Abstract
This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x 2)μ), where μ is a constant andx is the distance along the surface. It is shown that for μ > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when μ ⩽ -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely μ < -1/2 and μ = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of μ.
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Abbreviations
- g :
-
Gravitational acceleration
- k :
-
Thermal conductivity of the saturated porous medium
- K :
-
Permeability of the porous medium
- l :
-
Typical streamwise length
- q w :
-
Uniform heat flux on the wall
- Ra:
-
Rayleigh number, =gβK(q w /k)l/(αv)
- T :
-
Temperature
- Too:
-
Temperature far from the plate
- u, v :
-
Components of seepage velocity in the x and y directions
- x, y :
-
Cartesian coordinates
- α:
-
Thermal diffusivity of the fluid saturated porous medium
- β:
-
The coefficient of thermal expansion
- γ:
-
An undetermined constant
- φ:
-
Porosity of the porous medium
- η:
-
Similarity variable, =y(1+x μ)μ/3/x 1/3
- μ:
-
A preassigned constant
- ν:
-
Kinematic viscosity
- θ:
-
Nondimensional temperature, =(T − T ∞)Ra1/3 k/qw
- τ:
-
Similarity variable, = =y(loge x)1/3/x 2/3
- ξ:
-
Similarity variable, =y/x 2/3
- ψ:
-
Stream function
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Wright, S.D., Ingham, D.B. & Pop, I. On natural convection from a vertical plate with a prescribed surface heat flux in porous media. Transp Porous Med 22, 181–193 (1996). https://doi.org/10.1007/BF01143514
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DOI: https://doi.org/10.1007/BF01143514