Abstract
The non-similar problems associated with a non-isothermal vertical flat plate embedded in a fluid-saturated porous medium were considered to assess the performances of the two distinctive boundary layer solution methods, namely, the local similarity solution and the integral method. The results generated from these two approximate solution methods are compared against the results from a two-point finite difference and those based on a Merk-type series expansion. Comparison of the results reveals that both integral and local similarity methods perform excellently. Especially, the accuracy acquired by the local similarity solution is so high that the difference between the results from the local similarity solution and those from the two-point finite difference and local non-similarity solution methods is hardly discernible for the case of monotonic increasing wall temperature.
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Abbreviations
- f :
-
dimensionless stream function
- g :
-
acceleration due to gravity
- g x :
-
component of the acceleration due to gravity in thex-direction
- I :
-
function defined in equation (12)
- K :
-
intrinsic permeability of the porous media
- m t :
-
parameter associated with the wall temperature, defined in equation (17)
- Nu x :
-
local Nusselt number
- r :
-
function representing wall geometry
- r*:
-
1 for plane flow andr for axisymmetric flow
- Ra x :
-
local Rayleigh number, defined in equation (13)
- T :
-
temperature
- T e :
-
ambient constant temperature
- T w :
-
variable wall surface temperature
- ΔT :
-
temperature difference equal toT w −Te
- u, v :
-
Darcian or superficial velocity components
- x, y :
-
boundary layer coordinates
- α :
-
equivalent thermal diffusivity of the fluid-saturated porous medium
- β :
-
expansion coefficient of the fluid
- δ T :
-
thermal boundary layer thickness
- η :
-
similarity variable defined in equation (11)
- ϑ :
-
dimensionless temperature difference
- ν :
-
fluid kinematic viscosity
- λ :
-
exponent introduced in equation (23)
- μ :
-
fluid viscosity
- ρ :
-
fluid density
- ÷ :
-
stream function
- e :
-
ambient
- w :
-
wall
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Nakayama, A., Hossain, M.A. Free convection in a saturated porous medium beyond the similarity solution. Appl. Sci. Res. 52, 133–145 (1994). https://doi.org/10.1007/BF00868055
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DOI: https://doi.org/10.1007/BF00868055