Thermal Viscous Dissipative Couette-Poiseuille Flow in a Porous Medium Saturated Channel

Abstract

A Couette-Poiseuille flow between parallel plates saturated with porous medium is studied with emphasis on viscous dissipation effect on the temperature field; assuming a fully developed flow, with both plates subjected to unequal and uniform heat flux. Temperature field and Nusselt number are derived as a function of Brinkman number and porous medium shape factor. By specifying the ratio of wall to mean velocity as one, the resulting velocity and temperature fields attribute to a significant increase in Nusselt number for the moving wall as the permeability of porous medium increases. Increased permeability signifies competing effect between enhanced convection in the proximity of the moving wall and higher local viscous dissipation. When the former effect dominates, heat transfer coefficient increases. Effects of Reynolds number on the temperature field is elucidated, including a comparison between a microchannel and conventional duct to evaluate the characteristic length scale effect. As Reynolds number goes up in a microchannel, heat generation in the form of viscous dissipation intensifies and overrides the convection effect, causing an increase in the highest temperature along the duct on the contrary to the findings in conventional duct.

1. Introduction

The fluid flow and heat transfer in a Couette-Poiseuille flow in a channel play a vital role in a wide variety of materials processing applications which includes sheet metal forming, extrusion, wire and fiber glass drawing, continuous casting, and more, whereby in all of these applications, heat is continuously transferred to fluid from moving plane [1]. Considerably less amount of research has been directed at Couette-Poiseuille flow as compared to studies on Couette flow or Poiseuille flow. Laminar heat transfer problems in Poiseuille flow for Newtonian fluids in parallel plates have been solved [2] while Aydin and Avci [3] subsequently studied Poiseuille flow in laminar heat convection for two different thermal boundary conditions—constant heat flux and constant wall temperature respectively—accounting for the effect of viscous dissipation and highlighted the importance of viscous dissipation. Lin [4] numerically investigated the effects of viscous dissipation and pressure gradient on non-Newtonian Couette flow. On the other hand, a number of investigations have also been performed on Couette-Poiseuille flow in clear fluid between parallel plates. Aydin and Avci [5] looked into the heat transfer in a Couette-Poiseuille flow for both hydrodynamically and thermally fully developed flow between two parallel plates. Sheela-Francisca et al. [6] examined heat transfer in Couette-Poiseuille flow under asymmetric wall heat fluxes with viscous dissipation effect and obtained the closed form temperature field and Nusselt number expression. Chan et al. [7] explored the effect of viscous dissipation on the thermal aspect of a power-law fluid for a Couette-Poiseuille flow subjected to asymmetric thermal boundary conditions. Hashemabadi et al. [8] solved forced convective heat transfer problem of non-linear viscoelastic fluid flow between parallel plates for a Couette-Poiseuille flow analytically and remarked the significant effects of Brinkman number on heat convection coefficient. Davaa et al. [1] solved heat convection problem for a non-Newtonian Couette-Poiseuille flow numerically, stressing on the significance of viscous dissipation effect on temperature distribution and Nusselt number.

Unlike clear fluid, there is remarkably less research endeavours on Couette-Poiseuille flow in porous medium. Aydin and Avci [9] investigated the effect of viscous dissipation on the heat transfer rate for Couette-Poiseuille flow in a saturated porous medium between two plane parallel plates. The study concurred on the significance of viscous dissipation on Couette-Poiseuille flow but did not further any explorations on the thermal boundary condition implementation at the fixed boundary nor the length scale effect of the parallel plate channel on dimensional temperature field.

Hence, this study would like to fill in the gap by looking into the thermal viscous dissipative effects on a Couette-Poiseuille flow in a saturated porous medium, subjected to unequal and uniform heat flux applied at both plates, assuming a steady, laminar and fully developed flow with local thermal equilibrium inside the porous medium. The study would also compare the significance of viscous dissipation to the temperature field in microchannel and conventional size channel for different $Re$.

2. Problem Formulation and Analytical Solution

Figure 1 is a schematic diagram of the problem where there is a steady, laminar, hydrodynamically, and thermally fully-developed flow through porous medium between two plates separated by a gap of height $H$. The lower plate moves at a constant velocity $u_w,$ while the upper plate is stationary. Uniform heat flux is applied to both plates whereby $q″{}_1$ and $q″{}_2$ are applied to the moving plate and stationary plate, respectively. In solving the governing thermal energy equation, temperature field is first derived subjected to a uniform heat flux $q^{″}$ at a moving wall. By defining a heat flux ratio $R=\frac{{q^{″}}_2}{{q^{″}}_1+{q^{″}}_2}$ and rewriting $q^{″}$ as ${q^{″}}_1+{q^{″}}_2$, the temperature field for the prior solved single heated wall only solution can then be transformed to temperature field having both boundaries subjected to uniform heat flux. The details is provided in Section 2.1.

2.1. Governing Equation

This study adopts the Brinkman-extended Darcy equation solved by [9] as the governing momentum equation. By defining the following dimensionless variables,

$$Y= \frac{y}{H} ,U=\frac{{\mu }_{eff}u}{\gamma H^2},U_w=\frac{{\mu }_{eff}{u}_w}{\gamma H^2},M=\frac{{\mu }_{eff}}{\mu }, Da=\frac{K}{H^2}, S=\frac{1}{\sqrt{MDa}}$$

(1)

The Brinkman-extended Darcy Equation is non-dimensionalized as follows,

$$\frac{{\mathrm{d}}^{2}U}{\mathrm{d}{Y}^2}-S^{2}U+1=0$$

(2)

and is subjected to the dimensionless form of the boundary conditions, as follows:

$$Y=0, U=U_w$$

(3)

$$Y=1, U=0$$

(4)

Solving Equation (2) alongside boundary conditions (3) and (4) yields the dimensionless velocity, $u^{*}$ defined and expressed as [9]

$$u^{*}=\frac{U}{U_m}=\frac{S\left(1-\cos \mathrm{h}\left(SY\right)+S^2\cos \mathrm{ech}\left(S\right)\sin \mathrm{h}\left(S-SY\right)U_w+\sin \mathrm{h}\left(SY\right)\tan \mathrm{h}\left(S/2\right)\right)}{S+\left(-2+S^2{U}_w\right)\tan \mathrm{h}\left(S/2\right)}$$

(5)

The governing thermal energy equation is,

$${\rho }_f{c}_{p,f}u\frac{\partial T}{\partial x}=k_{eff}\frac{{\partial }^{2}T}{\partial y^2}+\frac{\mu u^2}{K}+{\mu }_{eff}{\left(\frac{\mathrm{d}u}{\mathrm{d}y}\right)}^2$$

(6)

In Equation (6), Al-Hadhrami’s model [10] is used to represent the viscous dissipation term, where the second and third term on the right hand side of Equation (6) are termed as internal heating (I.H.) and frictional heating (F.H.) respectively.

The problem is first solved for heat flux applied to moving wall only, where the thermal boundary conditions are

$$y=0,T=T_w ,$$

(7)

$$y=H, \frac{\partial T}{\partial y}=0 .$$

(8)

In the axial direction, the temperature gradient is constant under thermally fully-developed condition, hence

$$\frac{\partial T}{\partial x}=\frac{\partial T_m}{\partial x}.$$

(9)

By incorporating $q^{″}=-k_{eff}\frac{\partial T_A}{\partial y} at y=0$ and integrating Equation (6) throughout the height of the channel, $\frac{\partial T_m}{\partial x}$ is obtained as

$$\frac{\partial T_m}{\partial x}= \frac{1}{{\rho }_f{c}_{p,f}H{u}_m} \left[q^{″}+\frac{\mu }{K}\underset{0}{\overset{H}{{\displaystyle \int }}}{u}^{2}dy+ {\mu }_{eff}\underset{0}{\overset{H}{{\displaystyle \int }}}{\left(\frac{\mathrm{d}u}{\mathrm{d}y}\right)}^{2}dy\right] .$$

(10)

Introducing the following dimensionless variables,

$$\theta =\frac{k_{eff}\left(T-T_w\right)}{q″H}, Br=\frac{\mu u_m{}^{2}H}{q″K}, BrMDa=\frac{Br}{S^2}=B{r}^{\prime }=\frac{{\mu }_{eff}{u}_m{}^2}{q″H},$$

(11)

and incorporating Equation (10) in to Equation (6), the dimensionless governing thermal energy equation is obtained as

$$\frac{{\partial }^2\theta }{\partial Y^2}=u^{*}A-Br{u}^{*}{}^2-\frac{Br}{S^2}{\left(\frac{\mathrm{d}{u}^{*}}{\mathrm{d}Y}\right)}^2 ,$$

(12)

where

$$A=\left[1+Br\underset{0}{\overset{1}{{\displaystyle \int }}}{u}^{*}{}^{2}dY+\frac{Br}{S^2}\underset{0}{\overset{1}{{\displaystyle \int }}}{\left(\frac{\mathrm{d}{u}^{*}}{\mathrm{d}Y}\right)}^{2}dY\right] .$$

(13)

subjected to

$$Y=0, \theta =0 ,$$

(14)

$$Y=1, \frac{\partial \theta }{\partial Y}=0 .$$

(15)

Solving Equation (12) alongside the boundary conditions gives rise to the exact solution,

$$\theta \left(Y\right)={\mathrm{C}}_{1}Y+{\mathrm{C}}_2{Y}^2+{\mathrm{C}}_3\cosh \left(SY\right)+{\mathrm{C}}_4\cosh \left(2SY\right)+{\mathrm{C}}_5\sinh \left(SY\right)+{\mathrm{C}}_6\sinh \left(2SY\right)+{\mathrm{C}}_7$$

(16)

where ${\mathrm{C}}_1-{\mathrm{C}}_7$ are the coefficients defined in the Appendix A.

In our attempt to obtain the temperature field subjected to uniform heat flux at both plates, $T_B,$ the temperature field obtained in Equation (16) may be made use by equating

$$T_B=T_A+\frac{q″{}_2}{k_{eff}}y$$

(17)

where $T_A$ is the temperature solved subjected to uniform heat flux at moving wall only while $T_B$ denotes the temperature field subjected to uniform heat flux at both plates. Notably, $T_B$ fulfills the governing thermal energy equation, Equation (6) duly.

From Equation (17),

$$\frac{\partial T_B}{\partial y}=\frac{\partial T_A}{\partial y}+\frac{q″{}_2}{k_{eff}} .$$

(18)

By rewriting the boundary conditions for $T_A$ in Equation (7) as

$$y=0, -k_{eff}\frac{\partial T_A}{ \partial y}={q^{″}}_1+{q^{″}}_2,$$

(19)

$$y=H, -k_{eff}\frac{\partial T_A}{ \partial y}=0 .$$

(20)

The boundary conditions for $T_B$ may be written as

$$y=0, -k_{eff}\frac{\partial T_B}{ \partial y}={q^{″}}_1,$$

(21)

$$y=H, k_{eff}\frac{\partial T_B}{ \partial y}={q^{″}}_2.$$

(22)

Recasting Equation (17) in dimensionless temperature gives

$${\theta }_1=\theta +RY ,$$

(23)

where $\theta =\theta \left(Y\right)$=$\frac{k_{eff}\left(T_A-T_w\right)}{H\left({q^{″}}_1+{q^{″}}_2\right)}$ as solved in Equation (16).

The ratio of heat fluxes, R, is defined as,

$$R=\frac{q_2″}{{q^{″}}_1+{q^{″}}_2},$$

$${\theta }_1=\frac{k_{eff}\left(T_B-T_w\right)}{H\left({q^{″}}_1+{q^{″}}_2\right)}$$

Hence, ${\theta }_1$ can be written as

$${\theta }_1=RY+{\mathrm{C}}_{1}Y+{\mathrm{C}}_2{Y}^2+{\mathrm{C}}_3\cosh \left(SY\right)+{\mathrm{C}}_4\cosh \left(2SY\right)+{\mathrm{C}}_5\sinh \left(SY\right)+{\mathrm{C}}_6\sinh \left(2SY\right)+{\mathrm{C}}_7 .$$

(24)

Defining ${\theta }_2$ as

$${\theta }_2=\frac{k_{eff}\left(T_B-T_w\right)}{H{q^{″}}_1} ,$$

(25)

relates

$${\theta }_2={\theta }_1\times \frac{1}{1-R} ,$$

(26)

and solves the temperature field subjected to uniform heat fluxes at both boundaries as

$${\theta }_2=\frac{1}{1-R}\left[RY+{\mathrm{C}}_{1}Y+{\mathrm{C}}_2{Y}^2+{\mathrm{C}}_3\cosh \left(SY\right)+{\mathrm{C}}_4\cosh \left(2SY\right)+{\mathrm{C}}_5\sinh \left(SY\right)+{\mathrm{C}}_6\sinh \left(2SY\right)+{\mathrm{C}}_7\right] .$$

(27)

2.2. Nusselt Number

Nusselt number defined based on the moving wall temperature is,

$$Nu=\frac{{q^{″}}_{1}H}{k_{eff}\left(T_w-T_m\right)} .$$

(28)

Computing the bulk mean temperature, $Nu$ can be expressed as:

$$\begin{array}{cc}Nu\hfill & =\{1/6\left(R-1\right)S^2[S+\left(S^2{U}_w-2\right)\tanh \left(S/2\right)]\}\{-S\{3S\left[-{\mathrm{C}}_5+{\mathrm{C}}_6+\left({\mathrm{C}}_1-{\mathrm{C}}_3+2{\mathrm{C}}_7+R\right)S\right]\hfill \\ & +2{\mathrm{C}}_2\left(6+S^2\right)+\left(3{\mathrm{C}}_5-4{\mathrm{C}}_6\right)S\cos \mathrm{h}\left(S\right)+{\mathrm{C}}_{6}S\cos \mathrm{h}\left(2S\right)+S\left[3{\mathrm{C}}_3-4{\mathrm{C}}_4+2{\mathrm{C}}_4\cos \mathrm{h}\left(S\right)\right]\sin \mathrm{h}\left(S\right)\}\hfill \\ & +S^3\{-6\left({\mathrm{C}}_1+R\right)+\left(3{\mathrm{C}}_5+4{\mathrm{C}}_6\right)S-3{\mathrm{C}}_3{S}^2-4{\mathrm{C}}_{6}S\cos \mathrm{h}\left(S\right)+S\left(2{\mathrm{C}}_4-6{\mathrm{C}}_7-3{\mathrm{C}}_{5}S\right)\cot \mathrm{h}\left(S\right)\hfill \\ & +2S\left[3\left({\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_7+R\right)-{\mathrm{C}}_4\cos \mathrm{h}\left(2S\right)\right]\cos \mathrm{ech}\left(S\right)\}{U}_w+[S^2\left(6{\mathrm{C}}_1-4{\mathrm{C}}_4+12{\mathrm{C}}_7+6R+3{\mathrm{C}}_{5}S\right)\hfill \\ & +6{\mathrm{C}}_2\left(4+S^2\right)-12{\mathrm{C}}_2{S}^2{U}_w]\tanh \left(S/2\right)\}\hfill \end{array}$$

(29)

2.3. Nusselt Number Verification

Reducing and comparing the solutions to cases for Poiseuille flow and Couette-Poiseuille flow in clear fluid, as presented in

Table 1. Comparison of Nusselt number with the literature.

$\mathit{B}\mathit{r}$	$\mathit{R}$	$\mathit{S}$	${\mathit{U}}_{\mathit{w}}/{\mathit{U}}_{\mathit{m}}$	Nu, Present Study	Nu, Chen et al. [7]	Nu, Aydin et al. [9]	Nu, Davaa et al. [11]	Nu*, Tso et al. [12]	Nu, Tan and Chen [13]
0	0	$1/\sqrt{10}$	0	5.385	5.385	5.385	5.385	5.385	---
0	0	$1/\sqrt{10}$	1	7.238	---	7.241	---	---	---
0.2	0	$1/\sqrt{10}$	0	3.805	3.804	3.804	3.804	---	---
0.2	0	$1/\sqrt{10}$	1	9.992	10	---	10	---	---
0	0.5	$1/\sqrt{10}$	0	8.237	8.235	---	---	8.235	8.235
0.5	0.5	$1/\sqrt{10}$	0	3.183	3.182	---	---	---	3.182

, shows, that $Nu$ in this study is in excellent agreement with the literature, as presented in Table 1.

3. Results and Discussion

3.1. Velocity Profile

In order to facilitate the discussion of Couette-Poiseuille flow in a saturated porous medium, the velocity field in the channel for different porous medium shape factors $S$ is depicted in Figure 2 for $U_w/U_m=1$, that results in profile with a maximum velocity. Velocity gradient in the vicinity of a moving wall is higher for low $S$ porous medium while larger $S$ porous medium gives rise to steeper gradient in the vicinity of the fixed wall. Flow velocity is also higher for lower $S$ porous medium in the vicinity of the moving wall.

Figure 3 illustrates the viscous dissipation profile at $Br=0.1$, for $S=1$, and $S=10$, respectively, in order to elucidate the effects of local viscous dissipation on the temperature field in Section 3.2. It is noteworthy to point out that overall viscous dissipation corresponding to $S=1$ drops to a minimum and picks up tremendously towards the fixed wall, revealing a much lower viscous dissipation at the moving wall than the fixed wall. On the other hand, for $S=10$, viscous dissipation is dominated by internal heating, which varies in a similar pattern to the velocity profile. The average viscous dissipation is computed and tabulated in

Table 2. Effects of viscous dissipation, $Br=0.1$ (I.H = Internal Heating, F.H = Frictional Heating).

$\mathbf{Porous}\mathbf{Medium}\mathbf{Shape}\mathbf{Factor},\mathit{S}$	$\mathit{I}.\mathit{H}/{\mathit{q}}^{″}.\mathit{H}$	$\mathit{F}.\mathit{H}/{\mathit{q}}^{″}.\mathit{H}$
1	0.1129	0.4001
$\sqrt{10}$	0.1106	0.0412
10	0.1048	0.0064
$10\sqrt{10}$	0.1015	0.0016
100	0.1005	0.0005

at a fixed $Br$ for the velocity field in Figure 2. Table 2 indicates that smaller $S$ porous medium has a much higher overall viscous dissipation due to the more significant frictional heating towards fixed wall at a fixed $Br$, defined based on the relative magnitudes of internal heating to heat applied at wall.

3.2. Temperature Distribution

The transverse dimensionless temperature profile in the channel is depicted in Figure 4a,b for various shape factor $S$ and heat flux ratio. Both figures show increasing transverse dimensionless temperature with increasing $S$ in the vicinity of the heated wall. In the presence of viscous dissipation, Figure 4b indicates a more significant increase in temperature field for lower $S$ due to gradually more dominant convection effect over heat generation in the adjoining region of the moving wall. Hence, viscous dissipation is more significant to lower $S$ porous medium shape factor at a fixed $Br$ and specified $U_w/U_m=1$. Expectedly, at a fixed $Br$, temperatures increase in both figures as $R$ increases, for an increase in fluid temperature adjoining the fixed wall. In Figure 4b, the temperature at fixed wall rises to 0.3 when $R=0.5$, reflecting that increasing $R$ causes a higher surface temperature at the fixed wall than moving wall, due to a higher heat transfer coefficient at the moving wall than the fixed wall, as depicted by the velocity distribution in Figure 2, when heat flux is applied at both boundaries.

3.3. Nusselt Number Variation

Figure 5 illustrates the variation of Nusselt number defined at the moving wall versus Brinkman number with shape factor $S$ being the parameter, indicating its heavy reliance on $S.$ There is an expected increase in $Nu$ for $S=1$, particularly with a large $R$, attributed by a reduced heat flux at the moving wall, and therefore, a decrease in the temperature difference between moving wall temperature and bulk mean temperature when both plates are heated, as reflected in Figure 4b. While higher $Br$ indicates more intense viscous dissipation, competing effect between the more dominant convection and heat generation leads to a higher $Nu.$ Such effect is caused by enhanced heat convection effect at a specified $Br$, which is more significant for low $S$. As $S$ approaches infinitely large value, $Br$ has no noticeable effect on $Nu$, indicating the heat generation effect being offset by convection due to the velocity and viscous dissipation distribution depicted in Figure 2 and Figure 3 at a fixed $Br$, a higher $Br$ is accompanied by heat convection enhancement at the moving wall.

3.4. Temperature Contour Plots

3.4.1. $R=0$ (Heat Flux Applied to the Moving Plate Only)

Two dimensional temperature field is plotted to compare the effects of viscous dissipation on a conventional channel and a microchannel. A microchannel of height, $H=50 \mathsf{\mu }\mathrm{m}$ and length, $L=30 \mathrm{mm}$ is filled with water saturated porous medium made up of silicon.

Table 3. Thermophysical properties of fluid and porous medium in a microchannel [11].

Fluid	Water
Solid	Silicon
Porosity, $ϵ$	0.9
Density of fluid, ${\rho }_f$ (kg/m3)	997
Specific heat of fluid, $c_{p,f}$ (J/kg·K)	4179
Viscosity of fluid, ${\mu }_f$ (N·s/m2)	8.55 × 10-4
Heat flux, $q″$(W/m2)	1 × 104
Moving wall temperature, $T_w$ (K)	300
Thermal conductivity of fluid, $k_f$ (W/m·K)	0.613
Thermal conductivity of solid, $k_s$ (W/m·K)	148
Effective thermal conductivity of porous material, $k_{eff}$ (W/m·K)	15.3

shows the thermophysical properties of fluid and porous medium for computation. The properties of porous medium and geometry size are derived from Ting et. al. [14] who applied porous medium modeling to solve a forced convection problem in microchannel. The resulting porous medium shape factor is calculated to be $S=30$. The temperature contour plots are based on a constant $Re$, from which the corresponding flow velocity and $Br$ are computed. Inlet temperature is specified as 300 K and heat flux applied is taken as 1 × 10⁴ W/m² for all the computation hereinafter. Figure 6a–d depict the temperature contour plots in a microchannel with flow velocity,$u=1.90 \mathrm{m}/\mathrm{s}$ and $u=0.952 \mathrm{m}/\mathrm{s},$ respectively and $Re$ fixed at 100 and 50, respectively. Viscous dissipation causes an increase in the axial temperature as reflected by Figure 6b,d, corresponding to $Br=6.20$ and $Br=1.55$. The transverse temperature profile is flatter at a higher $Re$ indicating an improved heat transfer coefficient. Notwithstanding the larger heat convection coefficient with increasing $Re$, a higher $Br$ elevates the axial temperature along the axial direction slightly as reflected by Figure 6b,d, when viscous dissipation is accounted for, reflecting the significance of viscous dissipation to a microchannel.

In order to illustrate the effects of channel size on the temperature distribution, two-dimensional temperature distribution is also computed for a conventional size channel of height, $H=10\mathrm{mm}$ and length, $L=50 \mathrm{mm}$ based on another set of thermophysical properties in

Table 4. Thermophysical properties of fluid and porous medium in a conventional channel [12].

Fluid	Air
Solid	Sintered Bronze Beads
Porosity, $ϵ$	0.37
Permeability, $K$	0.422 × 10−9
Density of fluid, ${\rho }_f$ (kg/m3)	1.177
Specific heat of fluid, $c_{p,f}$ (J/kg·K)	1005
Viscosity of fluid, ${\mu }_f$ (N·s/m2)	1.846 × 10−5
Heat flux, $q″$(W/m2)	0.8 × 104
Moving wall temperature,$T_w$ (K)	300
Thermal conductivity of fluid, $k_f$ (W/m·K)	26.14 × 10−3
Thermal conductivity of solid, $k_s$ (W/m·K)	10.287
Effective thermal conductivity of porous material,$k_{eff}$ (W/m·K)	6.5

. The properties of porous medium are obtained from Hwang and Chao [15] who performed experimental investigation on the heat transfer measurement of sintered porous channel. Figure 7a–d show the temperature profile for $S=296$ at $Re=250$ and $Re=150$, with velocities, $u=1.06 \mathrm{m}/\mathrm{s}$ and $u=0.636 \mathrm{m}/\mathrm{s}$ respectively. For a conventional channel duct, Figure 7a,c indicate marked change in temperature difference along the axial direction where the highest temperature declines with larger$Re$. Compared to the effects of $Br$ on a microchannel at a fixed $Re$, the effects of viscous dissipation on the temperature difference diminishes in a conventional channel by Figure 7b,d, corresponding to $Br=0.061$ and $Br=0.022$ As Reynolds number goes up, the effect of heat generation is withered by a more marked convection, hence causing the decrease in the highest temperature along axial direction.

3.4.2. $R=0.5$ (Equal Heat Flux Applied to Both Plates)

Figure 8 and Figure 9 represent two-dimensional temperature field for heat flux applied at both boundaries for microchannel and conventional channel ducts, respectively. The same thermophysical properties of fluid and porous material are applied, as given in Table 3 and Table 4, respectively. Figure 8a,c depict the same range of temperature achieved in the channel as $R=0$ in Figure 6a,c but shows much higher temperature in transverse direction due to an increase in heat transfer coefficient, similar to Figure 9a,c as $R$ increases from $0 \mathrm{to} 0.5$. Likewise, Figure 8b,d reflect significant viscous dissipation effect on the axial temperature variation whereas for conventional channel in Figure 9b,d, the change in transverse temperature is more noticeable than the microchannel. In a microchannel, the surface temperature approaches the mean temperature due to its apparently much higher heat transfer coefficient than a conventional channel. Expectedly, a more gradual temperature change is noted in the proximity of the moving wall due to its higher heat transfer coefficient than the fixed wall. A decreasing temperature field with increasing Reynolds number indicates the enhancement in convection.

4. Conclusions

The temperature field for a Couette-Poiseuille flow characterized by a maximum velocity between two parallel plates in a saturated porous medium is solved with emphasis on the effects of viscous dissipation. The temperature distribution is subject to heat flux applied at both boundaries, whereby a heat flux ratio is defined as the heat flux applied to the fixed boundary to the total heat flux applied to both boundaries. The velocity field at lower porous medium shape factor $S$ in particular, envisages the competing effect between convection and heat generation in the form of viscous dissipation. Due to the more significant viscous dissipation effect particularly in the vicinity of the fixed wall, Nusselt number depends and varies significantly with $Br$ for smaller porous medium shape factor $S$. The two-dimensional temperature contour plots based on the properties derived from the literature show that viscous dissipation causes appreciable temperature hike in a microchannel along the axial direction at fixed Reynolds numbers. The transverse temperature change is however more apparent in a conventional channel due to a comparatively lower heat transfer coefficient at the wall for its correspondingly larger hydraulic diameter. Therefore, while the effect of viscous dissipation is significant to the enthalpy change for flows in a microchannel, its effect on the transverse temperature profile becomes more obvious as the characteristic size of the channel increases. As Reynolds number goes up, the attendant increase in viscous dissipation is more significant than the inertial effect in a microchannel, while the contrary is true in a conventional duct.