Summary
The free convection boundary layer on an impermeable vertical cylinder embedded in a saturated porous medium is considered. It is shown that a numerical solution of the governing equations fails to predict the flow at large distances along the cylinder with any real accuracy, though is accurate at moderate distances from the leading edge. An asymptotic solution is derived and it is shown that a simple approximate method can better represent the whole flow region than the full numerical solution.
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Abbreviations
- a :
-
radius of the cylinder
- g :
-
acceleration of gravity
- K :
-
permeability of the porous medium
- Q :
-
non-dimensional heat transfer at the cylinder
- g :
-
wall temperature gradient defined byq=Q/2µ
- r :
-
radial co-ordinate
- \(\bar r\) :
-
non-dimensional radial co-ordinate
- R a :
-
Rayleigh number
- T :
-
temperature of the convective fluid
- T w :
-
temperature of the cylinder
- T o :
-
ambient temperature
- ΔT :
-
temperature difference=T w -T 0
- u :
-
velocity in thex-direction
- v :
-
velocity in ther-direction
- x :
-
axial co-ordinate
- X :
-
non-dimensional axial co-ordinate
- α:
-
equivalent thermal diffusivity
- β:
-
coefficient of thermal expansion
- θ:
-
non-dimensional temperature=(T-T 0)/ΔT
- μ:
-
viscosity of the convective fluid
- ϱ:
-
density of the convective fluid
- Ψ:
-
stream function
- µ:
-
co-ordinate defined by µ=2(2X)1/2
References
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Minkowycz, W. J., Cheng, P.: Free convection about a vertical cylinder embedded in a porous medium. Int. J. Heat Mass Transfer19, 805–813 (1976).
Merkin, J. H.: Free convection boundary layers on axi-symmetric and two-dimensional bodies of arbitrary shape in a saturated porous medium. Int. J. Heat Mass Transfer22, 1461–1462 (1979).
Merkin, J. H.: Free convection boundary layers in a saturated porous medium with lateral mass flux. Int. J. Heat Mass Transfer21, 1499–1504 (1978).
Smith, G. D.: Numerical solution of partial differential equations. Oxford University Press 1965.
Kuicken, H. K.: The thick free-convection boundary layer along a semi-infinite isothermal vertical cylinder. ZAMP25, 497–514 (1974).
Karnis, J., Pechoc, V.: The thermal laminar boundary layer on a continuous cylinder. Int. J. Heat Mass Transfer21, 43–47 (1978).
Glauert, M. B., Lighthill, M. J.: The axisymmetric boundary layer on a long thin cylinder. Proc. Roy Soc.A 230, 188–203 (1955).
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Merkin, J.H. Free convection from a vertical cylinder embedded in a saturated porous medium. Acta Mechanica 62, 19–28 (1986). https://doi.org/10.1007/BF01175850
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DOI: https://doi.org/10.1007/BF01175850