Abstract
The stability and onset of convection in a rotating fluid saturated porous layer subject to a centrifugal body force and placed at an offset distance from the center of rotation is investigated analytically. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number for different values of the parameter representing the dimensionless offset distance from the center of rotation. At the limit of an infinite distance from the center of rotation the results are identical to the convection resulting from heating a porous layer from below subject to the gravitational body force. At the other limit, when the parameter controlling the offset distance approaches zero, the results converge to previously found solutions for the convection in a porous layer adjacent to the axis of rotation. The results provide the stability map for all positive values of the parameter controlling the offset distance from the center of rotation, hence bridging the gap between the two extreme limit cases.
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Abbreviations
- H :
-
the front aspect ratio of the porous layer, equals H */L *
- W :
-
the top aspect ratio of the porous layer, equals W s*/L s*
- ê x :
-
unit vector in the x-direction
- ê y :
-
unit vector in the y-direction
- ê z :
-
unit vector in the z-direction
- ê n :
-
unit vector normal to the boundary, positive outwards
- H * :
-
the height of the layer
- k * :
-
permeability of the porous domain
- L * :
-
the length of the porous layer
- M f :
-
a ratio between the heat capacity of the fluid and the effective heat capacity of the porous domain
- p :
-
reduced pressure generalized to include the constant component of the centrifugal term (dimensionless)
- q :
-
dimensionless filtration velocity vector, equals uê x + vê y + wê y
- Raω :
-
porous media centrifugal Rayleigh number related to the contribution of the horizontal location within the porous layer to the centrifugal acceleration, equals β *ΔT c ω 2* L * 2 k * M f/αe* v *
- Raω :
-
porous media centrifugal Rayleigh number related to the contribution of the offset distance from the rotation center to the centrifugal acceleration, equals β*ΔTcω 2* L 2* k*Mf/αe*v*
- R :
-
scaled centrifugal Rayleigh number, equals Raω/π2
- R 0 :
-
scaled centrifugal Rayleigh number, equals Raω0/π2
- T :
-
dimensionless temperature, equals (T * − T C )/T H − T C )
- T C :
-
coldest wall temperature
- T H :
-
hottest wall temperature
- u :
-
horizontal x component of the filtration velocity
- v :
-
horizontal y component of the filtration velocity
- w :
-
vertical component of the filtration velocity
- W * :
-
the width of the layer
- x 0 :
-
the dimensionless offset distance from the rotation center, equals x 0*/L *
- x :
-
horizontal length coordinate
- y :
-
horizontal width coordinate
- z :
-
vertical coordinate
- α:
-
a parameter related to the wave number, equals κ2/κ2
- α e* :
-
effective thermal diffusivity
- \:
-
a parameter, equals (1 + η/2)
- β* :
-
thermal expansion coefficient
- λ2 :
-
a parameter, equals 256η2/81χ4
- δij :
-
Kronecker delta function
- η:
-
the reciprocal of the offset distance from the rotation center, equals 1/x 0 = Raω/Raω0
- ϕ:
-
porosity
- ω* :
-
angular velocity of the rotating box
- ν* :
-
fluid's dynamic viscosity
- κ:
-
wave number
- µ* :
-
fluid's dynamic viscosity
- ψ:
-
stream function
- ΔT c :
-
characteristic temperature difference
- *:
-
dimensional values
- c :
-
characteristic values
- cr:
-
critical values
- C :
-
related to the coldest wall
- H :
-
related to the hottest wall
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Vadasz, P. Stability of free convection in a rotating porous layer distant from the axis of rotation. Transp Porous Med 23, 153–173 (1996). https://doi.org/10.1007/BF00178124
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DOI: https://doi.org/10.1007/BF00178124