The stability of second sound waves in a rotating Darcy-Brinkman porous layer in local thermal non-equilibrium

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Abstract

The linear and nonlinear stabilities of second sound waves in a rotating porous Darcy-Brinkman layer in local thermal non-equilibrium are studied when the heat flux in the solid obeys the Cattaneo law. The simultaneous action of the Brinkman effect (effective viscosity) and rotation is shown to destabilise the layer, as compared to either of them acting alone, for both stationary and overstable modes. The effective viscosity tends to favour overstable modes while rotation tends to favour stationary convection. Rapid rotation invokes a negative viscosity effect that suppresses the stabilising effect of porosity so that the stability characteristics resemble those of the classical rotating Benard layer. A formal weakly nonlinear analysis yields evolution equations of the Landau-Stuart type governing the slow time development of the amplitudes of the unstable waves. The equilibrium points of the evolution equations are analysed and the overall development of the amplitudes is examined. Both overstable and stationary modes can exhibit supercritical stability; supercritical instability, subcritical instability and stability are not possible. The dependence of the supercritical stability on the relative values of the six dimensionless parameters representing thermal non-equilibrium, rotation, porosity, relaxation time, thermal diffusivities and Brinkman effect is illustrated as regions in regime diagrams in the parameter space. The dependence of the heat transfer and the mean heat flux on the parameters of the problem is also discussed.

Original languageEnglish
Article number045512
JournalFluid Dynamics Research
Volume49
Issue number4
DOIs
Publication statusPublished - Jul 21 2017

Fingerprint

sound waves
Acoustic waves
Viscosity
viscosity
Heat flux
heat flux
Porosity
porosity
Thermal diffusivity
Nonlinear analysis
thermal diffusivity
Hot Temperature
Relaxation time
convection
relaxation time
heat transfer
diagrams
Heat transfer

Keywords

  • Brinkman effect
  • Cattaneo effect
  • nonlinear evolution
  • Nusselt number
  • rotation
  • stability

ASJC Scopus subject areas

  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Fluid Flow and Transfer Processes

Cite this

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title = "The stability of second sound waves in a rotating Darcy-Brinkman porous layer in local thermal non-equilibrium",
abstract = "The linear and nonlinear stabilities of second sound waves in a rotating porous Darcy-Brinkman layer in local thermal non-equilibrium are studied when the heat flux in the solid obeys the Cattaneo law. The simultaneous action of the Brinkman effect (effective viscosity) and rotation is shown to destabilise the layer, as compared to either of them acting alone, for both stationary and overstable modes. The effective viscosity tends to favour overstable modes while rotation tends to favour stationary convection. Rapid rotation invokes a negative viscosity effect that suppresses the stabilising effect of porosity so that the stability characteristics resemble those of the classical rotating Benard layer. A formal weakly nonlinear analysis yields evolution equations of the Landau-Stuart type governing the slow time development of the amplitudes of the unstable waves. The equilibrium points of the evolution equations are analysed and the overall development of the amplitudes is examined. Both overstable and stationary modes can exhibit supercritical stability; supercritical instability, subcritical instability and stability are not possible. The dependence of the supercritical stability on the relative values of the six dimensionless parameters representing thermal non-equilibrium, rotation, porosity, relaxation time, thermal diffusivities and Brinkman effect is illustrated as regions in regime diagrams in the parameter space. The dependence of the heat transfer and the mean heat flux on the parameters of the problem is also discussed.",
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AU - Eltayeb, I. A.

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N2 - The linear and nonlinear stabilities of second sound waves in a rotating porous Darcy-Brinkman layer in local thermal non-equilibrium are studied when the heat flux in the solid obeys the Cattaneo law. The simultaneous action of the Brinkman effect (effective viscosity) and rotation is shown to destabilise the layer, as compared to either of them acting alone, for both stationary and overstable modes. The effective viscosity tends to favour overstable modes while rotation tends to favour stationary convection. Rapid rotation invokes a negative viscosity effect that suppresses the stabilising effect of porosity so that the stability characteristics resemble those of the classical rotating Benard layer. A formal weakly nonlinear analysis yields evolution equations of the Landau-Stuart type governing the slow time development of the amplitudes of the unstable waves. The equilibrium points of the evolution equations are analysed and the overall development of the amplitudes is examined. Both overstable and stationary modes can exhibit supercritical stability; supercritical instability, subcritical instability and stability are not possible. The dependence of the supercritical stability on the relative values of the six dimensionless parameters representing thermal non-equilibrium, rotation, porosity, relaxation time, thermal diffusivities and Brinkman effect is illustrated as regions in regime diagrams in the parameter space. The dependence of the heat transfer and the mean heat flux on the parameters of the problem is also discussed.

AB - The linear and nonlinear stabilities of second sound waves in a rotating porous Darcy-Brinkman layer in local thermal non-equilibrium are studied when the heat flux in the solid obeys the Cattaneo law. The simultaneous action of the Brinkman effect (effective viscosity) and rotation is shown to destabilise the layer, as compared to either of them acting alone, for both stationary and overstable modes. The effective viscosity tends to favour overstable modes while rotation tends to favour stationary convection. Rapid rotation invokes a negative viscosity effect that suppresses the stabilising effect of porosity so that the stability characteristics resemble those of the classical rotating Benard layer. A formal weakly nonlinear analysis yields evolution equations of the Landau-Stuart type governing the slow time development of the amplitudes of the unstable waves. The equilibrium points of the evolution equations are analysed and the overall development of the amplitudes is examined. Both overstable and stationary modes can exhibit supercritical stability; supercritical instability, subcritical instability and stability are not possible. The dependence of the supercritical stability on the relative values of the six dimensionless parameters representing thermal non-equilibrium, rotation, porosity, relaxation time, thermal diffusivities and Brinkman effect is illustrated as regions in regime diagrams in the parameter space. The dependence of the heat transfer and the mean heat flux on the parameters of the problem is also discussed.

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