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Analytical Investigation of the Effect of Viscous Dissipation on Couette Flow in a Channel Partially Filled with a Porous Medium

Abstract

An analytical study of viscous dissipation effect on the fully developed forced convection Couette flow through a parallel plate channel partially filled with porous medium is presented. A uniform heat flux is imposed at the moving plate while the fixed plate is insulated. In the fluid-only region the flow field is governed by Navier–Stokes equation while the Brinkman-extended Darcy law relationship is considered in the fully saturated porous medium. The interface conditions are formulated with an empirical constant β due to the stress jump boundary condition. Fluid properties are assumed to be constant and the longitudinal heat conduction is neglected. A closed-form solution for the velocity and temperature distributions and also the Nusselt number in the channel are obtained and the viscous dissipation effect on these profiles is briefly investigated.

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Abbreviations

A :

Dimensionless parameter defined in Eq. 17

B :

Dimensionless parameter defined in Eq. 20

Br:

Brinkman number, \({\mu_{\rm f} u_w^2/(q^{{\prime}{\prime}}H)}\)

c :

Specific heat of the fluid [J/(kg K)]

C :

Dimensionless parameter defined in Eq. 26

D :

Dimensionless parameter defined in Eq. 27

Da:

Darcy number, K/H 2

E :

Dimensionless parameter defined in Eq. 30

F :

Dimensionless parameter defined in Eq. 17

G :

Dimensionless parameter defined in Eq. 33

H :

Total thickness of the composite channel, L + δ [m]

I :

Dimensionless parameter defined in Eq. 32

J :

Dimensionless parameter defined in Eq. 31

k f :

Thermal conductivity of the fluid [W/m K]

K :

Permeability of the porous medium [m2]

L :

Thickness of the porous region [m]

Nu:

Nusselt number at the isoflux plate

q′′:

Constant wall heat flux imposed to the lower [W m−2]

R :

Thermal conductivity ratio, k eff/k f

Re:

Reynolds number, ρ f u w H/μ f

T :

Fluid temperature [K]

u :

Fluid velocity [m s−1]

\({\overline u}\) :

Mean flow velocity, \({1/H\mathop \smallint_{-\delta}^L u{\rm d}y}\), [m s−1]

U :

Dimensionless fluid velocity, u/u w

\({\overline u}\) :

Dimensionless mean fluid velocity, \({\mathop \smallint_{-\delta/H}^{L/H} U\theta {\rm d}Y}\)

W :

Dimensionless parameter defined in Eq. 36

x :

Longitudinal coordinate of the channel [m]

X :

Dimensionless longitudinal coordinate

y :

Transverse coordinate of the channel [m]

Y :

Dimensionless transverse coordinate

β :

The adjustable coefficient in the stress jump boundary condition

δ :

Thickness of the clear fluid region [m]

γ :

Absolute viscosity ratio, (μ eff/μ f)1/2

μ :

Absolute viscosity [kg m−1s−1]

ρ :

Density [kg m−3]

θ :

Dimensionless fluid and wall temperature difference, (TT w)/(T mT w)

eff:

Effective (porous medium)

f:

Clear fluid

m:

Mean flow

w:

Isoflux plate

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Correspondence to Osameh Ghazian.

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Ghazian, O., Rezvantalab, H. & Ashjaee, M. Analytical Investigation of the Effect of Viscous Dissipation on Couette Flow in a Channel Partially Filled with a Porous Medium. Transp Porous Med 89, 1–13 (2011). https://doi.org/10.1007/s11242-011-9755-4

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Keywords

  • Viscous dissipation
  • Couette flow
  • Partially filled
  • Channel
  • Porous medium