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 fluidonly region the flow field is governed by Navier–Stokes equation while the Brinkmanextended 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 closedform 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 [m^{2}]
 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^{−1}s^{−1}]
 ρ :

Density [kg m^{−3}]
 θ :

Dimensionless fluid and wall temperature difference, (T − T _{w})/(T _{m} − T _{w})
 eff:

Effective (porous medium)
 f:

Clear fluid
 m:

Mean flow
 w:

Isoflux plate
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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/s1124201197554
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Keywords
 Viscous dissipation
 Couette flow
 Partially filled
 Channel
 Porous medium