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Non-Darcian flow and heat transfer along a permeable vertical surface with nonlinear density temperature variation

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Abstract

The non-Darcy free convection flow on a vertical flat plate embedded in a fluid-saturated porous medium in the presence of the lateral mass flux with prescribed constant surface temperature is considered. The coupled nonlinearities generated by the density variation with temperature, inertia, and viscous dissipation are included in the present study. In particular, we analyze a system of nonlinear ODEs describing self-similar solutions to the flow and heat transfer problem. These transformed equations are integrated numerically by a second-order finite difference scheme known as the Keller box method. Furthermore, some analytical results are provided to establish relationships between the physical invariants in the problem, and also to validate the numerical method. One of the important findings of our study is that an increase in the Rayleigh number increases the velocity boundary layer thickness, while the opposite is true for the thermal boundary layer thickness.

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Abbreviations

a :

Constant in Eq. (7a)

C p :

Specific heat at constant pressure

d :

The characteristic length scale

C :

Inertia coefficient

Ec:

Eckert number

f :

Dimensionless stream function

f w :

Lateral mass flux parameter

g :

Acceleration due to gravity

K :

Permeability

k :

Thermal conductivity of the saturated porous medium

k 1 :

Inertia parameter

Nu :

Nusselt number

P :

Pressure

Pr:

Prandtl number

Ra:

Rayleigh number

T :

Temperature

T w :

Temperature of the plate

T ∞ :

Ambient temperature

u, v :

Velocity components in the x and y directions

x, y :

Cartesian coordinates

α :

Effective thermal conductivity

β 0, β 1 :

Constants in Eq. (6)

γ :

Kinematic viscosity of the fluid

δ :

Constant in (9) known as NDT parameter

σ :

Permeability parameter

η :

Similarity variable

θ :

Dimensionless temperature

μ :

Viscosity

ρ :

Density

ψ :

Stream function

∞:

Condition at infinity

′:

Derivative with respect to η

w :

Condition at the wall

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bangalore, 560001, India

    K. V. Prasad

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    K. V. Prasad, K. Vajravelu & Robert A. van Gorder

Authors
  1. K. V. Prasad
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  2. K. Vajravelu
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  3. Robert A. van Gorder
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Corresponding author

Correspondence to Robert A. van Gorder.

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Prasad, K.V., Vajravelu, K. & van Gorder, R.A. Non-Darcian flow and heat transfer along a permeable vertical surface with nonlinear density temperature variation. Acta Mech 220, 139–154 (2011). https://doi.org/10.1007/s00707-011-0474-2

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  • DOI: https://doi.org/10.1007/s00707-011-0474-2

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