Abstract
A simple mathematical theory is proposed to investigate the development of the flow field which is the response of a fluid to the buoyancy force due to the existence of a temperature gradient in a hemispherical fluid-saturated porous medium, assuming the validity of the Brinkman model. The induced flow is assumed to be slow, and Stokes approximation is invoked. It is shown, at all times, the induced fluid motion occurs in the form of eddies on either side of the axis of symmetry. In the steady state, the behavior of the fluid motion on the free surface is similar to that of axial fluid flow.
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Abbreviations
- \(a\) :
-
Radius of the hemisphere [L]
- \(c_{\mathrm{p}}\) :
-
Specific heat at constant pressure [\(\hbox {L}^{2}\,\hbox {T}^{2}\,\Omega ^{-1}\)]
- \(D_{\mathrm{a}}\) :
-
Darcy number, \(Ka^{-2}\)
- \(\hbox {e}_{n}\) :
-
Coefficients in the series expansion (Eq. 12)
- \(f\) :
-
Function of \(\varphi \) (Eq. 12)
- \(f_{n}\) :
- \(G_{n}\) :
-
Gegenbauer polynomial of order \(n\) (Eq .15)
- \(g\) :
-
Acceleration due to gravity
- \(J_{m}\) :
-
Bessel functions of the first kind of odd order \(m\)
- \(K\) :
-
Permeability of the porous medium [\(\hbox {L}^{2}\)]
- n :
-
Unit vector along the \(\varphi = 0\) axis
- Pr :
-
Prandtl number, \(\nu /\alpha \)
- \(P\) :
-
Fluid pressure in excess of its hydrostatic value [\(\hbox {L M}^{-1}\,\hbox {T}^{2}\)]
- \(P_{n}\) :
-
Legendre polynomial of order n of the first kind
- q :
-
Fluid velocity \((u, v, w)\) [\(\hbox {L T}^{-1}\)]
- \(R\) :
-
Non-dimensional radial co-ordinate
- \(R_{\mathrm{a}}\) :
-
Thermal Rayleigh number \((\beta {gKa}/\alpha \nu )\hbox {T}_{0}\)
- \(r\) :
-
Radial co-ordinate [L]
- \(T\) :
-
Temperature [\(\Omega \)]
- \(T_{0}\) :
-
Temperature of the free surface [\(\Omega \)]
- \(t\) :
-
Time [T]
- \(U\) :
-
Non-dimensional radial component of velocity
- \(U_\mathrm{ax}\) :
-
Steady-state axial velocity
- \(u\) :
-
Radial component of velocity [\(\hbox {L T}^{-1}\)]
- \(V\) :
-
Non-dimensional transverse component of velocity
- \(\nu \) :
-
Transverse component of velocity [\(\hbox {L T}^{-1}\)]
- \(\alpha \) :
-
Effective thermal diffusivity of the fluid-porous matrix [\(\hbox {L}^{2}\,\hbox {T}^{-1}\)]
- \(\beta \) :
-
Thermal expansion coefficient [\(\Omega ^{-1}\)]
- \(\lambda \) :
-
Square root of the reciprocal of the Darcy number
- \(\varepsilon \) :
-
Porosity of the porous medium [\(\hbox {ML}^{-1}\Omega \)]
- \(\theta \) :
-
Azimuthal angle
- \(\varphi \) :
-
Meridian angle
- \(\eta _{k}\) :
-
\(k\)th positive zero of \(J_{2n+3/2}(\eta )\)
- \(\zeta \) :
-
\(\hbox {cos}\varphi \)
- \(\mu \) :
-
Coefficient of viscosity of the fluid [\(\hbox {ML}^{-1}\,\hbox {T}^{-1}\)]
- \(\mu ^*\) :
-
Effective viscosity in the porous medium [\(\hbox {ML}^{-1}\hbox {T}^{-1}\)]
- \(\nu \) :
-
Kinematic viscosity of the fluid [\(\hbox {L}^{2}\hbox {T}^{-1}\)]
- \(\nu ^*\) :
-
Effective kinematic viscosity of the fluid in the medium [\(\hbox {L}^{2}\hbox {T}^{-1}\)]
- \(\rho \) :
-
Fluid density [\(\hbox {ML}^{-3}\)]
- \(\rho _{0}\) :
-
Fluid density when the temperature is \(\hbox {T}_{0}\,[\hbox {ML}^{-3}]\)
- \(\psi \) :
-
Stream function [\(\hbox {L}^{3}\hbox {T}^{-1}\)]
- \(\varPsi \) :
-
Non-dimensional stream function
- \(\varPsi _{0}\) :
-
First convective correction to the flow field
- \(\sigma \) :
-
Heat capacity ratio, \(\varepsilon + (1- \varepsilon )(\rho \hbox {c}_{\mathrm{p}})_{s}/(\rho \hbox {c}_{\mathrm{p}})_{f}\)
- \(\omega \) :
-
\(\lambda (\hbox {s} + 1)^{1/2}\)
- \(\chi _{n}\) :
-
Function of \((R,s)\) (Eq. 18)
- \(\tau \) :
-
Non-dimensional time
- \(\varTheta \) :
-
Non-dimensional temperature
- \(\varTheta _{0}\) :
-
Conduction state solution for the temperature distribution
- \(\varPhi _{n}\) :
-
Laplace transform of \(\varPsi _{n}(R,\tau )\)
- \(\varpi \) :
-
Absolute difference between the temperature at the deepest point of the hemisphere and the free surface
- \({\nabla }\) :
-
The gradient vector
- \({\nabla }^{2}\) :
-
The Laplacian operator
- \(n\) :
-
\(n\)th term
- f:
-
Fluid phase
- s:
-
Solid phase
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Acknowledgments
The author is thankful to the referees for their many useful suggestions which led to a definite improvement of the paper. The support of the Principal, MAM College of Engineering (Anna University), Tiruchirapalli is gratefully acknowledged.
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Ganapathy, R. Thermal Convection in a Non-darcy Hemispherical Porous Medium. Transp Porous Med 105, 105–115 (2014). https://doi.org/10.1007/s11242-014-0362-z
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DOI: https://doi.org/10.1007/s11242-014-0362-z