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
References
Nield, D. A. and Bejan, A.:Convection in Porous Media, Springer, New York, 1992.
Cheng, P. and Minkowycz, W. J.: 1977, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike,J. Geophys. Res. 82 2040–2044.
Cheng, P.: Combined free and forced convection flow about inclined surfaces in porous media,Int. J. Heat Mass Transfer 20, 807–814.
Hsiech, J. J. C., Chen, T. S., and Armaly, B. F.: 1993, Nonsimilar solutions for mixed convection from vertical surfaces in porous media: variable surface temperature or heat flux,Int. J. Heat Mass Transfer 36, 1485–1493.
Merkin, J. H.: 1980, Mixed convection boundary layer flow on a vertical surface in a saturated porous medium,J. Engng. Math. 14, 301–313.
Pop, I., Lesnic, D., and Ingham, D. B.: Mixed convection on a vertical surface in a porous medium,Int. J. Heat Mass Transfer (to be published).
Cheng, P. and Hsu, C. T.: 1984, Higher-order approximations for Darcian free convective flow about a semi-infinite vertical flat plate,J. Heat Transfer 106, 143–151.
Joshi, Y. and Gebhart, B.: 1984, Vertical natural convection flows in porous media: calculations of improvement accuracy,Int. J. Heat Mass Transfer 27, 69–75.
Ingham, D. B. and Brown, S. N.: 1986, Flow past a suddenly heated vertical plate in a porous medium,Proc. Roy Soc London Ser A 403, 51–80.
Merkin, J. H. and Mahmood, T.: 1990, On the free convection boundary layer on a vertical plate with prescribed surface heat flux,J. Engng. Math. 24, 95–107.
Hunt, R. and Wilks, G.: 1981, Continuous transformation computations of boundary layer equations between similarity regimes.J. Compnt. Phys. 40, 478–490.
Kuiken, H. K.: 1981, On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small IMA,J. Appl. Math. 27, 387–405.
Keller, H. B.: ANumerical Solution for Partial Differential Equations. Hubbard, New York, 1971.
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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