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

DOI: http://dx.doi.org/10.5772/intechopen.80301

5.3.2 Modified flow structures

Figure 18.

Figure 19.

33

structures above over the m-n-plane.

Figure 17.

Bifurcation curves for the onset of wTVF (blue) and wSPI (orange) in parameter space for different ζ values (a) ζ ¼ 0, (b) ζ ¼ 0:4, and (c) ζ ¼ 0:8 with an applied transversal magnetic field (T) sx ¼ 0:6. ((a) is a replot of Figure 7(b) for comparison).

with moving the γ-point to the left toward more negative values in Re2. Consider larger values, e.g., ζ ¼ 0:8 (Figure 15(c)), one sees that the co-dimension-two γ-point is even shifted to Re2>50 out of the considered parameter space. As a result, the primary instability is always the helical SPI, even with the two cylinders in strong co-rotation. This observation is crucially different from classical TCS, both without magnetic fields and also with assumption of an applied uniform magnetic field. Thus, the radial field dependence, induced by the susceptibility of the ferrofluid-filled annulus, is seen to cause major changes in the quantitative and qualitative characteristics of the flow instabilities.

Consider a transversal magnetic field, field strength sx ¼ 0:6 (Figure 17) instead of an axial one, the variation by increasing ζ remains virtually and qualitatively the same, aside that now all structures are wavy-like modulated (2-CCF, wTVF, wSPI). One finds the same preference, in the sense of stronger upshifts for toroidally closed wTVF with the same result of a shift in the γ-point to the right to larger values in Re2. Thus also here, for sufficiently strong ζ, e.g., ζ ¼ 0:8 in Figure 17(c), the helical wSPI is the primary stable bifurcating solution in all presented parameter space. Comparing both, pure axial and pure transversal field (at same field strength), the up-shift, i.e., the stabilization is slightly larger in axial fields. This holds for toroidally closed (w)TVF and helical (w)SPI solutions.

Finally, this effect is even more pronounced when an oblique magnetic field is applied as illustrated in Figure 18 for field strength sx ¼ 0:6, sz ¼ 0:6. Note also the different scale on the ordinate in Figure 17(c) highlighting the stronger shifts due to such field configuration. One can speculate that this enforcement is due to the axial component of the oblique field leading to a base state with w having axial shear, and the radial field dependence ζ enhances the w component of the 2-CCF base state. The combination of the base state azimuthal shear and this axial shear

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

Figure 18.

with moving the γ-point to the left toward more negative values in Re2. Consider larger values, e.g., ζ ¼ 0:8 (Figure 15(c)), one sees that the co-dimension-two γ-point is even shifted to Re2>50 out of the considered parameter space. As a result, the primary instability is always the helical SPI, even with the two cylinders in strong co-rotation. This observation is crucially different from classical TCS, both without magnetic fields and also with assumption of an applied uniform magnetic field. Thus, the radial field dependence, induced by the susceptibility of the ferrofluid-filled annulus, is seen to cause major changes in the quantitative and

Bifurcation curves for the onset of wTVF (blue) and wSPI (orange) in parameter space for different ζ values (a) ζ ¼ 0, (b) ζ ¼ 0:4, and (c) ζ ¼ 0:8 with an applied transversal magnetic field (T) sx ¼ 0:6. ((a) is a

Bifurcation curves for the onset of TVF (blue) and SPI (orange) in Re2 � Re1 diagram for different ζ-values (a) ζ ¼ 0, (b) ζ ¼ 0:4 and (c) ζ ¼ 0:8 with an applied axial magnetic field (A) with sz ¼ 0:6. ((a) is a replot of Figure 7(c) for comparison). Note the different scale on the ordinate in (c). Simulation results of full nonlinear equations (11) consider a short periodic TCS with axial wavenumber k ¼ 3:927 (same hold for

Consider a transversal magnetic field, field strength sx ¼ 0:6 (Figure 17) instead of an axial one, the variation by increasing ζ remains virtually and qualitatively the same, aside that now all structures are wavy-like modulated (2-CCF, wTVF, wSPI). One finds the same preference, in the sense of stronger upshifts for toroidally closed wTVF with the same result of a shift in the γ-point to the right to larger values in Re2. Thus also here, for sufficiently strong ζ, e.g., ζ ¼ 0:8 in Figure 17(c), the helical wSPI is the primary stable bifurcating solution in all presented parameter space. Comparing both, pure axial and pure transversal field (at same field strength), the up-shift, i.e., the stabilization is slightly larger in axial fields. This

Finally, this effect is even more pronounced when an oblique magnetic field is applied as illustrated in Figure 18 for field strength sx ¼ 0:6, sz ¼ 0:6. Note also the different scale on the ordinate in Figure 17(c) highlighting the stronger shifts due to such field configuration. One can speculate that this enforcement is due to the axial component of the oblique field leading to a base state with w having axial shear, and the radial field dependence ζ enhances the w component of the 2-CCF base state. The combination of the base state azimuthal shear and this axial shear

qualitative characteristics of the flow instabilities.

Pattern Formation and Stability in Magnetic Colloids

Figure 16.

Figure 17.

32

replot of Figure 7(b) for comparison).

Figures 15 and 16).

holds for toroidally closed (w)TVF and helical (w)SPI solutions.

Bifurcation curves for the onset of wTVF (blue) and wSPI (orange) in Re2 � Re1 diagram for different ζvalues (a) ζ ¼ 0, (b) ζ ¼ 0:4, and (c) ζ ¼ 0:8 with an applied oblique magnetic field (O) with sx ¼ 0:6 ¼ s � z. ((a) is a replot of Figure 7(d) for comparison). Note different scaling in (c).

favors the helical instability. Close to the onset, the variation of the maxima of all flow components u, v, w is almost linear.

### 5.3.2 Modified flow structures

Figure 19 illustrates the flow structural properties of wTVF and wSPI in either a transverse magnetic field T (a,b) and an oblique magnetic field O (c,d), both with assumed radial field dependence ζ ¼ 0:8. Note, the flow structures of TVF and SPI under a pure axial applied magnetic field are not shown, as they virtually do not differ from the ones in absence of a magnetic field. Only the absolute values (e.g. in η) are different. Comparing these flow structures to the ones for ζ ¼ 0 cases as earlier presented in Figure 6 (Section 3.3.3), one sees the modulations in all structures to be enhanced/enforced. In general, the waviness (axial variation in azimuthal direction) of

#### Figure 19.

(See also Figures 2 and 6 for comparison to flow states without radial field dependence. The radial field dependence is considered to be ζ ¼ 0:8. Top row: Azimuthal vorticity isosurfaces η over two axial wavelengths with pure transversal applied magnetic field (T): (a) wTVF at Re2 ¼ 0, Re1 ¼ �130, (b) wSPI at Re2 ¼ �150, Re1 ¼ 190; and with oblique applied magnetic field (O): (c) wTVF at Re2 ¼ 0, Re1 ¼ �150, and (f) wSPI at Re2 ¼ �150, Re1 ¼ �200 Red (yellow) isosurfaces correspond to positive (negative) values as indicated. The bottom row shows the mode amplitudes j j um,n of the radial velocity field u corresponding to the structures above over the m-n-plane.

all flow structures is enforced/enhanced by the radial dependent magnetic field. However, the field-induced wTVF always remains a stationary, nonrotating, and phase-pinned structure, irrespective of whether the radial field dependence is considered or not. On the other hand, wSPIs bifurcate as degenerated solutions, either as a left- or right-winding SPI, depending on initial conditions. Their corresponding frequencies remain mainly constant over a wide range and do not differ much (about 20% or less) from those of SPI. Comparing the stimulated modes for flow structures when ζ ¼ 0:8 (bottom in Figure 19) to those with ζ ¼ 0 (bottom in Figure 6), one sees general larger mode amplitudes in the additional stimulated modes for consideration of an internal field dependence ð Þ ζ ¼6 0 . These larger amplitudes in the field induced modes are also the reason for the increase in waviness.

reason for this is computational high-expensive simulations, which even increase with continuation in refining the models to make them more realistic/appropriate, e.g., finite size effects, internal field modifications, etc. However, these model improvements are essential and necessary to uncover the full potential of

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

DOI: http://dx.doi.org/10.5772/intechopen.80301

ferrofluids, which already are used in a wide field of applications. The use in the car industry within antishocks to provide adjustable damper settings is probably one of the most prominent examples. However, either experiments and numerics are essential to mutually prove each other and combined to provide new inside of

This chapter deals with a rotating ferrofluid between two concentric cylinders,

i.e., ferrofluid in Taylor-Couette geometry. Therefor we consider different improvements in the model describing such magnetic particles in a flow under external applied field. Concluding the discussions, the main points can be summa-

• Any applied magnetic field changes the bifurcation thresholds of primary

• Magnetic fields with a finite transversal component break the system

develop belly shape structure at specific azimuthal positions.

solutions—the thresholds of both primary solutions, TVF and SPI, are shifted to larger parameter values—the basic state becomes stabilized against perturbations.

symmetry and renders all flow states to be inherently 3D. As a result, the pure flow states TVF and SPI cease to exist and are substituted by their wave-like cousins, wTVF and wSPI. Thereby, the generated wTVF differs crucially from the classical ones, which rotate in azimuthal direction. Instead, the magnetic field induced wTVFs are nonrotational, phase-pinned flow solutions and

• Assuming elongational flow, in order to consider finite-size effect of the magnetic particles and their interaction (agglomeration, particle-particle interaction and chain formation procedures), one detects further changes in the bifurcation threshold of the solutions. Also, this can result in further stabilization or destabilization. However, the effect of such elongational flow is never big enough to over-compensate the general, field-induced stabilization. Only the magnitude/amplitude varies. The flow states remain structural as in absence of

• Accounting for the changes in ferroluid's susceptibility in the container due to an external magnetic field, the real internal field can be approximated by 1=r2. As before the flow, structural properties remain qualitative similar. Bifurcation thresholds are shifted upward, toward larger Re1, whereby toroidally closed (w)TVFs are in general preferred and shifted more than helical (w)SPI. The

There are various different ways to solve the ferrohydrodynamic equations of

motions (Eqs. (5) and (10)), the latter including elongational flow). All here presented results have been obtained by a code (G1D3 [6, 7]), which combines a finite difference method of second order in (r,z) and time (explicit) with spectral

result is an increase in the regime of primary bifurcating (w)SPI.

potential applications.

elongational flow.

A. Appendix

35

A.1 Numerical method

decomposition in θ [6]:

rized as follows:

## 5.4 Resume

Applying a magnetic field across a container of ferrofluid changes the ferrofluid's susceptibility and results in modification of the magnetic field structure within the container. So far, most theories commonly neglect such modification, either for convenience or simplicity, and at the end, there is also possibility of numerical simulations in appropriate time or feasibility at all.

However, it is a matter of fact that an external applied magnetic field is modified, depending on the magnetic susceptibility. Experimental studies [10, 38] confirmed the modifications also for susceptibilities of commonly used ferrofluids. Consider such modifications to the imposed magnetic field effects the basic state, its stability, as well as the primary (supercritical) bifurcating solutions. This holds for any field configuration, with only differences in the strength of modifications, i.e., the stabilization (its magnitude; up-shift, move of the bifurcation thesholds to larger control parameters). On the other hand, the flow structural properties remain qualitative unaffected by such an radial field dependence, only change in quantitative manner. They are determined only by the general kind and orientation of the applied magnetic field. Thus, a pure axial field does not have any effect on the flow structural properties, leaving them as in absence of any field, i.e., pure TVF and pure SPI. As described before, the presence of a finite transversal field component results in a wavy-like modulated flow state. This is independent/untouched if the magnetic field is considered to be homogeneous or inhomogeneous with a radial field dependence. However, consider stronger modifications to the imposed magnetic field, i.e., larger values of the radial field dependence parameter ζ have the effect of further strengthening the already present waviness of the structures. This holds for (w)TVF and (w)SPI alike. An important observation is the different influence of the radial field dependence ð Þ ζ ¼6 0 onto topological closed and helical flow states. In general, the effect of modification in the primary bifurcation thresholds is larger for toroidally closed flow structures compared to helical ones. As a result, their intersection point, i.e., codimension γ-point is moved toward positive Re2, which is just the opposite direction, as it moves with increasing the magnetic field strength (for any given field configuration, e.g. Figure 7). The parameter space, the area for which (w)SPI bifurcate primary stable out of the basic state, is growing, even into region of co-rotating cylinders, i.e. Re2>0, making wSPI the favorable flow state in the system.

## 6. Conclusions

Although there has been significant progress in the understanding of ferrofluidic flows in the past decades, their fully theoretical understanding is still limited. A key

## Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

all flow structures is enforced/enhanced by the radial dependent magnetic field. However, the field-induced wTVF always remains a stationary, nonrotating, and phase-pinned structure, irrespective of whether the radial field dependence is considered or not. On the other hand, wSPIs bifurcate as degenerated solutions, either as a left- or right-winding SPI, depending on initial conditions. Their corresponding frequencies remain mainly constant over a wide range and do not differ much (about 20% or less) from those of SPI. Comparing the stimulated modes for flow structures when ζ ¼ 0:8 (bottom in Figure 19) to those with ζ ¼ 0 (bottom in Figure 6), one sees general larger mode amplitudes in the additional stimulated modes for consideration of an internal field dependence ð Þ ζ ¼6 0 . These larger amplitudes in the field

Applying a magnetic field across a container of ferrofluid changes the ferrofluid's susceptibility and results in modification of the magnetic field structure within the container. So far, most theories commonly neglect such modification, either for convenience or simplicity, and at the end, there is also possibility of

However, it is a matter of fact that an external applied magnetic field is modified, depending on the magnetic susceptibility. Experimental studies [10, 38] confirmed the modifications also for susceptibilities of commonly used ferrofluids. Consider such modifications to the imposed magnetic field effects the basic state, its stability, as well as the primary (supercritical) bifurcating solutions. This holds for any field configuration, with only differences in the strength of modifications, i.e., the stabilization (its magnitude; up-shift, move of the bifurcation thesholds to larger control parameters). On the other hand, the flow structural properties remain qualitative unaffected by such an radial field dependence, only change in quantitative manner. They are determined only by the general kind and orientation of the applied magnetic field. Thus, a pure axial field does not have any effect on the flow structural properties, leaving them as in absence of any field, i.e., pure TVF and pure SPI. As described before, the presence of a finite transversal field component results in a wavy-like modulated flow state. This is independent/untouched if the magnetic field is considered to be homogeneous or inhomogeneous with a radial field dependence. However, consider stronger modifications to the imposed magnetic field, i.e., larger values of the radial field dependence parameter ζ have the effect of further strengthening the already present waviness of the structures. This holds for (w)TVF and (w)SPI alike. An important observation is the different influence of the radial field dependence ð Þ ζ ¼6 0 onto topological closed and helical flow states. In general, the effect of modification in the primary bifurcation thresholds is larger for toroidally closed flow structures compared to helical ones. As a result, their intersection point, i.e., codimension γ-point is moved toward positive Re2, which is just the opposite direction, as it moves with increasing the magnetic field strength (for any given field configuration, e.g. Figure 7). The parameter space, the area for which (w)SPI bifurcate primary stable out of the basic state, is growing, even into region of co-rotating cylinders, i.e. Re2>0, making wSPI the favorable flow state in the system.

Although there has been significant progress in the understanding of ferrofluidic flows in the past decades, their fully theoretical understanding is still limited. A key

induced modes are also the reason for the increase in waviness.

Pattern Formation and Stability in Magnetic Colloids

numerical simulations in appropriate time or feasibility at all.

5.4 Resume

6. Conclusions

34

reason for this is computational high-expensive simulations, which even increase with continuation in refining the models to make them more realistic/appropriate, e.g., finite size effects, internal field modifications, etc. However, these model improvements are essential and necessary to uncover the full potential of ferrofluids, which already are used in a wide field of applications. The use in the car industry within antishocks to provide adjustable damper settings is probably one of the most prominent examples. However, either experiments and numerics are essential to mutually prove each other and combined to provide new inside of potential applications.

This chapter deals with a rotating ferrofluid between two concentric cylinders, i.e., ferrofluid in Taylor-Couette geometry. Therefor we consider different improvements in the model describing such magnetic particles in a flow under external applied field. Concluding the discussions, the main points can be summarized as follows:

## A. Appendix

## A.1 Numerical method

There are various different ways to solve the ferrohydrodynamic equations of motions (Eqs. (5) and (10)), the latter including elongational flow). All here presented results have been obtained by a code (G1D3 [6, 7]), which combines a finite difference method of second order in (r,z) and time (explicit) with spectral decomposition in θ [6]:

Pattern Formation and Stability in Magnetic Colloids

$$f(r, \theta, z, t) = \sum\_{m = -m\_{\text{max}}}^{m\_{\text{max}}} f\_m(r, z, t) e^{ink\theta},\tag{20}$$

where f denotes one of f g u; v; w; p . For the parameter regimes investigated here, mmax ¼ 8 provides adequate accuracy. For discretization, a forward time, centered space (FTCS) algorithm is used on staggered grids in the r � z plane following the procedure of Hirt et al. [45]. In particular, a homogeneous grid with discretization length δr ¼ δz ¼ 0:05 and time steps δt < 3800 has been used. Here, δt defines the time step between two iterations via FTCS algorithm of the system of coupled equations for the amplitudes f mð Þ r; z; t of the azimuthal normal modes �mmax ≤ m ≤ mmax. For diagnostic purposes, the complex mode amplitudes f m,nð Þ r; t were calculated obtained from a second axial Fourier decomposition:

$$f\_m(r, z, t) = \sum\_n f\_{m,n}(r, t)e^{inkx},\tag{21}$$

Author details

37

Sebastian Altmeyer

de Catalunya, Barcelona, Spain

provided the original work is properly cited.

Castelldefels School of Telecom and Aerospace Engineering, Universitat Politècnica

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to<EMAIL_ADDRESS>
Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

DOI: http://dx.doi.org/10.5772/intechopen.80301

where k ¼ 2π=λ is the axial wavenumber.

## Acknowledgments/Funding

S. Altmeyer is a Serra Húnter Fellow. This work was supported by the Spanish Government grant FIS2017–85794-P.

## Conflict of interest

The author declares no conflict of interest.

## Nomenclature and abbreviations

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

## Author details

f rð Þ¼ ; θ; z; t ∑

equations for the amplitudes f mð Þ r; z; t of the azimuthal normal modes

were calculated obtained from a second axial Fourier decomposition:

f mð Þ¼ r; z; t ∑

where k ¼ 2π=λ is the axial wavenumber.

Pattern Formation and Stability in Magnetic Colloids

The author declares no conflict of interest.

 For the parameter regimes investigated here, mmax ¼ 8 provides adequate accuracy. For discretization, a forward time, centered space (FTCS) algorithm is used on staggered grids in the r � z plane following the procedure of Hirt et al. [45]. In particular, a homogeneous grid with discretization length δr ¼ δz ¼ 0:05 and time steps δt < 3800 has been used. Here, δt defines the time step between two iterations via FTCS algorithm of the system of coupled

�mmax ≤ m ≤ mmax. For diagnostic purposes, the complex mode amplitudes f m,nð Þ r; t

n

S. Altmeyer is a Serra Húnter Fellow. This work was supported by the Spanish

f m,nð Þ r; t e

f mð Þ r; z; t e

inkθ

, (20)

inkz, (21)

Sebastian Altmeyer Castelldefels School of Telecom and Aerospace Engineering, Universitat Politècnica de Catalunya, Barcelona, Spain

\*Address all correspondence to<EMAIL_ADDRESS>
© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

[1] Rosensweig RE. Ferrohydrodynamics. Cambridge University Press, Cambridge; 1985. 344 pp. Journal of Fluid Mechanics, 200, 597-599 DOI: 10.1017/S00221120 89220773

[2] Hart JE. Dynamics of Atmospheres and Oceans. 2006;41. DOI: 10.1016/j. dynatmoce.2006.03.001

[3] Hart JE, Kittelman S. Dynamics of Atmospheres and Oceans. 2006;41:139. DOI: 10.1017/ S0022112075001309

[4] Niklas M. Zeitschrift fuer Physik B: Condensed Matter. 1987;68:493. DOI: 10.1007/BF01471080

[5] Niklas M, Müller-Krumbhaar H, Lücke M. Journal of Magnetism and Magnetic Materials. 1989;81:29. DOI: 10.1016/0304-8853(89)90225-4

[6] Altmeyer S, Hoffmann C, Leschhorn A, Lücke M. Physical Review E. 2010;82: 016321. DOI: 10.1103/PhysRevE.82. 016321

[7] Leschhorn A, Lücke M, Hoffmann C, Altmeyer S. Physical Review E. 2009;79: 036308. DOI: 10.1103/PhysRevE.79. 036308

[8] Reindl M, Leschhorn A, Lücke M, Odenbach S. Journal of Physics: Conference Series. 2009;149:012109. DOI: 10.1088/1742-6596/149/1/012109

[9] Reindl M, Odenbach S. Physics of Fluids. 2011;23:093102. DOI: 10.1063/ 1.3633341

[10] Odenbach S, Müller HW. Journal of Magnetism and Magnetic Materials. 2005;289:242. DOI: 10.1016/j. jmmm.2004.11.069

[11] Mahle S, Ilg P, Liu M. Physical Review E. 2008;77:016305. DOI: 10.1103/PhysRevE.77.016305

[12] Müller HW, Liu M. Physical Review E. 2001;64:061405. DOI: 10.1103/ PhysRevE.64.061405

1995;152(123). DOI: 10.1016/0304-8853

DOI: http://dx.doi.org/10.5772/intechopen.80301

Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow

[36] King GP, Li Y, Lee W, Swinney HL, Marcus PS. Journal of Fluid Mechanics.

1984;141:365. DOI: 10.1017/ S0022112084000896

38/S06

[37] Altmeyer S, Lopez JM, Do Y. Physical Review E. 2013;88:013003. DOI: 10.1103/PhysRevE.88.013003

[38] Odenbach S, Müller HW. Physical Review Letters. 2002;89:037202. DOI: 10.1103/PhysRevLett.89.037202

[39] Müller HW, Hahn D, Liu M. Journal of Physics: Condensed Matter. 2006;18: S2623. DOI: 10.1088/0953-8984/18/

[40] Chang MH, Chen CK, Weng HC. International Journal of Engineering Science. 2003;41:103. DOI: 10.1016/

S0020-7225(02)00183-0

jmmm.2004.10.123

[41] Singh J, Bajaj R. Journal of Magnetism and Magnetic Materials. 2005;294:53. DOI: 10.1016/j.

[42] Singh J, Bajaj R. International Journal of Mathematics and

10.1155/IJMMS.2005.3727

053010. DOI: 10.1103/ PhysRevE.87.053010

s00348-010-0940-y

1975STIN...7532418H

Mathematical Sciences. 2005;3727. DOI:

[44] Reindl M, Odenbach S. Experiments in Fluids. 2010;50:375. DOI: 10.1007/

[45] Hirt CW, Nichols BD, Romero NC. NASA STI/Recon Technical Report N; 1975. http://adsabs.harvard.edu/abs/

[43] Altmeyer S, Leschhorn A, Ch H, Lücke M. Physical Review E. 2013;87:

[24] Altmeyer S, Lopez JM, Do Y. Physical Review E. 2012;85:066314. DOI: 10.1103/PhysRevE.85.066314

[25] Zubarev AY, Skakova Y. Physical Review E. 2000;61:5415. DOI: 10.1103/

[26] Golubitsky M, Stewart I, Schaeffer

D. Singularities and Groups in Bifurcation Theory II. New York: Springer; 1988. ISBN: 978-1-4612-4574-2

17:249. DOI: 10.1137/0517023

[27] Golubitsky M, Stewart I. SIAM Journal on Mathematical Analysis. 1986;

[28] Ch H, Altmeyer S, Pinter A, Lücke M. New Journal of Physics. 2009;11: 053002. DOI: 10.1088/1367-2630/11/5/

[29] Davey A, Di Prima RC, Stuart JT. Journal of Fluid Mechanics. 1968;31:17. DOI: 10.1017/S0022112068000029

[30] Iooss G. Journal of Fluid Mechanics.

[31] Altmeyer S, Do Y, Marques F, Lopez JM. Physical Review E. 2012;86:046316. DOI: 10.1103/PhysRevE.86.046316

[32] Coles D. Journal of Fluid Mechanics.

[33] Wereley ST, Lueptow RM. Journal of Fluid Mechanics. 1998;364:59. DOI:

Mechanics. 1981;102:249. DOI: 10.1017/

[35] Akonur A, Lueptow RM. Physics of Fluids. 2003;15:947. DOI: 10.1063/

1986;173:273. DOI: 10.1017/ S0022112086001179

1965;21:385. DOI: 10.1017/ S0022112065000241

10.1017/S0022112098008969

[34] Jones CA. Journal of Fluid

S0022112081002620

1.1556615

39

(95)00442-4

PhysRevE.61.5415

053002

[13] Debye PJW. Polar Molecules. New York: Dover; 1929 https://hdl.handle. net/2027/uc1.\$b45989

[14] McTague JP. The Journal of Chemical Physics. 1969;51:133. DOI: 10.1063/1.1671697

[15] Martsenyuk MA, Raikher YL, Shliomis MI. Soviet Physics - JETP. 1974;38:413 http://www.jetp.ac.ru/cgibin/dn/e\_038\_02\_0413.pdf

[16] Gilles de GP, Prost J, editors. The Physics of Liquid Crystals. Oxford: Clarendon; 1983. ISBN:<PHONE_NUMBER>856

[17] Leschhorn A, Embs JP, Lücke M. Journal of Physics: Condensed Matter. 2006;18:S2633. DOI: 10.1088/ 0953-8984/18/38/S07

[18] Embs JP, May S, Wagner C, Kityk AV, Leschhorn A, Lücke M. Physical Review E. 2006;73:036302. DOI: 10.1103/PhysRevE.73.036302

[19] Taylor GI. Philosophical Transactions of the Royal Society of London A. 1923;223:289. DOI: 10.1098/ rsta.1923.0008

[20] Chossat P, Iooss G. The Couette-Taylor Problem. Berlin: Springer; 1994. DOI: 10.1007/978-1-4612-4300-7

[21] Langevin P. Annales de Chimie Physique. 1905;5(70)

[22] Embs J, Müller HW, Wagner C, Knorr K, Lücke M. Physical Review E. 2000;61:R2196. DOI: 10.1103/ PhysRevE.61.R2196

[23] Odenbach S, Gilly H. Journal of Magnetism and Magnetic Materials. Interaction of Magnetic Fields on Ferrofluidic Taylor-Couette Flow DOI: http://dx.doi.org/10.5772/intechopen.80301

1995;152(123). DOI: 10.1016/0304-8853 (95)00442-4

[24] Altmeyer S, Lopez JM, Do Y. Physical Review E. 2012;85:066314. DOI: 10.1103/PhysRevE.85.066314

References

89220773

[1] Rosensweig RE. Ferrohydrodynamics. Cambridge University Press, Cambridge; 1985. 344 pp. Journal of Fluid Mechanics, 200, 597-599 DOI: 10.1017/S00221120

Pattern Formation and Stability in Magnetic Colloids

[12] Müller HW, Liu M. Physical Review E. 2001;64:061405. DOI: 10.1103/

[13] Debye PJW. Polar Molecules. New York: Dover; 1929 https://hdl.handle.

[14] McTague JP. The Journal of Chemical Physics. 1969;51:133. DOI:

[15] Martsenyuk MA, Raikher YL, Shliomis MI. Soviet Physics - JETP. 1974;38:413 http://www.jetp.ac.ru/cgi-

[16] Gilles de GP, Prost J, editors. The Physics of Liquid Crystals. Oxford: Clarendon; 1983. ISBN:<PHONE_NUMBER>856

[17] Leschhorn A, Embs JP, Lücke M. Journal of Physics: Condensed Matter.

[18] Embs JP, May S, Wagner C, Kityk AV, Leschhorn A, Lücke M. Physical Review E. 2006;73:036302. DOI: 10.1103/PhysRevE.73.036302

2006;18:S2633. DOI: 10.1088/

[19] Taylor GI. Philosophical

rsta.1923.0008

Physique. 1905;5(70)

PhysRevE.61.R2196

Transactions of the Royal Society of London A. 1923;223:289. DOI: 10.1098/

[20] Chossat P, Iooss G. The Couette-Taylor Problem. Berlin: Springer; 1994. DOI: 10.1007/978-1-4612-4300-7

[21] Langevin P. Annales de Chimie

[22] Embs J, Müller HW, Wagner C, Knorr K, Lücke M. Physical Review E.

[23] Odenbach S, Gilly H. Journal of Magnetism and Magnetic Materials.

2000;61:R2196. DOI: 10.1103/

0953-8984/18/38/S07

bin/dn/e\_038\_02\_0413.pdf

PhysRevE.64.061405

net/2027/uc1.\$b45989

10.1063/1.1671697

[2] Hart JE. Dynamics of Atmospheres and Oceans. 2006;41. DOI: 10.1016/j.

[3] Hart JE, Kittelman S. Dynamics of Atmospheres and Oceans. 2006;41:139. DOI: 10.1017/ S0022112075001309

[4] Niklas M. Zeitschrift fuer Physik B: Condensed Matter. 1987;68:493. DOI:

[5] Niklas M, Müller-Krumbhaar H, Lücke M. Journal of Magnetism and Magnetic Materials. 1989;81:29. DOI: 10.1016/0304-8853(89)90225-4

[6] Altmeyer S, Hoffmann C, Leschhorn A, Lücke M. Physical Review E. 2010;82: 016321. DOI: 10.1103/PhysRevE.82.

[7] Leschhorn A, Lücke M, Hoffmann C, Altmeyer S. Physical Review E. 2009;79: 036308. DOI: 10.1103/PhysRevE.79.

[8] Reindl M, Leschhorn A, Lücke M, Odenbach S. Journal of Physics: Conference Series. 2009;149:012109. DOI: 10.1088/1742-6596/149/1/012109

[9] Reindl M, Odenbach S. Physics of Fluids. 2011;23:093102. DOI: 10.1063/

[10] Odenbach S, Müller HW. Journal of Magnetism and Magnetic Materials. 2005;289:242. DOI: 10.1016/j.

[11] Mahle S, Ilg P, Liu M. Physical Review E. 2008;77:016305. DOI: 10.1103/PhysRevE.77.016305

dynatmoce.2006.03.001

10.1007/BF01471080

016321

036308

1.3633341

38

jmmm.2004.11.069

[25] Zubarev AY, Skakova Y. Physical Review E. 2000;61:5415. DOI: 10.1103/ PhysRevE.61.5415

[26] Golubitsky M, Stewart I, Schaeffer D. Singularities and Groups in Bifurcation Theory II. New York: Springer; 1988. ISBN: 978-1-4612-4574-2

[27] Golubitsky M, Stewart I. SIAM Journal on Mathematical Analysis. 1986; 17:249. DOI: 10.1137/0517023

[28] Ch H, Altmeyer S, Pinter A, Lücke M. New Journal of Physics. 2009;11: 053002. DOI: 10.1088/1367-2630/11/5/ 053002

[29] Davey A, Di Prima RC, Stuart JT. Journal of Fluid Mechanics. 1968;31:17. DOI: 10.1017/S0022112068000029

[30] Iooss G. Journal of Fluid Mechanics. 1986;173:273. DOI: 10.1017/ S0022112086001179

[31] Altmeyer S, Do Y, Marques F, Lopez JM. Physical Review E. 2012;86:046316. DOI: 10.1103/PhysRevE.86.046316

[32] Coles D. Journal of Fluid Mechanics. 1965;21:385. DOI: 10.1017/ S0022112065000241

[33] Wereley ST, Lueptow RM. Journal of Fluid Mechanics. 1998;364:59. DOI: 10.1017/S0022112098008969

[34] Jones CA. Journal of Fluid Mechanics. 1981;102:249. DOI: 10.1017/ S0022112081002620

[35] Akonur A, Lueptow RM. Physics of Fluids. 2003;15:947. DOI: 10.1063/ 1.1556615

[36] King GP, Li Y, Lee W, Swinney HL, Marcus PS. Journal of Fluid Mechanics. 1984;141:365. DOI: 10.1017/ S0022112084000896

[37] Altmeyer S, Lopez JM, Do Y. Physical Review E. 2013;88:013003. DOI: 10.1103/PhysRevE.88.013003

[38] Odenbach S, Müller HW. Physical Review Letters. 2002;89:037202. DOI: 10.1103/PhysRevLett.89.037202

[39] Müller HW, Hahn D, Liu M. Journal of Physics: Condensed Matter. 2006;18: S2623. DOI: 10.1088/0953-8984/18/ 38/S06

[40] Chang MH, Chen CK, Weng HC. International Journal of Engineering Science. 2003;41:103. DOI: 10.1016/ S0020-7225(02)00183-0

[41] Singh J, Bajaj R. Journal of Magnetism and Magnetic Materials. 2005;294:53. DOI: 10.1016/j. jmmm.2004.10.123

[42] Singh J, Bajaj R. International Journal of Mathematics and Mathematical Sciences. 2005;3727. DOI: 10.1155/IJMMS.2005.3727

[43] Altmeyer S, Leschhorn A, Ch H, Lücke M. Physical Review E. 2013;87: 053010. DOI: 10.1103/ PhysRevE.87.053010

[44] Reindl M, Odenbach S. Experiments in Fluids. 2010;50:375. DOI: 10.1007/ s00348-010-0940-y

[45] Hirt CW, Nichols BD, Romero NC. NASA STI/Recon Technical Report N; 1975. http://adsabs.harvard.edu/abs/ 1975STIN...7532418H

Chapter 3

Abstract

1. Introduction

41

Equilibrium Structures and

Ricardo Peredo Ortíz, Martin Hernández Contreras

Colloidal Fluids

and Raquel Hernández Gómez

rium in colloidal ferrofluids is discussed.

Stationary Patterns on Magnetic

In this chapter, we review the experimental and theoretical modeling of struc-

Presently, several prototype experimental systems are very well characterized. We survey the different models, which help to reach a comprehensive knowledge of these complex magnetic fluids. One prime example is the ongoing investigation of the realistic interparticle potentials that drive the formation of the different phase states observed experimentally. Further, a stochastic equation approach for the description of tracer diffusion, viscoelasticity, and dielectric relaxation at equilib-

tural and dynamical properties of colloidal magnetic fluids at equilibrium.

Keywords: ferrofluids, viscoelasticity, colloidal magnetic fluids,

magnetohydrodynamics, dielectric relaxation, diffusion stochastic dynamics

magnetic dipole pair interaction potential is symmetric. Consequently, a corresponding change of physical units leads to an equivalent description of the magnetic fluid phase diagrams. Novel Monte Carlo simulation methods have

Colloidal magnetic fluids are made up of nanometer and micron-size ferromagnetic particles dispersed either in electrolytes or in organic solvents [1]. Due to their easy manipulation with magnetic or electric fields, they are having a significant impact on diverse technological applications ranging from biomedicine [2] and photonic devices [3, 4] up to fundamental studies that motivate the development of modern theories of nonequilibrium condensed matter [5–10]. In this chapter, we review the studies of static structural and dynamical properties in colloidal magnetic fluids, which provide a comprehensive description of their bulk phase behavior at thermal equilibrium. Presently, model magnetic colloidal systems can be prepared with tunable interaction among particles of the hard sphere and longrange dipolar types [11–14], especially the colloidal system constituted of micronsize polystyrene spheres that are electrically charged and dispersed in organic solvents [11]. In such a system, particles acquire an induced electric dipole moment under the external static electric field. Confocal microscopy has allowed the determination of its three-dimensional bulk phase diagram as a function of applied electric field and by taking into account the presence of a 1:1 salt. The electric and

## Chapter 3