# Pressure-driven Nitrogen Flow in Divergent Microchannels with Isothermal
Walls

Amin Ebrahimi  Department of Materials Science and Engineering, Faculty of
Mechanical, Maritime and Materials Engineering, Delft University of
Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Vahid Shahabi
Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi
University of Mashhad, Mashhad, P.O. Box 91775-1111, Khorasan Razavi, Iran
Ehsan Roohi  State Key Laboratory for Strength and Vibration of Mechanical
Structures, International Center for Applied Mechanics, School of Aerospace
Engineering, Xi’an Jiaotong University, Xi’an, China

###### Abstract

Gas flow and heat transfer in confined geometries at micro and nano scales
differ considerably from those at macro-scales, mainly due to nonequilibrium
effects such as velocity slip and temperature jump. The nonequilibrium effects
enhance with a decrease in the characteristic length-scale of the fluid flow
or the gas density, leading to the failure of the standard Navier-Stokes-
Fourier (NSF) equations in predicting thermal and fluid flow fields. The
direct simulation Monte-Carlo (DSMC) method is employed in the present work to
investigate pressure-driven nitrogen flow in divergent microchannels with
various divergence angles and isothermal walls. The thermal fields obtained
from numerical simulations are analysed for different inlet-to-outlet pressure
ratios ($1.5\leq\Pi\leq 2.5$), tangential momentum accommodation coefficients
and Knudsen numbers ($0.05\leq\mathrm{Kn}\leq 12.5$), covering slip to free-
molecular rarefaction regimes. The thermal field in the microchannel is
predicted, heat-lines are visualised, and the physics of heat transfer in the
microchannel is discussed. Due to the rarefaction effects, the direction of
heat flow is largely opposite to that of the mass flow. However, the interplay
between thermal and pressure gradients, which are affected by geometrical
configurations of the microchannel and applied boundary conditions, determines
the net heat flow direction. Additionally, the occurrence of thermal
separation and cold-to-hot heat transfer (also known as anti-Fourier heat
transfer) in divergent microchannels is explained.

Keywords: Poiseuille micro-flow; Thermal field analysis; Heat flow; Divergent
microchannel; Direct Simulation Monte-Carlo (DSMC)

##  1 Introduction

Gas flow in micro- and Nano-channels with non-uniform cross-sections offers
opportunities to develop small devices with novel applications (see for
example, [1, 2, 3, 4, 5, 6]). Understanding heat and fluid flow in
microchannels is essential to engineer novel micro-electromechanical systems
(MEMS) [7, 8, 9]. However, this is a challenging task since the rate of
collisions between the gas molecules and solid walls reduces with decreasing
the characteristic length scale of the fluid flow or the gas density,
affecting the random movement of molecules, which is commonly known as
nonequilibrium effects [10, 11]. Nonequilibrium molecular transport processes
cause velocity slip and temperature jump at walls and influence thermal and
fluid flow fields [12, 13]. The Knudsen number (Kn), which is the ratio of the
molecular mean free path $\lambda$ to a characteristic length scale
$\mathscr{L}$, is often employed as an indicator of deviation from the
equilibrium condition. The higher the Knudsen number, the higher the deviation
from the equilibrium condition. Four different rarefaction regimes have been
defined based on the Knudsen number [14], and are commonly named as free
molecular ($\mathrm{Kn}>10$), transition ($10^{-1}\leq\mathrm{Kn}\leq 10$),
slip ($10^{-3}\leq\mathrm{Kn}\leq 10^{-1}$) and continuum ($\mathrm{Kn}\leq
10^{-3}$) regimes.

Micro-devices usually operate in the slip and transition rarefaction regimes
at the standard pressure and temperature [12]. It is widely acknowledged that
the standard Navier-Stokes-Fourier (NSF) equations fail to predict thermal and
fluid flow fields accurately when the nonequilibrium effects are significant
[15, 16, 17]. The Boltzmann equation governs fluid flow at micro-scales in the
entire range of Knudsen number [18]. However, direct numerical solution of the
Boltzmann equation is generally expensive with available computational
capabilities and are limited to simple applications because of the high
dimensionality of the Boltzmann equation and the complexity of the collision
integral. Moreover, the application of the Boltzmann equation is limited to
dilute gasses due to the assumption of binary intermolecular collisions [19].
The direct simulation Monte-Carlo (DSMC) method is a numerical technique that
has been developed based on the kinetic theory to approximate the solution of
the Boltzmann equation [18].

Zheng et al. [20] compared the numerical predictions obtained from DSMC
simulations with the solutions of the NSF equations for a Poiseuille gas flow
between two parallel plates, and reported quantitative and qualitative
differences between the predictions. Deviations between numerical predictions
obtained using the DSMC method and the NSF equations increases with increasing
the Knudsen number [20, 21, 22, 23]. Using the regularised 13-moment
equations, Torrilhon and Struchtrup [24] achieved a better agreement with the
DSMC results up to $\mathrm{Kn}\approx 1$ in comparison with the NSF-based
solutions. Varoutis et al. [25] studied Poiseuille gas flow in channels with
finite lengths using the DSMC method, and stated that the nonequilibrium
effects are the most significant factor affecting rarefied gas flow in
channels.

The majority of published literature on subsonic internal gas flows at micro-
scales are limited primarily to flow passages with uniform cross-sections [26,
27, 28]. Moreover, previous studies have mostly focused on characterising
pressure loss, mass flow rate, and velocity field [29]. There is a crucial
lack of detailed investigations concerning the thermal field analysis of gas
flow in micro- and Nano-channels with non-uniform cross-sections in which
nonequilibrium effects play an important role [30]. Using the NSF equations
with the first-order slip boundary condition, Varade et al. calculated
temperature variations in divergent [31] and convergent [32] microchannels for
both continuum and slip flow regimes. Using the same approach, Hemadri et al.
[33, 34] calculated temperature distribution in convergent microchannels in
the slip and early-transition rarefaction regimes. Milićev and Stevanović [35]
proposed an analytical solution to the one-dimensional NSF equations to
describe steady-state pressure-driven isothermal gas flow in microchannels
with variable cross-sections. Ohwada et al. [36] stated that heat and mass
flow are in opposite directions for Poiseuille flow between two parallel
plates for $\mathrm{Kn}>10^{-1}$ while they are in the same direction adjacent
to the walls at low Knudsen numbers. It should be noted that only the
tangential heat flow profiles were considered in their study while both normal
and tangential heat fluxes contribute to the net heat flow. John et al. [37]
numerically studied the influence of pressure ratio and surface accommodation
coefficient on the thermal field of a Poiseuille gas flow between two parallel
plates. The DSMC method has been recently employed to investigate temperature
variations in divergent microchannels [38, 39, 40]. Characterising the thermal
field in microchannels with non-uniform cross-sections is emerging for micro-
and Nano-system engineering and further investigations are essential.

Despite the extensive interest in studying fluid flow at the micro- and Nano-
scales, numerical investigations of gas flow under nonequilibrium conditions
are deficient in thermal analysis. Previous studies on thermal field analysis
in microfluidics systems are mostly limited to the channels with uniform
cross-sections (see for instance, [41, 42, 37, 43, 44, 45, 46, 47, 48]).
Therefore, there is an indispensable need for detailed investigations on heat
and fluid flow at the micro- and Nano-scales under nonequilibrium conditions.
The goal of the present study is to enhance our understanding of heat and
fluid flow in divergent microchannels, which are applicable in micro-devices
such as micro-pumps, micro-actuators and micro-thrusters. Numerical
simulations based on the DSMC method are performed to explore the effects of
divergence angle, inlet-to-outlet pressure ratio, rarefaction, and tangential
momentum accommodation coefficient on thermal and fluid flow fields. Heat and
fluid flow under nonequilibrium conditions are considerably different from its
equilibrium counterpart, providing the motivation to conduct the present
study.

##  2 Problem description

A probabilistic numerical approach based on the direct simulation Monte-Carlo
method is employed to study steady-state, pressure-driven, Poiseuille gaseous
nitrogen flow in divergent microchannels shown schematically in Figure 1. The
divergence angle of the channel ($\phi$) is defined as follows [39]:

$\phi=\frac{100\cdot\left(H_{\mathrm{o}}-H_{\mathrm{i}}\right)}{L},$ (1)

where, $H_{\mathrm{i}}$ and $H_{\mathrm{o}}$ are the height of the channel at
the inlet and the outlet, respectively. The length of the channel ($L$) is
twenty times larger than its inlet height (i.e. $L=20\times
H_{\mathrm{i}}=$8\text{\times}{10}^{-6}\text{\,}\mathrm{m}$$). The problem is
assumed to be two-dimensional, which is a valid assumption for channels with
considerably large depths [43, 49]. The molecular properties of gaseous
nitrogen are summarised in Table 1, where $d_{\mathrm{p}}$ is molecular
diameter, $m_{\mathrm{p}}$ molecular mass, $\omega$ the viscosity-temperature
index, $T_{\mathrm{ref}}$ the reference temperature in the viscosity-
temperature relation, and $\mu_{\mathrm{0}}$ the viscosity at the reference
temperature.

Figure 1: Schematic of the divergent microchannel considered in the present
study. The hatched region was considered for calculations.

Due to the symmetric configuration of the channel and the flow field, one-half
of the physical domain (the hatched region in Figure 1) is considered for
calculations. Heat and fluid flow in the channel are described in a Cartesian
coordinate system. The temperature of the solid walls ($T_{\mathrm{w}}$) and
the gas at the inlet ($T_{\mathrm{i}}$) is constant equal to
$300\text{\,}\mathrm{K}$. The Knudsen number is defined based on the inlet
height, i.e. $\mathrm{Kn}=\lambda/H_{\mathrm{i}}$. The inlet-to-outlet
pressure ratio ($\Pi$) is defined as $p_{\mathrm{i}}/p_{\mathrm{o}}$ and the
gas pressure at the channel inlet ($p_{\mathrm{i}}$) is calculated based on
the Knudsen number and the ideal gas law. The effects of tangential momentum
accommodation coefficient ($\alpha$) on thermal and fluid flow fields are also
studied in the present work. The tangential momentum accommodation coefficient
ranges between zero (specular reflection) and one (fully-diffused reflections)
depending on internal gas molecule structure, the molecular mass of the gas,
and surface material [50, 51].

Table 1: Molecular properties of gaseous nitrogen. $d_{\mathrm{p}}$ [$\text{\,}\mathrm{m}$] |  $m_{\mathrm{p}}$ [$\text{\,}\mathrm{kg}$] |  $DOF_{\mathrm{rot}}$ [$\text{\,}\mathrm{-}$] |  $\omega$ [$\text{\,}\mathrm{-}$] |  $T_{\mathrm{ref}}$ [$\text{\,}\mathrm{K}$] |  $\mu_{\mathrm{0}}$ [$\text{\,}\mathrm{N}\text{\,}\mathrm{s}\text{\,}{\mathrm{m}}^{-2}$]
---|---|---|---|---|---
$4.17\text{\times}{10}^{-10}\text{\,}$ | $4.65\text{\times}{10}^{-26}\text{\,}$ | $2$ | $7.4\text{\times}{10}^{-1}\text{\,}$ | $300$ | $1.656\text{\times}{10}^{-5}\text{\,}$

##  3 Numerical procedure

An extended version of the dsmcFoam solver [52], which has been developed
within the framework of an open-source solver (OpenFOAM [53]), was employed in
the present study. This solver has been meticulously validated for a variety
of benchmark cases [54, 55, 52] including subsonic Poiseuille micro-flows. The
variable hard sphere (VHS) model was selected to treat the binary inter-
molecular collisions. The energy exchange between both translational and
rotational modes was allowed. The no-time-counter (NTC) [18] collision
sampling model was chosen. Additionally, the transient adaptive sub-celling
(TAS) scheme [56] was utilised to divide collision cells every time-step
according to the number of particles inside each collision cell to have at
least two simulator particles in each subcell. This results in the selection
of collision partners that are located within a relative distance smaller than
the local molecular mean free path.

Oran et al. [57] suggested a cell size of $\lambda/3$ to capture the gradients
in flow fields. In the case of low-speed micro-flows, larger grids might be
employed in the stream-wise direction due to insignificant flow gradients [58,
59, 60, 61]. After a grid independence test, a grid with cell sizes of
$\lambda/4$ and $\lambda/2$ in the normal and the stream-wise directions,
respectively, was selected. To capture the flow gradients for the cases with
$\mathrm{Kn}>$2.5\text{\times}{10}^{-1}\text{\,}$$, the grid size was the same
as that for the cases with $\mathrm{Kn}=$2.5\text{\times}{10}^{-1}\text{\,}$$.
All simulations were initialised with $50$ particles per cell to minimise
statistical relations between particles [62] and to have at least $15$
particles in the computational cells adjacent to the channel’s outlet. The
time-step was selected sufficiently smaller than the local mean collision time
so particles stay inside a cell for multiple time-steps. The results of the
grid, time-step size and particle-per-cell independence tests are presented in
Figure 3. To return the relevant macroscopic data, the results were sampled
over $3\text{\times}{10}^{6}\text{\,}$ time-steps after achieving the steady-
state condition. The sample size was sufficiently large to let the solution
procedure continues even after equating the inlet and outlet mass flow rates;
leading to a converged solution with negligible (less than $1\%$) statistical
noises [63, 64, 65]. Each simulation was carried out in parallel on four cores
of an Intel Core i7-3520M processor, and took roughly $200\text{\,}\mathrm{h}$
to complete.

##  4 Model validation and verification

The Poiseuille argon flow in a microchannel with a uniform cross-section was
considered to verify the reliability of the present model. The microchannel
length is $15\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and its height is
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Gaseous argon enters the channel
with a constant temperature of $300\text{\,}\mathrm{K}$, and the inlet-to-
outlet pressure ratio $\Pi$ was set to 3. The channel walls were assumed to be
fully diffuse (i.e. $\alpha=1$), and their temperature was set to
$300\text{\,}\mathrm{K}$. Four different cases with different Knudsen numbers
are studied, which covers the slip and transition rarefaction regimes. For
each case about $1.5\text{\times}{10}^{6}\text{\,}$ DSMC simulator particles
were utilised. The predicted pressure and Mach number (Ma) distributions along
the microchannel centreline are shown in Figure 2. The Mach number is defined
as follows:

$\mathrm{Ma}=\frac{\lVert\vec{V}\rVert}{\sqrt{\gamma RT}},$ (2)

where, $\vec{V}$ is the fluid velocity vector, $\gamma$ the ratio of specific
heat of nitrogen at a constant pressure to its specific heat at a constant
volume, $R$ the specific gas constant, and $T$ the temperature. The results
are in reasonable agreement with the data reported by White et al. [65].
Further comparison of the results obtained from the present model with
experimental, analytical and numerical data was performed by the authors for
gaseous nitrogen flow in divergent microchannels, and can be found in [38, 39,
31].

Figure 3 shows the influence of grid size, number of simulator particles per
cell (PPC), and time-step size on the results obtained from the present DSMC
simulations. For this study, nitrogen flow in a microchannel with uniform
cross-section ($\phi=1$), $\mathrm{Kn}_{\mathrm{i}}=0.1$ (transition
rarefaction regime), and the inlet-to-outlet pressure ratio $\Pi$ of 2.5 was
considered. To determine an appropriate time-step size $\Delta t$ within which
the molecular movement and collision are distinguishable, the following
correlations were employed:

$\Delta
t_{\mathrm{c}}=\frac{0.2\lambda}{\sqrt{\frac{2\kappa_{\mathrm{b}}T}{m_{\mathrm{p}}}}},$
(3)

$\Delta t_{\mathrm{t}}=\frac{0.5\Delta
x}{\sqrt{\frac{2\kappa_{\mathrm{b}}T}{m_{\mathrm{p}}}}},$ (4)

where, $\Delta t_{\mathrm{c}}$ is the mean collision time, $\Delta
t_{\mathrm{t}}$ the mean transit time and $\kappa_{\mathrm{b}}$ the Boltzmann
constant. The time-step size $\Delta t$ was defined as a fraction of the
minimum value of $\Delta t_{\mathrm{c}}$ and $\Delta t_{\mathrm{t}}$ (i.e.
$\Delta t$ = $\xi\cdot\min\left(\Delta t_{\mathrm{c}},\Delta
t_{\mathrm{t}}\right)$ and $\xi\leq 1$). For the range of parameters studied
in the present work, the results are practically insensitive to the grid size,
number of simulator particles per cell, and time-step size.

Figure 2: Comparison of the numerical results obtained from the present model
(solid lines) with the data reported by White et al. [65] (symbols) for
pressure-driven rarefied argon flow in a microchannel with uniform cross-
section. Profiles of (a) pressure, and (b) Mach number along the microchannel
centreline.  Figure 3: The effects of (a and b) the computational grid size,
(c and d) number of simulator particles per cell, and (e and f) time-step size
on the predicted profiles of temperature and Mach number along the channel
centreline. $\mathrm{Kn}_{\mathrm{i}}=0.1$, $\phi=1$ and $\Pi=2.5$. Curves
shown with symbols indicate the chosen values.

##  5 Results and Discussion

The influence of the microchannel divergence angle on variations of
temperature and Mach number along the microchannel centreline is shown in
Figure 4. For $\phi<2.5$, the gas temperature decreases along the channel
because of the augmentation of the gas velocity towards the channel outlet.
The increase in kinetic energy of the gas results in a reduction of its
internal energy and thus its temperature. The gas velocity remains almost
constant along the channel with $\phi=2.5$ and the gas temperature hardly
changes along the channel, except in regions close to the inlet and outlet
boundaries. Further increase in the divergence angle $\phi$ results in a
decrease of gas velocity along the channel, which is attributed to the
increased cross-sectional area of the channel as well as the increased shear
stress at the channel walls [39, 31]. The kinetic energy of the gas converts
into internal energy as the fluid velocity decreases towards the channel
outlet for the cases with $\phi>2.5$, leading to an increase in gas
temperature. Additionally, the bulk gas temperature reduces with an increase
in divergence angle of the channel because of the increased bulk gas velocity
and the reduced rate of molecular collision. Because of the development of a
viscous boundary layer close to the channel inlet [39, 65], the gas flow
accelerates in the entrance region; however, the flow decelerates after
passing this region moving towards the channel outlet to conform to the
pressure prescribed at the outlet. Close to the channel outlet ($x/L>0.9$),
the gas temperature drops that is attributed to the rapid gas expansion in
that region, which is known as ‘the expansion cooling phenomenon’ [66, 20]. A
similar behaviour close to the channel outlet has been reported by others [28,
37, 45, 43, 44] for Poiseuille micro-flows in microchannels with uniform
cross-sections.

Figure 4: Distribution of (a) gas temperature and (b) Mach number along the
microchannel centreline for different divergence angles $\phi$
($\mathrm{Kn}_{\mathrm{i}}=0.05$ and $\Pi=2.5$). The inlet temperature
($T_{\mathrm{i}}$) is used to non-dimensionalise the predicted temperatures.

The overall heat flow direction, for the problem considered in the present
work, is determined by both the thermal and pressure gradients in the
microchannel [67]. Heat flow direction and contours of dimensionless
temperature ($T/T_{\mathrm{i}}$) obtained from the DSMC simulations are shown
in Figure 5 for microchannels with different divergence angles at
$\mathrm{Kn}_{\mathrm{i}}=0.05$ and $\Pi=2.5$. Heat flows generally towards
the channel inlet except in regions adjacent to the walls in the Knudsen
layer, where the heat flow is dominated by viscous heating. For the cases with
$\phi>2.5$, the amount of heat induced by viscous dissipation decreases along
the microchannel due to the reduction of wall shear stress and slip velocity
[39], decreasing the influence of viscous dissipation on net heat flow
direction in regions close the walls. In contrast, the amount of heat
generated by viscous dissipation increases as the gas moves along the channels
with $\phi\leq 2.5$, increasing its influence on net heat flow direction in
regions adjacent to the walls. Accordingly, it can be argued that the heat
flow is dominated by the pressure gradient (known as a rarefaction effect
[67]) in microchannels with a divergence angle $\phi\leq 2.5$. Due to the
rarefaction effects, heat flow from cold to hot regions is observed, which is
not predictable using the NSF equations because of the neglect of high-order
rarefaction terms [68]. Heat flow in the central region of the channel is
dominated by the mass flow driven by the pressure gradient. However, the
magnitude of thermal gradient increases with an increase in divergence angle
$\phi$, affecting the net heat flow direction. For the cases with $\phi>2.5$,
a thermal separation occurs at certain location in the channel beyond which
heat flows towards the channel outlet. Thermal separation for pressure-driven
gas flow in microchannels of variable cross-sections is attributed to the
enhanced contribution of thermal gradient to the total heat transfer [69]. The
position of thermal separation point moves towards the inlet with an increase
in the divergence angle of the channel. It should be noted that no
hydrodynamic separation was observed for the cases considered in the present
study [39, 38]. These observations can also be explained through the
asymptotic analysis of the Boltzmann equation [67] for small Knudsen numbers
(i.e. slip and early-transition regimes). However, this type of analysis,
which is based on the continuum theory, fails to predict heat transfer
accurately at high Knudsen numbers [38].

Figure 5: Contours of temperature overlaid with heatlines (coloured by the
magnitude of heat flux) obtained from the simulations for microchannels with
different divergence angles $\phi$. ($\mathrm{Kn}_{\mathrm{i}}=0.05$ and
$\Pi=2.5$)

Numerical predictions of the gas temperature and Mach number along the channel
centreline are presented in Figure 6 for different inlet Knudsen numbers
$\mathrm{Kn}_{\mathrm{i}}$ at $\phi=7$ and $\Pi=2.5$. The bulk gas velocity
along the channel and variations in the gas velocity decreases with an
increase in the Knudsen number. However, this reduction in the velocity
magnitude and its variations becomes insignificant for
$\mathrm{Kn}_{\mathrm{i}}\geq 1$. Variation of gas velocity along the channel
and the molecular collision frequency decreases with an increase in Knudsen
number, limiting the variation of gas temperature along the channel. The
magnitude of temperature drop close to the channel outlet also decreases with
an increase in the inlet Knudsen number. Moreover, the predicted gas
temperatures close to the channel inlet are slightly higher than the inlet gas
temperature $T_{\mathrm{i}}$, which is because of the augmentation of
temperature jump with increasing the Knudsen number [37].

Figure 6: The effect of rarefaction on the profiles of (a) gas temperature and
(b) Mach number along the microchannel centreline ($\phi=7.00$ and $\Pi=2.5$).
The inlet temperature ($T_{\mathrm{i}}$) is used to non-dimensionalise the
predicted temperatures.

The influence of the Knudsen number on the thermal field and the heat flow
direction in a divergent channel with $\phi=7$ and $\Pi=2.5$ is shown in
Figure 7. The temperature gradient in the channel decreases with an increase
in Knudsen number, decreasing the contribution of the Fourier heat transfer in
net heat flow. According to Figure 7 (c) and (d), at sufficiently large
$\mathrm{Kn}_{\mathrm{i}}$ the heat flow in the channel is merely towards the
channel inlet that demonstrates the domination of the pressure gradient in the
net heat flow direction, intensifying the counter-gradient cold-to-hot heat
transfer (also known as anti-Fourier heat transfer). Moreover, the total heat
flow rate decreases with increasing the Knudsen number. The distance between
the thermal separation point and the channel outlet increases with a decrease
in $\mathrm{Kn}_{\mathrm{i}}$.

Figure 7: Heatlines (coloured by the magnitude of heat flux) overlaid on
contours of dimensionless temperature ($T/T_{\mathrm{i}}$) predicted for
different inlet Knudsen numbers $\mathrm{Kn}_{\mathrm{i}}$. $\phi=7.00$ and
$\Pi=2.5$.

The bulk gas velocity and temperature in the channel decrease with a decrease
in the inlet-to-outlet pressure ratio $\Pi$, as shown in Figure 8. Increasing
the inlet-to-outlet pressure ratio increases the gas kinetic energy and
decreases its internal energy. The thermal field as well as the heat lines are
shown in Figure 9 for channels with a divergence angle of $\phi=7$ and Knudsen
number of 0.05. The magnitude of thermal gradient close to the channel outlet
increases with decreasing the pressure ratio $\Pi$, which enhances the
contribution of the thermal gradient to the net heat flow; this affects the
net heat flow direction and leads to the occurrence of thermal separation
farther away from the channel outlet. The heat generated in the Knudsen layer
due to viscous dissipation diminishes with reducing the inlet-to-outlet
pressure ratio $\Pi$, decreasing the contribution of thermal gradients to the
net heat flow close to the channel walls.

Figure 8: The effect of the inlet-to-outlet pressure ratio $\Pi$ on the
profiles of (a) gas temperature and (b) Mach number along the microchannel
centreline ($\phi=7.00$ and $\mathrm{Kn}=0.05$). The inlet temperature
($T_{\mathrm{i}}$) is used to non-dimensionalise the predicted temperatures.
Figure 9: The influence of the inlet-to-outlet pressure ratio $\Pi$ on thermal
field (contours) and heat flow direction (coloured by the magnitude of heat
flux). $\mathrm{Kn}_{\mathrm{i}}=0.05$ and $\phi=7$.

The influence of the tangential accommodation coefficient ($\alpha$) on the
profiles of temperature and velocity along divergent channels is shown in
Figure 10 for $\Pi=2.5$ and $\mathrm{Kn}_{\mathrm{i}}=0.05$. For all the cases
considered in the present work, decreasing the tangential accommodation
coefficient weakens the interactions between the gas molecules and the channel
walls, leading to an increase in the gas velocity. A similar behaviour has
also been observed both experimentally and numerically for rarefied gas flows
in microchannels with uniform cross-sections [37, 51, 50, 70]. As expected,
the augmentation of gas velocity in the channel results in an increase in the
kinetic energy and a decrease in internal energy and thus the gas temperature.
The results presented in Figure 10 show that the gas reaches supersonic
velocities when $\alpha=0$ (i.e. specular reflection); this happens for the
case with $\phi=7.0$ only when $\alpha=0.5$ and does not happen when
$\alpha=1$ (i.e. fully-diffused reflection).

Figure 10: The effect of the tangential accommodation coefficient $\alpha$ on
the profiles of (a) gas temperature and (b) Mach number along the microchannel
centreline ($\Pi=2.5$ and $\mathrm{Kn}_{\mathrm{i}}=0.05$). The inlet
temperature ($T_{\mathrm{i}}$) is used to non-dimensionalise the predicted
temperatures. Figure 11: The influence of the tangential accommodation
coefficient $\alpha$ on thermal field (contours) and heat flow direction
(coloured by the magnitude of heat flux). $\Pi=2.5$ and $\phi=7$.

The results presented in Figure 10 show that heat and fluid flow patterns in
the channel are sensitive to the value of the tangential accommodation
coefficient $\alpha$, particularly for low values of the Knudsen numbers. The
number of gas molecules contributing to the adiabatic exchange of energy
increases with decreasing the value of the tangential accommodation
coefficient $\alpha$, increasing the contribution of thermal gradients to the
net heat flow. Reducing the value of the tangential accommodation coefficient
to $\alpha=0$ as an extreme condition, it is observed that the heat flow
converges to a sink point, where pressure gradients balance thermal gradients.
Heat lines are directed towards the channel outlet in regions where thermal
gradients dominate the net heat flow.

##  6 Conclusions

Two-dimensional numerical simulations based on the direct simulation Monte-
Carlo (DSMC) method were performed to predict thermal and flow fields of
pressure-driven nitrogen flow in divergent microchannels. A wide range of
Knudsen numbers was studied covering the slip to the free molecular
rarefaction regimes. Moreover, the influence of microchannel divergence angle,
inlet-to-outlet pressure ratio and incomplete surface accommodation were
investigated. The following conclusions are drawn based on the present study.

the influence of thermal gradient on net heat flow direction enhances with an
increase in the divergence angle of the microchannel and can lead to the
occurrence of thermal separation even when hydrodynamic separation is absent.
The contribution of the pressure gradient in determining the heat flow
direction enhances with an increase in the Knudsen number. At sufficiently
large Knudsen numbers, the heat flow is merely towards the channel inlet.
Cold-to-hot heat transfer (also known as anti-Fourier heat transfer) was
observed in the microchannel as a result of nonequilibrium effects. Viscous
dissipation in the Knudsen layer diminishes with a decrease in applied
pressure ratio, which results in the domination of the pressure gradient
influence on heat flow close to the channel walls.

Further studies may focus on realising the effects of the molecular weight and
structure, or different gas mixtures on thermal and fluid flow fields inside
micro- and Nano-channels with with complex geometries.

## Author Contributions

Conceptualization, A.E. and E.R.; methodology, A.E.; software, A.E. and V.S.;
validation, A.E.; formal analysis, A.E.; investigation, A.E.; resources, A.E.,
V.S. and E.R; data curation, A.E. and V.S.; writing—original draft
preparation, A.E.; writing—review and editing, A.E., V.S. and E.R.;
visualization, A.E. and V.S.; supervision, A.E. and E.R.; project
administration, A.E. All authors have read and agreed to the published version
of the manuscript.

## Conflict of interest

The authors declare no conflict of interest.

## Data availability

The data presented in this study are available on request from the
corresponding author.

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