Abstract
The buoyancy-induced flows of non-Newtonian fluids over non-isothermal bodies of arbitrary shape within saturated porous media have been treated using the boundary layer approximations and the power-law model to characterize the non-Newtonian fluid behavior. Upon introducing a general similarity transformation which considers both the geometrical effect and the wall temperature effect on the development of the boundary layer length scale, the governing equations for a non-isothermal body of arbitrary shape have been reduced to those for a vertical flat plate. The transformed equations reveal that a plane or axisymmetric body of arbitrary shape possesses its corresponding family of the wall temperature distributions which permit similarity solutions. Numerical integrations were carried out using the Runge-Kutta-Gill method, and the results of the heat transfer function were presented once for all plane and axisymmetric bodies. As illustrations, local wall heat flux distributions were discussed for wedges, cones, spheres, circular cylinders and other geometries. Furthermore, an approximate formula based on the Karman-Pohlhausen integral relation has been presented for speedy and sufficiently accurate estimation of heat transfer rates.
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- d :
-
particle diameter
- f :
-
dimensionless stream function
- F :
-
heat transfer function, Eq. (30)
- g, g x :
-
acceleration due to gravity and its streamwise component
- I :
-
function adjusting the boundary layer length scale, Eq. (11c)
- k :
-
effective thermal conductivity of a porous medium
- K*:
-
modified permeability
- L r :
-
reference length
- m :
-
function associated with wall temperature, Eq. (15)
- n :
-
power law index
- Nu x :
-
local Nusselt number
- p :
-
pressure
- q w ,q * w :
-
local wall heat flux and its dimensionless form, Eq. (33)
- r :
-
function representing a body shape
- r*:
-
1 for plane flow andr for axisymmetric flow
- Ra x :
-
modified local Rayleigh number
- T :
-
temperature
- u, v :
-
Darcian velocity components
- x, y :
-
boundary layer coordinates
- z :
-
vertical distance measured from the lower stagnation point
- α :
-
effective thermal diffusivity of a porous medium
- β :
-
coefficient of thermal expansion
- γ :
-
apex half angle of a wedge or a cone
- δ :
-
boundary layer thickness
- ε :
-
porosity
- ζ :
-
exponent associated with the temperature profile
- η :
-
similarity variable
- θ :
-
dimensionless temperature
- λ :
-
exponent associated with the wall temperature distribution
- μ*:
-
power law constant
- ξ :
-
transformed variable, Eq. (18)
- ρ :
-
fluid density
- φ :
-
peripheral angle measured from the lower stagnation point
- ψ :
-
stream function
- ω :
-
parameter representing a family of plane geometries
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Nakayama, A., Koyama, H. Buoyancy-induced flow of non-Newtonian fluids over a non-isothermal body of arbitrary shape in a fluid-saturated porous medium. Appl. Sci. Res. 48, 55–70 (1991). https://doi.org/10.1007/BF01998665
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DOI: https://doi.org/10.1007/BF01998665