Abstract
We examine the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection induced by a vertical heated surface embedded in a fluid-saturated porous medium. We consider the boundary-layer regime where the Darcy-Rayleigh number, Ra, is very large, and assume that the surface waves have O(1) amplitude and wavelength. The resulting boundary-layer equations are found to be nonsimilar only when the surface is nonuniform and inertia effects are present; self-similarity results when either or both effects are absent. Detailed results for the local and global rates of heat transfer are presented for a range of values of the inertia parameter and the surface wave amplitude.
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Abbreviations
- a :
-
amplitude of the wavy surface
- d :
-
particle diameter
- f :
-
reduced streamfunction
- g :
-
acceleration due to gravity
- Gr* :
-
modified Gashof number
- k c :
-
effective thermal conductivity
- K :
-
permeability
- \(\tilde K\) :
-
material parameter
- l :
-
half-wavelength, or lengthscale associated with the surface
- \(\mathcal{L}^2\) :
-
differential operator; see Equation (17)
- n :
-
unit vector normal to the wavy surface
- Nu:
-
local Nusselt number
- p :
-
pressure
- q :
-
rate of heat flux
- Q :
-
nondimensional velocity; see Equation (11)
- Ra:
-
Darcy-Rayleigh number based onl
- s :
-
surface length
- T :
-
temperature
- u, v :
-
fluid velocities in thex andy directions, respectively
- v :
-
velocity vector
- x, y :
-
streamwise and cross-stream Cartesian coordinates
- α :
-
thermal diffusivity of the porous medium
- Β :
-
coefficient of thermal expansion
- ε :
-
porosity
- ξ, η :
-
pseudo-similarity variables
- σ :
-
surface profile; see Equation (1)
- θ :
-
dimensionless temperature
- Μ :
-
dynamic viscosity
- v :
-
kinematic viscosity
- ρ :
-
density
- ψ :
-
streamfunction
- ¯:
-
dimensional variables
- ∼:
-
transformed variables (ξ<1)
- ^:
-
boundary-layer variables
- ′:
-
differentiation with respect toη
- g :
-
global
- x :
-
differentiation with respect tox
- w :
-
condition at the wall
- ∞:
-
condition at infinity
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Rees, D.A.S., Pop, I. Non-Darcy natural convection from a vertical wavy surface in a porous medium. Transp Porous Med 20, 223–234 (1995). https://doi.org/10.1007/BF01073173
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DOI: https://doi.org/10.1007/BF01073173