**4. Structural characterization of perovskites**

The synthetic methodology and the characterization of the perovskite often go hand in hand in the sense that not one but a series of reaction mixtures are prepared and subjected to heat treatment. The stoichiometry is typically varied in a systematic way to find which ones will lead to new solid compounds or to solid solutions between known ones. A prime method to characterize the reaction products is X-ray powder diffraction (XRD), because many solid state reactions will produce polycristalline powders. Thus, powder diffraction will facilitate the identification of known phases in the mixture. If a pattern is found that is not known in the diffraction data libraries an attempt can be made to index the pattern, i.e. to identify the symmetry and the size of the unit cell. Obviously, if the product is not crystalline enough the characterization is typically much more difficult.

Once the unit cell of a new phase is known, the next step is to establish the stoichiometry of the phase. This can be done in a number of ways. Sometimes the composition of the original mixture will give a clue, if one finds only one product -a single powder pattern- or if one was trying to make a phase of a certain composition by analogy to known materials but this is rare. Often considerable effort in refining the synthetic methodology is required to obtain a pure sample of the new material. If it is possible to separate the product from the rest of the reaction mixture elemental analysis can be used. Another ways involves SEM and the generation of characteristic X-rays in the electron beam.

The easiest way to solve the structure is by using single crystal X-ray diffraction. The latter often requires revisiting and refining the preparative procedures and that is linked to the question which phases are stable at what composition and what stoichiometry. In other words what does the phase diagram looks like. An important tool in establishing this is thermal analysis techniques like DSC or DTA and, increasingly also, the synchrotron temperaturedependent powder diffraction. Increased knowledge of the phase relations often leads to further refinement in synthetic procedures in an iterative way. New phases are thus characterized by their melting points and their stoichiometric domains. The latter is important for the many solids that are non-stoichiometric compounds. The cell parameters obtained from XRD are particularly helpful to characterize the homogeneity ranges of the latter.

Structural Characterization of New Perovskites 115

Magnetic susceptibility can be measured as a function of temperature to establish whether the material is a *para-*, *ferro-*, *ferri-* or *antiferro*- magnet, among others. Again the information obtained pertains to the bonding in the material. This is particularly important for transition metal compounds. In the case of magnetic order neutron diffraction can be used to

Magnetic measurements are usually carried out with a SQUID magnetometer under different applied magnetic field. Figure 7 shows a picture of one of those SQUID

As an example, Figure 8 shows the magnetic susceptibility versus temperature data for the

Fig. 8. Magnetic susceptibility versus temperature for Sr2Ru0.5Co1.5O6.

**4.1.3 Magnetic properties**

magnetometers.

determine the magnetic structure.

Fig. 7. SQUID magnetometer.

perovskite Sr2Ru0.5Co1.5O6.

In order to analyse the different morphological and surface characteristics of particles in the perovskites, SEM (scanning electron microscopy) can be used. Figure 6 shows a micrograph obtained for La0.75Sr0.25Cr0.2Fe0.8O3 (the figure shows the texture and relief created by the elimination of volatile substances produced in the combustion of organic compounds during thermal treatment).

Fig. 6. SEM for La0.75Sr0.25Cr0.2Fe0.8O3.


*3.2. Volatile Fraction*

In this investigation SPME-GC–MS, a simple and effective analytical method, was performed to obtain qualitative and semi-quantitative profiles of volatiles from EVOO samples. Figure 1 shows the chromatographic profile of the volatile compounds of a monovarietal EVOO (Dolce Agogia) sample.

**Figure 1.** High-resolution gas chromatography mass-spectrometry (HRGC-MS) profile of volatile fraction of a monovarietal EVOO (Dolce Agogia) sample. Peak numbers correspond to the compounds listed in Table 3.

The semi-quantitative results (% areas) of the considered EVOO samples are shown in Table 3.

Some differences have been highlighted among the cultivars, both as regards the percent content of the *trans*-2-hexenal, the compound most represented in all cultivars, and other minor components. The *trans*-2-hexenal, which is formed from the α-linolenic acid by the action of the lipoxygenase enzymes, hydroperoxide lyase and isomerase, has been found in less amounts in the Moraiolo cultivar with respect to the Frantoio variety (*p* < 0.01), and also to Leccino and Dolce Agogia (*p* < 0.01). The content of this unsaturated aldehyde was also significantly different between Dolce Agogia - Frantoio (*p* <0.01) and Leccino (*p* < 0.05) varieties. Other observations regard the percent content of the *trans*-2-hexen-1-ol. This compound is obtained from the *trans*-2-hexenal by alcohol dehydrogenase activity, and its contents show an opposite trend with respect to its precursor (*trans*-2-hexenal). It was more represented in Moraiolo variety, followed by Dolce Agogia, Leccino and Frantoio. The differences between the cultivars are also enhanced if the relationship between the two compounds was considered. Based

on the results obtained, it could be affirmed that the activity of the enzyme alcohol dehydrogenase is influenced by the variety.

EVOO cultivar discrimination by volatile analysis has been addressed by numerous other authors. In this regard, SPME-GC-MS technique has also been used to discriminate Italian monovarietal EVOO [34,35]. Moreover, in a recent paper several approaches for the varietal differentiation of monovarietal virgin olive oils are overviewed [36].

## *3.3. Discriminant Analysis*

Previous papers have shown that TAG stereospecific analysis coupled with multivariate statistical data analysis was successfully used to characterize vegetable [22,23,37] and animal [38–40] foods. Generally, LDA is the best known and more widely used method to highlight differences between groups and to classify them. Moreover, it is known that LDA is an important parametric method useful to discriminate samples when the sample allocation is just known [41].

In this study, in order to classify and discriminate EVOO samples of different cultivars (Dolce Agogia, Frantoio, Leccino, and Moraiolo), multivariate parametric LDA technique was used. To better evaluate the influence of FA (total and intrapositional) compositions and volatile fraction in the classification of the oils, the results of the analytical determinations were elaborated by LDA. The statistical elaborations were performed considering the total and intrapositional FA percent compositions in TAG positions (*sn*-1, *sn*-2, and *sn*-3) of monovarietal EVOO samples, reported in Tables 1 and 2, respectively. Afterwards, the statistical elaborations were performed considering the volatile percent compositions of monovarietal EVOO samples, reported in Table 3.

The statistical elaboration of the results of the stereospecific analysis of the TAG provided interesting results for the characterization of oil samples obtained from different varieties of *O. europea*; the theory that the intrapositional TAG compositions represent a fingerprint of the most represented fraction of the oils is confirmed. In fact, these compositions depend on the specificity of the acyltransferase involved in the process of biosynthesis of the TAG, and are therefore species-specific [42].

The selection of the most significant variables was performed by stepwise analysis. Table 4 shows the canonical Fisher's linear discriminant characteristics (eigenvalue, percentage of variance, and significance test) of the testing data (FA compositions and volatiles) from monovarietal EVOO samples. It can be emphasized that LDA performed on volatile data showed higher percentage of the variance explained (97.8 vs 54.8) and canonical correlation (1.000 vs 0.965) for the first discriminant function in respect to FA compositional data. The statistical significance of each discriminant function was also evaluated on the basis of the Wilks' lambda factor; it showed that the first two functions for volatile data were significant (*p* < 0.05). In fact, Wilks' lambda values of the first two discriminant functions (0.000) showed an optimal discriminant power of the model, with a better discriminant power of volatiles in respect to FA data. The significance values (0.000 for the first two functions) indicated that there was a highly significant difference between the group centroids for volatile data.

**Table 4.** Fisher's linear discriminant functions and functions at group centroids obtained from LDA analysis using FA or volatiles percent compositions of monovarietal EVOO samples.

Table 5 shows the standardized canonical discriminant function coefficients. According to standardized coefficients, oleic and linoleic acids for FA had the greatest impact on the discrimination for functions 1 and 2. As regards the volatiles, ethanol and 1-penten-3-one had the greatest impact on the discrimination for function 1. The values showed the impact of each variable on the discriminant function after "standardizing", putting each variable on the same platform. Figure 2 shows the plot of the first two discriminant functions, using FA percent composition of monovarietal EVOO samples, while Figure 3 shows the plot of the first two discriminant functions, using volatile percent composition.

**Table 5.** Standardized canonical discriminant function coefficients obtained from LDA analysis using total and positional TAG acidic compositions of monovarietal EVOO samples.

t, total FA% content in TAG fraction; *sn*-1, FA percent content in *sn*-1 position of TAG fraction; *sn*-2, FA percent content in *sn*-2 position of TAG fraction.

**Figure 2.** Discriminant function plot of the first two functions obtained using total and intrapositional FA percent composition of monovarietal EVOO samples (DA, Dolce Agogia; FR, Frantoio; LE, Leccino; MO, Moraiolo).

**Figure 3.** Discriminant function plot of the first two functions obtained using volatile percent composition of monovarietal EVOO samples (DA, Dolce Agogia; FR, Frantoio; LE, Leccino; MO, Moraiolo).

In the two-dimensional space defined by the first two discriminating functions, the samples belonging to the same cultivar are well discriminated, even if the Dolce Agogia and Moraiolo groups are better concentrated around the centroid of the group. However, the results of the classification show that the samples of all groups are correctly classified.


**2. Problem formulation and scale analysis**

The problem of combined heat and mass transfer over a heated semi-infinite inclined solid wall is considered. The fluid is assumed to be steady, Newtonian, viscous, and incompressible. It is assumed that the wall is maintained at uniform surface temperature *Tw* and concentration *Cw* and it is immersed in fluid reservoir at rest which is kept at uniform ambient temperature *T*∞ and concentration *C*∞ such that *Tw*>*T*∞ and *Cw*>*C*∞. Boundary layer flow over an inclined wall driven by both thermal gradient and concentration gradient, respectively, are thereby set up due to the difference between wall values and quiescent fluid values. Hence, it is called combined heat and mass transfer phenomenon over an inclined wall (**Figure 1**).

This problem is governed by the non-linear and coupled conservation equations. Using the Boussinesq approximation and boundary layer simplifications, we have the following:

#### **Continuity equation**

diffusion has been studied by few investigators over the years. The pioneer of this area of research is the work of Gebhart and Pera in 1971 [4] where they investigated the combined buoyancy effects of thermal and mass diffusion on natural convection flow. Also, Bejan and Khair [7] carried out some analysis on heat and mass transfer by natural convection in porous media. Furthermore, the Schmidt number is the appropriate number in the concentration equation for *Pr* < 1 regime, while in the *Pr* > 1, regime Lewis number is the appropriate dimensionless number for vertical walls and this extends to inclined walls. This important criterion is sometimes omitted from heat and mass transfer studies. Allain et al. [8] also considered the problem of combined heat and mass transfer convection flows over a vertical isothermal plate. These contributors used a combination of integral and scaling laws of Bejan for their investigations. Their work was restricted to cases where two buoyancy forces aid each other; however, it was observed that heat diffusion is always more efficient than mass diffusion meaning that Lewis number is always greater than unity in most cases. It has been recommended in some previous works that more numerical or experimental results covering a wide range

of Prandtl and Schmidt numbers are needed to be obtained by further

Some other research studies carried out were by Angirasa and Peterson [9], who considered free convection due to combined buoyancy forces for *N* = 2 in a thermally stratified medium, and, recently, other contributors have considered flow of power law fluids in saturated porous medium due to double diffusive free convection [10]. Other effects such as Soret and Dufour forces in a Darcy porous medium were considered by Krishna et al. [11]. The problem of mass transfer flow through an inclined plate has generated much interest from astrophysical, renewable energy system, and also hypersonic aerodynamics researchers for a number of decades [1]. It is important to note that combined heat and mass flow over an isothermal inclined wall has received little contributions from scholars [12, 13]. The key notable ones in the literature include the general model formulation of natural convection boundary layer flow over a flat plate with arbitrary inclination by Umemura and Law [14]. Their results showed that flow properties depend on both the degree of inclination and distance from the leading edge. Other investigations considered the problem of combined heat and mass transfer by MHD free convection from an inclined plate in the presence of internal heat generation of absorption [15], natural convection flow over a permeable inclined surface with variable temperature, momentum, and concentration [16], investigations on combined heat and mass transfer in hydro-magnetic dynamic boundary layer flow past an inclined plate with viscous dissipation in porous medium [17], a study on micro-polar fluid behavior in MHD-free convection with constant heat and mass flux [18] and investigations on mass transfer flow through an inclined plate with

However, research conducted to critically analyze fluid behavior with the effect of species concentration and thermal diffusion on heat and mass transfer particularly for low Prandtl flows past an inclined wall is very rare. This gap has been captured in this study. The objective of this research is to investigate the effect of combined heat and species concentration involving a low Prandtl number fluid flow over an inclined wall using the method of scale analysis in formulation of the model along with the similarity transformation technique to convert partial differential equations to ordinary differential equation. The resulting dimensionless coupled and non-linear equations are solved using differential transform method. The numerics of the computation are discussed for different values of

dimensionless parameters and are graphically presented.

investigations.

*Computational Fluid Dynamics Simulations*

porous medium [19].

**26**

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial \mathbf{y}} = \mathbf{0}, \tag{1}$$

#### **Momentum equation**

$$u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = \mathfrak{G}\frac{\partial^2 v}{\partial x^2} + \rho \mathfrak{g}\beta\_T (T - T\_{\Leftrightarrow}) \cos a + \rho \mathfrak{g}\beta\_c (\mathbf{C} - \mathbf{C}\_{\Leftrightarrow}) \cos a \tag{2}$$

**Energy equation**

$$
\mu \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial \mathbf{y}} = \alpha\_T \frac{\partial^2 T}{\partial \mathbf{x}^2} \tag{3}
$$

**Species concentration equation**

$$
u \frac{\partial \mathbf{C}}{\partial \mathbf{x}} + v \frac{\partial \mathbf{C}}{\partial \mathbf{y}} = D \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} \tag{4}$$

**Figure 1.** *Physical model of double diffusive free convection over vertical wall.*

Here, *u* is the velocity along x-axis, *v* is the velocity along y-axis or along the vertical wall,*T* is the temperature, and *C* is the concentration. These equations are subject to the boundary conditions given by

$$\mu = 0, \nu = 0, T = T\_w, \mathbf{C} = \mathbf{C}\_w \text{ at } \mathbf{x} = \mathbf{0} \tag{5}$$

$$\mu = 0, \nu = 0, T = T\_{\infty}, \mathbf{C} = \mathbf{C}\_{w} \text{ at } \mathbf{x} = \infty \tag{6}$$

Following the procedures as outlined by Khair and Bejan [5], it can be clearly shown that for low Prandtl number flows, the velocity, concentration, and temperature boundary layer scales (for N = 0) are:

$$
\delta\_v \sim HRa\_\* \, ^{-\frac{1}{4}} Pr^{\frac{1}{4}} \tag{7}
$$

*ξ* the similarity variable for concentration layer is defined as *ξ* ¼ *x*

boundary layer. The attendant stream function is obtained as *ψ* � *DRa*

3 4

3 4

In the outer layer, where there is inertia-buoyancy balance, *ξ* which is the

3 4

In further works, these equations will be solved asymptotically as *Pr*!0 to

The differential transform method is used to solve the non-linear similarity Eqs. (13)–(15) subject to boundary conditions in Eq. (16). The procedure to convert

3; *Z*5 ¼ *Z*0

*Z*3*Z*5 þ *NZ*7*cosα*

3 4 *LeFC*~0

; *Z*3 ¼ *F*; *Z*4 ¼ *F*0 ¼ *Z*0

Such that the governing equations of motion become:

5 ¼ *Z*1*cosα* � 1

2 *Z*2 4 þ 3 4

Le*:*Pr*:Z*3*Z*7*:*

cies concentration become:

�3 4 *FF* 00 þ

function obtained as *ψ* � *αRa*1

�3 4

obtain approximate analytical results.

the PDEs to ODEs is outlined below.

1 ¼ *θ*0

*Z*0 1 ¼ *Z*2

*Z*0 2 ¼ � 3 4 Pr*:Z*2*Z*3

*Z*0 3 ¼ *Z*4

*Z*0 4 ¼ *Z*5

*Z*0

*Z*0 6 ¼ *Z*7

*Z*0 7 ¼ � 3 4

**2.1 Method of solution**

Let *Z*1 ¼ *θ*; *Z*2 ¼ *Z*0

**29**

*FF*00 þ

*F* 0 � �2

*DOI: http://dx.doi.org/10.5772/intechopen.90896*

similarity variable for thermal layer is defined as *ξ* ¼ *x*

*F*0 ð Þ2

*=*4 ∗ *Pr*1*=*4*F*ð Þ*η* . The resulting dimensionless equations for low Prandtl number flow are:

where *δv* is the thin viscous layer closest to the wall and *δc* is the concentration

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass…*

The ordinary differential equations governing the momentum, energy, and spe-

2 ¼ �*LeScF*000ð Þþ *η θ η*ð Þ cos *α* þ *NC*ð Þ*η* cos *α* (17)

2 ¼ �*PrF*000ð Þþ *η* ~*θ η*ð Þcos*α* þ *NC*~ð Þ*η* cos*α* (20)

*Fθ*0 ¼ *Sc:θ*00 (18)

*FC*0 ¼ *Pr:C*00 (19)

*F*~*θ*0 ¼ ~*θ*00 (21)

¼ *C*~00 (22)

4 ¼ *F*00; *Z*6 ¼ *C*; *Z*7 ¼ *Z*0

9

>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>;

6 ¼ *C*0 *:*

(23)

(24)

*δc*

*δT* , and the associated stream

1*=*4 ∗ *Pr*�1*=*4*F*ð Þ*η :*

, *for δc* >*δv*

$$
\delta\_T \sim H \left( Ra\_\* \, ^{-\frac{1}{4}} Pr \right)^{-\frac{1}{4}} \tag{8}
$$

$$\delta \mathbf{c} \sim H \frac{D}{\infty} \text{Ra}^{-1\downarrow\_4} \text{Pr}^{-\text{3}\_{\%}} \tag{9}$$

While vertical velocity (*v*) scales as.

$$v \sim \frac{\infty}{\mathcal{Y}} \left( Ra\_\* \, ^{-\frac{1}{4}} Pr \right)^{\dagger\_{\hat{\Lambda}}} \tag{10}$$

The similarity variable for velocity layer scales as

$$\eta = \frac{\varkappa}{\delta\_v} = \frac{\varkappa}{\mathcal{Y}} \left( \frac{Ra\_\*}{Pr} \right)^{\frac{1}{4}} \tag{11}$$

And corresponding stream function *ψ* scales as

$$
\psi \sim a \text{Ra}\_\*^{1\_4} \text{Pr}^{\otimes 4} F(\eta) \tag{12}
$$

where *F*ð Þ*η* is the velocity function.

Boussinesq approximation is used in Eq. (2), and the PDEs are reduced to a set of coupled ODEs using similarity variable *η*. It can easily be shown that for the inner layer of low Prandtl number flows, the dimensionless momentum, energy and concentration equations give:

$$\frac{-3}{4}ff'' + \frac{\left(f'\right)^2}{2} = -f'''(\eta) + \theta(\eta)\cos a + NC(\eta)\cos a\tag{13}$$

$$\frac{3}{4}\text{Pr}\mathfrak{f}\theta'=\theta'\text{\AA}\tag{14}$$

$$\frac{3}{4} \text{Pr.Left}^{\circ} = \text{C}^{\prime} \tag{15}$$

Eqs. (13)–(15) are solved for temperature *θ*, velocity *f* 0 , and concentration *C*, respectively, subject to the boundary conditions in Eq. (16).

$$f'(\mathbf{0}) = f(\mathbf{0}) = \mathbf{0}, \theta(\mathbf{0}) = \boldsymbol{\gamma}(\mathbf{0}) = \mathbf{1} \text{ at } \boldsymbol{\eta} = \mathbf{0} \tag{16}$$

$$f'(\boldsymbol{\infty}) = \theta(\boldsymbol{\infty}) = \boldsymbol{\gamma}(\boldsymbol{\infty}) = \mathbf{0} \text{ at } \boldsymbol{\eta} = \boldsymbol{\infty}$$

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass… DOI: http://dx.doi.org/10.5772/intechopen.90896*

*ξ* the similarity variable for concentration layer is defined as *ξ* ¼ *x δc* , *for δc* >*δv* where *δv* is the thin viscous layer closest to the wall and *δc* is the concentration 1*=*4 *=*

boundary layer. The attendant stream function is obtained as *ψ* � *DRa* ∗ *Pr*�14*F*ð Þ*η :* The ordinary differential equations governing the momentum, energy, and species concentration become:

$$\frac{-3}{4}\overline{F}\overline{F}' + \frac{\left(\overline{F}\right)^2}{2} = -L\varepsilon \text{Sc}\overline{F}''(\eta) + \overline{\theta}(\eta)\cos a + N\overline{C}(\eta)\cos a\tag{17}$$

$$\frac{3}{4}\overline{F}\theta'=\text{Sc.}\overline{\theta}\prime \tag{18}$$

$$\frac{3}{4}\overline{F}C' = Pr.\overline{C}'\tag{19}$$

In the outer layer, where there is inertia-buoyancy balance, *ξ* which is the similarity variable for thermal layer is defined as *ξ* ¼ *x δT* , and the associated stream function obtained as *ψ* � *αRa*1*=*4 ∗ *Pr*1*=*4*F*ð Þ*η* .

The resulting dimensionless equations for low Prandtl number flow are:

$$\frac{-3}{4}FF'' + \frac{\left(F'\right)^2}{2} = -PrF''''(\eta) + \ddot{\theta}(\eta)\cos a + N\ddot{\mathcal{C}}(\eta)\cos a\tag{20}$$

$$\frac{3}{4}F\ddot{\theta}' = \ddot{\theta}''\tag{21}$$

$$\frac{3}{4}LeF\tilde{\mathbf{C}}' = \tilde{\mathbf{C}}'\tag{22}$$

In further works, these equations will be solved asymptotically as *Pr*!0 to obtain approximate analytical results.

#### **2.1 Method of solution**

Here, *u* is the velocity along x-axis, *v* is the velocity along y-axis or along the vertical wall,*T* is the temperature, and *C* is the concentration. These equations are

Following the procedures as outlined by Khair and Bejan [5], it can be clearly shown that for low Prandtl number flows, the velocity, concentration, and temper-

> �1 4*Pr*1

�1 4*Pr* �1

∝ *Ra*�1*=*4 ∗ *Pr*�3*=*

*Ra*∗ �1 4*Pr* 1

*δv* � *HRa*∗

*δT* � *H Ra*∗

*δC* � *H D*

*v* � ∝ *y*

*η* ¼ *x δv* ¼ *x y*

*ψ* � *αRa*

1*=*4 ∗ *Pr*3*=*

Boussinesq approximation is used in Eq. (2), and the PDEs are reduced to a set of coupled ODEs using similarity variable *η*. It can easily be shown that for the inner layer of low Prandtl number flows, the dimensionless momentum, energy and

ð Þ¼ 0 *f*ð Þ¼ 0 0, *θ*ð Þ¼ 0 *γ*ð Þ¼ 0 1 *at η* ¼ 0

ð Þ¼ ∞ *θ*ð Þ¼ ∞ *γ*ð Þ¼ ∞ 0 *at η* ¼ ∞

The similarity variable for velocity layer scales as

And corresponding stream function *ψ* scales as

where *F*ð Þ*η* is the velocity function.

4 *f f*00 þ *f*

*f* 0

**28**

0 2

2 ¼ �*f*000

3

3 4

Eqs. (13)–(15) are solved for temperature *θ*, velocity *f*

respectively, subject to the boundary conditions in Eq. (16).

*f* 0

concentration equations give:

�3

*u* ¼ 0, *v* ¼ 0, *T* ¼ *Tw*,*C* ¼ *Cw at x* ¼ 0 (5) *u* ¼ 0, *v* ¼ 0, *T* ¼ *T*∞,*C* ¼ *Cw at x* ¼ ∞ (6)

4

*=*2

*Ra*∗ *Pr* 1 4

4 (7)

4 (9)

4*F*ð Þ*η* (12)

ð Þþ *η θ η*ð Þ cos *α* þ *NC*ð Þ*η* cos *α* (13)

4 *Prf θ*0 ¼ *θ*00 (14)

*Pr:LefC*0 ¼ *C*00 (15)

0

, and concentration *C*,

(16)

(8)

(10)

(11)

subject to the boundary conditions given by

*Computational Fluid Dynamics Simulations*

ature boundary layer scales (for N = 0) are:

While vertical velocity (*v*) scales as.

The differential transform method is used to solve the non-linear similarity Eqs. (13)–(15) subject to boundary conditions in Eq. (16). The procedure to convert the PDEs to ODEs is outlined below.

$$\text{Let } \mathbf{Z}\_1 = \theta; \; \mathbf{Z}\_2 = \mathbf{Z}\_1' = \theta'; \; \mathbf{Z}\_3 = \mathbf{F}; \; \mathbf{Z}\_4 = \mathbf{F}' = \mathbf{Z}\_3'; \; \mathbf{Z}\_5 = \mathbf{Z}\_4' = \mathbf{F}''; \; \mathbf{Z}\_6 = \mathbf{C}; \; \mathbf{Z}\_7 = \mathbf{Z}\_6' = \mathbf{C}'.\tag{23}$$

Such that the governing equations of motion become:

$$\begin{aligned} Z\_1' &= Z\_2\\ Z\_2' &= -\frac{3}{4} \text{Pr} Z\_2 Z\_3\\ Z\_3' &= Z\_4\\ Z\_4' &= Z\_5\\ Z\_5' &= Z\_1 \cos \alpha - \frac{1}{2} Z\_4^2 + \frac{3}{4} Z\_3 Z\_5 + N Z\_7 \cos \alpha\\ Z\_6' &= Z\_7\\ Z\_7' &= -\frac{3}{4} \text{Le.Pr.} Z\_3 Z\_7. \end{aligned} \tag{24}$$

Due to limitation of convergence of the classical DTM which is only valid near *η* ¼ 0, the multi-step transformation is used. Carrying out multi-step differential transformations, we have:

$$F\_i(\eta) = \sum\_{i=0}^k \left(\frac{\eta}{Hi}\right)^i \bar{F}\_i(k)$$

$$\theta\_i(\eta) = \sum\_{i=0}^k \left(\frac{\eta}{Hi}\right)^i \bar{\theta}\_i(k)$$

$$C\_i(\eta) = \sum\_{i=0}^k \left(\frac{\eta}{Hi}\right)^i \bar{C}\_i(k) \tag{25}$$

**3. Results and discussion**

*DOI: http://dx.doi.org/10.5772/intechopen.90896*

respect to the angle of inclination of the wall.

increases as the buoyancy ratio is increased.

the angle of inclination of the wall.

ary layer thickness for all buoyancy ratio.

angle of inclination increases, *δc* increases.

compared to when N = 0.

increases.

**31**

respectively.

**Table 1** shows the expressions for the dimensionless parameters that are of interest in this work. The solutions to Eqs. (13)–(15) subjected to Eq. (16) solved using the multi-step DTM method are presented graphically in the figures below. The results shown are for Prandtl numbers of 0.01, 0.1, 0.5, and 0.72,

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass…*

**Figure 2** shows the profiles of local skin friction against Prandtl number for various angles of inclination at a constant Lewis number. It could be observed from the plots that the shearing stress decreases as the Prandtl number increases for all the plate angles considered. More importantly, it is illustrated by the graphs that the local skin friction also decreases with increase in the buoyancy ratio and also with

**Figure 3** shows the plots of local Nusselt number against Prandtl number for various angles of inclination at a constant Lewis number. It is clearly seen from the results that the rate of heat transfer (Nusselt number) increases as the Pr increases for all the wall inclination angles. Also to note is the fact that Nusselt number

**Figure 4** shows the results of how the local Sherwood number changes as the Lewis number is increased for various angles of inclination at a constant Prandtl number. It is noted from the graphs that the local Sherwood number increases as the Lewis number increases for all angles of inclination. Also, it is observed from the figures that the rate of increase of Sherwood number is dependent on N as well as

In **Figure 5**, the similarity profiles of the effect of the buoyancy ratio and the angles of inclination on the dimensionless velocities at fixed Prandtl and Lewis numbers is presented. It could be interpreted from the results for N = 0 that a maximum velocity is obtained for a vertical wall while at an angle of 60°, the minimum velocity. Also, it is clearly seen from the plots that aside from the vertical wall possessing the maximum velocity, its vertical velocity value for N = 1 is higher when compared to the case when N = 0. When the buoyancy ratio N is further increased to 1.5, the velocity for the vertical wall also increases. The trend in the figure also shows that increasing the inclination angle increases the velocity bound-

In **Figure 6**, the similarity profiles of dimensionless temperature for the wall angles of inclinations considered in this study at constant Lewis number are presented. The temperature profiles show that despite the varying buoyancy ratio, the thermal boundary layer thickness increases as the inclination angle of the wall

**Figure 7** presents the similarity profiles of dimensionless species concentration distributions for the angles of inclination of the wall at a constant Lewis number. It is clearly seen that as the buoyancy ratio is increased, the concentration boundary layer thickness has a decreasing trend under this same flow configuration but as the

**Figure 8** shows the plots of the coupled similarity profiles of dimensionless temperature, species concentration, and velocity for wall inclination angle of 60° under the constant Lewis number of 10. It can be seen from the results that

increasing N has negligible effect on the trio of velocity, concentration and temperature boundary layer thicknesses for a fixed wall angle. However, the effect of N is very noticeable in the vertical velocity values which is clearly higher when N = 1.5

Where *i* = 0, 1, 2, 3 … . n indicates the *i th* sub-domain; *k* = 0, 1, 2, … m represents the number of terms of the power series. *Hi*represents the sub-domain interval, and *F i*ð Þ*k* , *θi*ð Þ*k* ,*C i*ð Þ*k* are the transformed functions, respectively.

The transformation of the associated boundary conditions follows as:

*Z*1ð Þ¼ 0 1, *Z*2ð Þ¼ 0 *a*, *Z*3ð Þ¼ 0 0, *Z*4ð Þ¼ 0 0, *Z*5ð Þ¼ 0 *b*, *Z*6ð Þ¼ 0 1, *Z*7ð Þ¼ 0 *c* where *a*, *b*, and *c* are obtained by solving the system of algebraic simultaneous equations, and the results obtained are shown in **Table 1**.

$$\begin{aligned} Z\_1(k+1) &= \frac{Z\_2(k)}{k+1} \\ Z\_2(k+1) &= \frac{-\frac{3}{4}\text{Pr}}{k+1} \sum\_{i=0}^k (Z\_3(i)Z\_2(k-i)) \\ Z\_3(k+1) &= \frac{Z\_4(k)}{k+1} \\ Z\_4(k+1) &= \frac{Z\_5(k)}{k+1} \\ Z\_5(k+1) &= \frac{1}{k+1} \left[ Z\_1(k)\cos\alpha - \frac{1}{2} \sum\_{i=0}^k Z\_4(i)Z\_4(k-i) + \frac{3}{4} \sum\_{i=0}^k Z\_5(i)Z\_5(k-i) + N.Z\_7(k)\cos\alpha \right] \\ Z\_6(k+1) &= \frac{Z\_7\left(k\right)}{k+1} \\ Z\_7(k+1) &= \frac{-\frac{3}{4}\text{Pr}L\epsilon}{k+1} \sum\_{i=0}^k [Z\_3(i)Z\_7(k-i)] \end{aligned} \tag{26}$$

Parameters Domain: *Pr* < 1, *Le* ≫ 1, *Le* ≪ 1, and *N* ≥ 0

The local Nusselt number: *Nu* � ½ � *Gr* ∗ ð Þ*y* 1 4 *θ*0 ð Þ*η*¼0 The Local Sherwood number: *Sh* ¼ �½ � *Gr* ∗ ð Þ*y* 1 4 *C*0 ð Þ*η*¼0 The shearing stress on the plate: *τw* ¼ ∝ *y*2 *Ra*∗ ð Þ*y* 3*=*4 *Pr*1*=*4 *f* 00 � � *η*¼0 The coefficient of skin friction: *Cf* ¼ ∝*ϑPr*1*=*4*Ra*∗ ð Þ*y* 3*=*4 *u*2 *y*2 2 *f* 00 � � *η*¼0

#### **Table 1.**

*Important dimensionless parameters of interest.*

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass… DOI: http://dx.doi.org/10.5772/intechopen.90896*