Sharp nonlinear stability for centrifugal filtration convection in magnetizable media

Saravanan, S.; Brindha, D.

doi:10.1103/physreve.84.056318

Cited by 3 publications

(9 citation statements)

References 16 publications

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“…Here we restrict our attention to small disturbances alone and hence consider the linearized equations which are then subjected to the normal mode technique (Drazin and Reid [18]). As the analysis has already been discussed in our previous works [12, 17] we shall just have a brief look at it.…”

Section: Stability Analysismentioning

confidence: 99%

“…We then introduce some scales for the dimensional physical quantities as in refs. [12, 17], confine to two dimensional perturbations and separate

v_{x} false(x, z false) = U false(x false) e^{σ t + i k z}, T false(x, z false) = Θ false(x false) e^{σ t + i k z}, ϕ false(x, z false) = Φ false(x false) e^{σ t + i k z}

where U , Θ, Φ are the amplitudes of the respective perturbations with growth rate σ and wave number k . The linearized system of perturbation equations then reduces to

\begin{matrix} true \\ right (() \frac{1}{ζ} D^{2} - k^{2}) U - [R m + R c false(x - x_{0} false)] k^{2} normalΘ + R m k^{2} D normalΦ & = & left 0, \\ right ([] (D^{2} - k^{2} η) - P r σ) normalΘ + (N M_{2} - 1) U + P r M_{2} σ D normalΦ & = & left 0, \\ right false(D^{2} - M_{1} k^{2} false) normalΦ - D normalΘ & = & left 0 \end{matrix}

where

R c = \frac{{(ρ C)}_{2} α β ω^{2} K_{x} L^{3}}{ν k_{x}}

is the centrifugal Rayleigh number,

P r = {(ρ C)}_{1} ν / k_{x}

the Prandtl number,

R m = {(ρ C)}_{2} μ_{0<...>}

…”

Section: Stability Analysismentioning

confidence: 99%

“…We now nondimensionalize the system (5), using the known scales [12],[17] together with

μ ν / K_{x}

for pressure, as

\begin{matrix} true \\ right \nabla \cdot \overset{v}{true} & = & left 0, \\ right v_{x} \overset{i}{true} + \frac{1}{ζ} v_{z} \overset{k}{true} & = & left - \frac{1}{N} \nabla P - [R m + R c false(x - x_{0} false)] T \overset{i}{true} + R m \frac{\partial ϕ}{\partial x} \overset{i}{true} - R m P r T \nabla \frac{\partial ϕ}{\partial x} \\ right & left + \frac{R m P r}{1 + χ} ([] χ \frac{\partial ϕ}{\partial x} \nabla \frac{\partial ϕ}{\partial x} + \frac{M_{0}}{H_{0}} \frac{\partial ϕ}{\partial z} \nabla \frac{\partial ϕ}{\partial z}), \\ right \frac{\partial T}{\partial t} & = & left \frac{1}{P r} (N M_{2} - 1) v_{x} - ([] v_{x} \frac{\partial T}{\partial x} + v_{z} \frac{\partial T}{\partial z}) + \frac{1}{P r} ([] \frac{\partial^{2} T}{\partial x^{2}} + η \frac{\partial^{2} T}{\partial z^{2}}) \\ right & left + M_{2} ([] N v_{x} \frac{\partial^{2} ϕ}{\partial x^{2}} + N v_{z} \frac{\partial^{2} ϕ}{\partial z \partial x} + \frac{\partial^{2} ϕ}{\partial t \partial x}), \\ right M_{1} \frac{\partial^{2} ϕ}{\partial z} \end{matrix}

…”

Section: Stability Analysismentioning

confidence: 99%

“…They showed that the combined effects are identical to those of the case in which gravity effects are neglected. Recently we made nonlinear analyses of a centrifugally driven convective flow in a magnetic fluid filled porous medium involving thermal non‐equilibrium phenomenon [12, 13].…”

Section: Introductionmentioning

confidence: 99%

“…Hence, we shall now make a more complete investigation of this problem by performing a nonlinear analysis which in turn enables one to understand possible convective instabilities that could occur in magnetic fluid rotary shaft seals and related equipment. For this we pursue the ideas which we elaborated in our earlier work [12].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Anisotropic thermomagnetic effects on the universal stability of a diffusive state in porous media

2021

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A nonlinear study is made to find the influence of transverse anisotropy on the onset of thermomagnetic convection in a homogeneous and rotating porous medium. The medium is filled with a magnetic fluid and is subjected to centrifugal force with alternating directions. Unconditional nonlinear stability bounds are found by exploiting the variational principles. Effects of various parameters on the stability threshold are discussed. The overpredicting nature of the linear theory is demonstrated by isolating the subcritical region which cannot be obtained a priori.

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Section: Stability Analysismentioning

confidence: 99%

“…We then introduce some scales for the dimensional physical quantities as in refs. [12, 17], confine to two dimensional perturbations and separate

v_{x} false(x, z false) = U false(x false) e^{σ t + i k z}, T false(x, z false) = Θ false(x false) e^{σ t + i k z}, ϕ false(x, z false) = Φ false(x false) e^{σ t + i k z}

where U , Θ, Φ are the amplitudes of the respective perturbations with growth rate σ and wave number k . The linearized system of perturbation equations then reduces to

\begin{matrix} true \\ right (() \frac{1}{ζ} D^{2} - k^{2}) U - [R m + R c false(x - x_{0} false)] k^{2} normalΘ + R m k^{2} D normalΦ & = & left 0, \\ right ([] (D^{2} - k^{2} η) - P r σ) normalΘ + (N M_{2} - 1) U + P r M_{2} σ D normalΦ & = & left 0, \\ right false(D^{2} - M_{1} k^{2} false) normalΦ - D normalΘ & = & left 0 \end{matrix}

where

R c = \frac{{(ρ C)}_{2} α β ω^{2} K_{x} L^{3}}{ν k_{x}}

is the centrifugal Rayleigh number,

P r = {(ρ C)}_{1} ν / k_{x}

the Prandtl number,

R m = {(ρ C)}_{2} μ_{0<...>}

…”

Section: Stability Analysismentioning

confidence: 99%

“…We now nondimensionalize the system (5), using the known scales [12],[17] together with

μ ν / K_{x}

for pressure, as

\begin{matrix} true \\ right \nabla \cdot \overset{v}{true} & = & left 0, \\ right v_{x} \overset{i}{true} + \frac{1}{ζ} v_{z} \overset{k}{true} & = & left - \frac{1}{N} \nabla P - [R m + R c false(x - x_{0} false)] T \overset{i}{true} + R m \frac{\partial ϕ}{\partial x} \overset{i}{true} - R m P r T \nabla \frac{\partial ϕ}{\partial x} \\ right & left + \frac{R m P r}{1 + χ} ([] χ \frac{\partial ϕ}{\partial x} \nabla \frac{\partial ϕ}{\partial x} + \frac{M_{0}}{H_{0}} \frac{\partial ϕ}{\partial z} \nabla \frac{\partial ϕ}{\partial z}), \\ right \frac{\partial T}{\partial t} & = & left \frac{1}{P r} (N M_{2} - 1) v_{x} - ([] v_{x} \frac{\partial T}{\partial x} + v_{z} \frac{\partial T}{\partial z}) + \frac{1}{P r} ([] \frac{\partial^{2} T}{\partial x^{2}} + η \frac{\partial^{2} T}{\partial z^{2}}) \\ right & left + M_{2} ([] N v_{x} \frac{\partial^{2} ϕ}{\partial x^{2}} + N v_{z} \frac{\partial^{2} ϕ}{\partial z \partial x} + \frac{\partial^{2} ϕ}{\partial t \partial x}), \\ right M_{1} \frac{\partial^{2} ϕ}{\partial z} \end{matrix}

…”

Section: Stability Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Anisotropic thermomagnetic effects on the universal stability of a diffusive state in porous media

2021

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show abstract

Internal Natural Convection: Heating from Below

Nield

Bejan

2017

Convection in Porous Media

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Onset of centrifugal filtration convection: departure from thermal equilibrium

Saravanan

Brindha

2013

Proc. R. Soc. A.

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This paper deals with convective instability in a fluidsaturated, rotating porous layer subject to alternating direction of the centrifugal body force, when the layer fails to exhibit thermal equilibrium. The Darcy model is used to describe the flow, and a twofield model is used to take care of the energy exchange. The normal mode approach of the linear theory and the energy approach of the nonlinear theory are used to find the stability characteristics. Unconditional and sharp nonlinear thresholds are found. The study brings out the failure of the linear theory in describing the instability in most parts of the parameter space of interest where possible subcritical instabilities may arise. The stability boundaries are presented graphically and it is found that the interphase heat transfer coefficient has a significant effect in the thermal non-equilibrium regime.

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Sharp nonlinear stability for centrifugal filtration convection in magnetizable media

Cited by 3 publications

References 16 publications

Anisotropic thermomagnetic effects on the universal stability of a diffusive state in porous media

Anisotropic thermomagnetic effects on the universal stability of a diffusive state in porous media

Internal Natural Convection: Heating from Below

Onset of centrifugal filtration convection: departure from thermal equilibrium

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