Convective instabilities of Maxwell-Cattaneo fluids

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Abstract

Motivated by the need to understand better the dynamics of non-Fourier fluids, we examine the linear and weakly nonlinear stabilities of a horizontal layer of fluid obeying the Maxwell-Cattaneo relationship of heat flux and temperature using three different forms of the time derivative of the heat flux. Linear stability mode regime diagrams in the parameter plane have been established and used to summarize the linear instabilities. The energy balance of the system is used to identify the mechanism by which the Maxwell-Cattaneo effect (i) introduces overstability, (ii) leads to preferred stationary modes with the critical Rayleigh and wavelengths either both increasing or both decreasing, (iii) gives rise to instabilities in a layer heated from above, and (iv) enhances heat transfer. A formal weakly nonlinear analysis leads to evolution equations for the amplitudes of linear instability modes. It is shown that the amplitude of the stationary mode obeys an equation of the Landau-Stuart type. The two equally excitable overstable modes obey two equations of the same type coupled by an interaction term. The evolution of the different amplitudes leads to supercritical stability, supercritical instability or subcritical instability depending on the model and parameter values. The results are presented in regime diagrams.

Original languageEnglish
Article number20160712
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume473
Issue number2201
DOIs
Publication statusPublished - May 1 2017

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Keywords

  • Convection
  • Linear stability
  • Maxwell-Cattaneo fluids
  • Nonlinear stability
  • Objective derivative

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

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