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This study investigates a mixed convection boundary layer flow over a vertical wall embedded in a highly porous medium. The fluid viscosity is assumed to decrease exponentially with temperature. The boundary layer equations are transformed into a non-similar form using an appropriate non-similar variable ξ and a pseudo-similar variable η. The non-similar equations are solved using an efficient local non-similarity method. The effect of viscosity variation parameter on the heat transfer, skin friction and the velocity and temperature distribution within the boundary layer is investigated. The viscosity variation parameter, the viscous dissipation parameter and non-simi-larity variable are shown to have a significant effect on velocity and thermal boundary layer and also on the skin friction coefficient and heat transfer at the wall.

Mixed convection boundary layer flow through a porous medium is of primary importance due to its applications in industry such as geothermal operations (for example dynamics of hot springs), terrestrial heat flow through an aquifer, flow of moisture through porous industrial material etc. Engineering applications include porous thermal insulation and heat exchangers with fluidized beds etc.

Reviews on convective heat transfer in porous medium are well documented in the books by Nield and Bejan, [

Most fluids used in engineering have temperature varying viscosity. A study by Hossain and Munir [

The current study focuses on a boundary layer flow over a vertical wall embedded in a highly porous medium and with variable viscosity. The non-similar boundary layer equations are solved using the local non-similarity method due to Sparrow et al. [

A steady, two-dimensional flow of a viscous incompressible fluid over a heated semi-infinite vertical wall embedded in a highly porous medium is studied. The

The appropriate boundary conditions for the flow are:

Here

where

where

The equations are rendered non-dimensional by introducing a boundary layer pseudo-similarity variable

with

where

We use the scaling

where

From the equation of continuity:

with

Assuming power law variation in the free-stream velocity

where

The boundary conditions, in non-dimensional form, are:

The shear stress and the heat transfer at the wall can be represented using the local skin friction coefficient and the local Nusselt number, defined by

The system of Equations (2.4)-(2.6) is solved using the non-similar method of Sparrow et al. [

The first level of approximation involves solving the local similarity equations, where all terms involving the

At the second level of truncation, the method involves solving the system (2.4)-(2.6) together with the auxiliary system obtained by differentiating (2.4)-(2.6) partially with respect

The system (2.4)-(2.9) solved as if it is an ordinary differential system, with

An even better approximation can be obtained at the third order level of approximation, where the system (2.4)-(2.9) are solved together with an of auxiliary system obtained by further differentiating Equations (2.7)-(2.9) partially with respect to

We approximate the solution of (2.4)-(2.6) using the second level of approximation. The system of Equations (2.4)-(2.9) is solved using the fourth order Runge-Kutta method combined with a Newton root refinement scheme. Computations were carried out for the case when

The effect of varying the viscous dissipation parameter

Heat transfer at the wall (represented by the Nusselt number,

Similar results showing the effect of

The effect of varying

Effect of varying

Effect of varying viscosity variation parameter

The effect of varying

The variation of (a) skin friction coefficient and (b) Nusselt number with

cient and a decrease in the Nusselt number. The results for the heat transfer coefficient indicate that at a fixed value of

A non-Darcy mixed convection boundary layer flow has been investigated. The results have shown that the non-similarity parameter

The skin friction coefficient increases with increase in the viscosity variation parameter, the viscous dissipation parameter and the buoyancy-related parameter. For flows with viscous dissipation effects, there is a critical value of the buoyancy related parameter, above which the Nusselt number changes sign from positive (wall extracts heat from fluid) to negative (fluid extracts heat from wall).

The author would like to thank the East African Universities Mathematics Programme (EAUMP) for their financial support.