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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Analytical Solution

#### 2.1. Governing Equation

#### 2.2. Nusselt Number

#### 2.3. Nusselt Number Verification

## 3. Results and Discussion

#### 3.1. Velocity Profile

#### 3.2. Temperature Distribution

#### 3.3. Nusselt Number Variation

#### 3.4. Temperature Contour Plots

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

^{4}W/m

^{2}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.

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

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | Constant, defined in Equation (13) |

$Br$ | Brinkman number |

${\mathrm{C}}_{1}-{\mathrm{C}}_{7}$ | Coefficients in Equation (16), listed in Appendix A |

${c}_{p,f}$ | Specific heat of fluid, J/kg·K |

$Da$ | Darcy number, $K/{H}^{2}$ |

$H$ | Height of the channel, m |

$K$ | Permeability of the porous medium, m^{2} |

$k$ | Thermal conductivity, W/m·K |

${k}_{eff}$ | Effective thermal conductivity of porous medium, W/m·K |

$L$ | Length of the channel, m |

$M$ | Ratio of effective viscosity to viscosity, defined as, $\frac{{\mu}_{eff}}{\mu}$ |

$Nu$ | Nusselt number, defined as, $\frac{q\u2033H}{{k}_{eff}\left({T}_{w}-{T}_{m}\right)}$ |

${q}^{\u2033}$ | Heat flux, W/m^{2} |

$Re$ | Reynolds number, defined as $\frac{{\rho}_{f}{u}_{m}H}{{\mu}_{eff}}$ |

$R$ | The fraction of heat flux applied to fixed wall, defined as,$\frac{{{q}^{\u2033}}_{2}}{{{q}^{\u2033}}_{1}+{{q}^{\u2033}}_{2}}$ |

$S$ | Porous medium shape factor, defined as, $\frac{1}{\sqrt{MDa}}$ |

$T$ | Fluid temperature, K |

${T}_{m}$ | Mean temperature, K |

${T}_{w}$ | Wall temperature at lower plate, K |

$u$ | Fluid velocity, m/s |

${u}_{w}$ | Moving wall velocity, m/s |

u_{m} | Mean velocity, m/s |

${U}_{m}$ | Dimensionless mean velocity |

$U$ | Dimensionless velocity, defined as, $\frac{{\mu}_{eff}u}{\gamma {H}^{2}}$ |

${U}_{w}$ | Dimensionless wall velocity, defined as, $\frac{{\mu}_{eff}{u}_{w}}{\gamma {H}^{2}}$ |

${u}^{*}$ | Dimensionless velocity, defined as, $\frac{u}{{u}_{m}}=\frac{U}{{U}_{m}}$ |

$x$ | Axial coordinate of the channel, m |

$X$ | Dimensionless length, $\frac{x}{L}$ |

$Y$ | Dimensionless transverse distance, defined as, $\frac{y}{H}$ |

$y$ | Vertical coordinate, m |

$\gamma $ | Pressure gradient, N/m^{3} |

${\rho}_{f}$ | Density of the fluid, kg/m^{3} |

$\mu $ | Viscosity of the fluid, N·s/m^{2} |

${\mu}_{eff}$ | Effective viscosity of the porous medium, N·s/m^{2} |

$\theta $ | Dimensionless temperature,$\frac{{k}_{eff}\left({T}_{A}-{T}_{w}\right)}{(q{\u2033}_{1}+q{\u2033}_{2})H}$ |

${\theta}_{1}$ | Dimensionless temperature,$\frac{{k}_{eff}\left({T}_{B}-{T}_{w}\right)}{(q{\u2033}_{1}+q{\u2033}_{2})H}$ |

${\theta}_{2}$ | Dimensionless temperature, $\frac{{k}_{eff}\left({T}_{B}-{T}_{w}\right)}{q{\u2033}_{1}H}$ |

## Appendix A: List of coefficients

## References

- Davaa, G.; Shigechi, T.; Momoki, S. Effect of viscous dissipation on fully developed heat transfer of non-newtonian fluids in plane laminar poiseuille-couette flow. Int. Commun. Heat Mass Transf.
**2004**, 31, 663–672. [Google Scholar] [CrossRef] - Shah, R.K.; London, A.L. Laminar flow forced convention in ducts; Advances in heat transfer, Supplement 1; Academic press: New York, NY, USA, 1978. [Google Scholar] [CrossRef]
- Aydin, O.; Avci, M. Viscous-dissipation effects on the heat transfer in a Poiseuille flow. Appl. Energy
**2006**, 83, 495–512. [Google Scholar] [CrossRef] - Lin, S.H. Heat transfer to plane non-Newtonian Couette flow. Int. J. Heat Mass Transf.
**1979**, 22, 1117–1123. [Google Scholar] [CrossRef] - Aydin, O.; Avci, M. Laminar forced convection with viscous dissipation in a Couette-Poiseuille flow between parallel plates. Appl. Energy
**2006**, 83, 856–867. [Google Scholar] [CrossRef] - Sheela-Francisca, J.; Tso, C.P.; Rilling, D. Heat Transfer with Viscous Dissipation in Couette-Poiseuille Flow under Asymmetric Wall Heat Fluxes. Open J. Fluid Dyn.
**2012**, 2, 111–119. [Google Scholar] [CrossRef] - Chan, Y.H.; Chen, G.M.; Tso, C.P. Effect of Asymmetric Boundary Conditions on Couette–Poiseuille Flow of Power-Law Fluid. J. Thermophys. Heat Transf.
**2015**, 29, 496–503. [Google Scholar] [CrossRef] - Hashemabadi, S.H.; Etemad, S.G.; Thibault, J. Forced convection heat transfer of Couette-Poiseuille flow of nonlinear viscoelastic fluids between parallel plates. Int. J. Heat Mass Transf.
**2004**, 47, 3985–3991. [Google Scholar] [CrossRef] - Aydin, O.; Avci, M. Analytical Investigation of Heat Transfer in Couette–Poiseuille Flow Through Porous Medium. J. Thermophys. Heat Transf.
**2011**, 25, 468–472. [Google Scholar] [CrossRef] - Al-Hadhrami, A.K.; Elliott, L.; Ingham, D.B. A new model for viscous dissipation in porous media across a range of permeability values. Transp. Porous Media
**2003**, 53, 117–122. [Google Scholar] [CrossRef] - Davaa, G.; Shigechi, T.; Momoki, S. Effect of viscous dissipation on fully developed heat laminar heat transfer of Power-Law non-newtonian fluids in plane Couette-Poiseuille laminar flow. Rep. Fac. Eng. Nagasaki University
**2000**, 30, 97–104. [Google Scholar] - Tso, C.P.; Sheela-Francisca, J.; Hung, Y.M. Viscous dissipation effects of power-law fluid flow within parallel plates with constant heat fluxes. J. Nonnewton. Fluid Mech.
**2010**, 165, 625–630. [Google Scholar] [CrossRef] - Tan, L.Y.; Chen, G.M. Analysis of entropy generation for a power-law fluid in a microchannel. In Proceedings of the ASME 2013 4th International Conference on Micro/Nanoscale Heat and Mass Transfer, Hong Kong, China, 11–14 December 2013. [Google Scholar]
- Ting, T.W.; Hung, Y.M.; Guo, N. Viscous dissipative forced convection in thermal non-equilibrium nanofluid-saturated porous media embedded in microchannels. Int. Commun. Heat Mass Transf.
**2014**, 57, 309–318. [Google Scholar] [CrossRef] - Hwang, G.J.; Chao, C.H. Heat transfer measurement and analysis for sintered porous channels. J. Heat Transfer.
**1994**, 116, 456–464. [Google Scholar] [CrossRef]

**Figure 1.**Schematic Diagram of the Problem subject to unequal and uniform heat fluxes at both boundaries.

**Figure 2.**Velocity distribution in the channel for different porous medium shape factor, $S$ and ${U}_{w}/{U}_{m}=1$.

**Figure 3.**Viscous Dissipation (I.H = Internal Heating, F.H = Frictional Heating) in the channel for different porous medium shape factor, $S$ and$Br=0.1$.

**Figure 4.**Dimensionless temperature distribution subject to constant heat flux at both boundaries for different heat flux ratio, R. (

**a**) $Br=0$ and (

**b**) $Br=0.1$.

**Figure 6.**Temperature field in a microchannel, for$R=0,S=30$ and ${k}_{eff}=15.3$ W/m·K based on the thermal physical properties of Table 3. (

**a**) Re = 100 without viscous dissipation, (

**b**) Re = 100 with viscous dissipation, (

**c**) Re = 50 without viscous dissipation, (

**d**) Re = 50 with viscous dissipation.

**Figure 7.**Temperature field in a conventional duct for $R=0,S=296$ and ${k}_{eff}=6.5$ W/m·K based on the thermal physical properties of Table 4. (

**a**) Re = 250 without viscous dissipation, (

**b**) Re = 250 with viscous dissipation, (

**c**) Re = 150 without viscous dissipation, (

**d**) Re = 150 with viscous dissipation.

**Figure 8.**Temperature field in a microchannel, for $R=0.5,S=30$ and ${k}_{eff}=15.3$ W/m·K based on the thermal physical properties of Table 3. (

**a**) Re = 100 without viscous dissipation, (

**b**) Re = 100 with viscous dissipation, (

**c**) Re = 50 without viscous dissipation, (

**d**) Re = 50 with viscous dissipation.

**Figure 9.**Temperature field in a conventional duct for $R=0.5,S=296$ and ${k}_{eff}=6.5$ W/m·K based on the thermal physical properties of Table 4. (

**a**) Re = 250 without viscous dissipation, (

**b**) Re = 250 with viscous dissipation, (

**c**) Re = 150 without viscous dissipation, (

**d**) Re = 150 with viscous dissipation.

$\mathit{B}\mathit{r}\text{}$ | $\mathit{R}$ | $\mathit{S}$ | ${\mathit{U}}_{\mathit{w}}/{\mathit{U}}_{\mathit{m}}\text{}$ | 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 |

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

$\mathbf{Porous}\text{}\mathbf{Medium}\text{}\mathbf{Shape}\text{}\mathbf{Factor},\text{}\mathit{S}$ | $\mathit{I}.\mathit{H}/{\mathit{q}}^{\u2033}.\mathit{H}$ | $\mathit{F}.\mathit{H}/{\mathit{q}}^{\u2033}.\mathit{H}$ |
---|---|---|

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 |

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

Fluid | Water |

Solid | Silicon |

Porosity, $\u03f5$ | 0.9 |

Density of fluid, ${\rho}_{f}$ (kg/m^{3}) | 997 |

Specific heat of fluid, ${c}_{p,f}$ (J/kg·K) | 4179 |

Viscosity of fluid, ${\mu}_{f}$ (N·s/m^{2}) | 8.55 × 10^{-4} |

Heat flux, $q\u2033$(W/m^{2}) | 1 × 10^{4} |

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 |

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

Fluid | Air |

Solid | Sintered Bronze Beads |

Porosity, $\u03f5$ | 0.37 |

Permeability, $K$ | 0.422 × 10^{−9} |

Density of fluid, ${\rho}_{f}$ (kg/m^{3}) | 1.177 |

Specific heat of fluid, ${c}_{p,f}$ (J/kg·K) | 1005 |

Viscosity of fluid, ${\mu}_{f}$ (N·s/m^{2}) | 1.846 × 10^{−5} |

Heat flux, $q\u2033$(W/m^{2}) | 0.8 × 10^{4} |

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 |

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**MDPI and ACS Style**

Chen, G.M.; B., M.F.; Lim, B.K.; Tso, C.P.
Thermal Viscous Dissipative Couette-Poiseuille Flow in a Porous Medium Saturated Channel. *Symmetry* **2019**, *11*, 869.
https://doi.org/10.3390/sym11070869

**AMA Style**

Chen GM, B. MF, Lim BK, Tso CP.
Thermal Viscous Dissipative Couette-Poiseuille Flow in a Porous Medium Saturated Channel. *Symmetry*. 2019; 11(7):869.
https://doi.org/10.3390/sym11070869

**Chicago/Turabian Style**

Chen, G. M., M. Farrukh B., B. K. Lim, and C. P. Tso.
2019. "Thermal Viscous Dissipative Couette-Poiseuille Flow in a Porous Medium Saturated Channel" *Symmetry* 11, no. 7: 869.
https://doi.org/10.3390/sym11070869