Abstract
We present a model for convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from below, the thermosolutal convection problem. Instability thresholds are calculated for thermal convection with a dissolved salt field in a complex viscoelastic fluid of Navier–Stokes–Voigt type. The Kelvin–Voigt parameter is seen to play a very important role in acting as a stabilizing agent when the convection is of oscillatory type. The quantitative size of this effect is displayed. Nonlinear stability is also discussed, and it is briefly indicated how the global nonlinear stability limit may be increased, although there still remains a region of potential sub-critical instability, especially when the Kelvin–Voigt parameter increases.
Highlights
The Navier–Stokes equations for the flow of a linearly viscous, incompressible fluid are well known throughout the fields of applied mathematics and engineering
We have developed an analysis for the thermosolutal convection problem for a Navier– Stokes–Voigt viscoelastic fluid
The thermal convection problem for a Navier–Stokes–Voigt fluid without a salt field was suggested by Sukacheva and Matveeva [21], Sukacheva and Kondyukov [23]
Summary
The Navier–Stokes equations for the flow of a linearly viscous, incompressible fluid are well known throughout the fields of applied mathematics and engineering. For a Navier–Stokes fluid the response of stress to change in the velocity gradient is instantaneous This special response is not enjoyed by all real life fluids and, in particular, by many classes of viscoelastic and complex fluids. These fluids possess a stress which does not react instantaneously and instead they remember the velocity gradient history. Such fluids possess fading memory where the history dependence diminishes as the time advances into the past. Theoretical work on such fluids is vast, see e.g. Amendola and Fabrizio [1], Amendola et al [2], Anand et al [3], Anand and Christov [4], Christov and Christov [5], Fabrizio et al [6], Franchi et al [7,8,9], Gatti et al [10], Applied Mathematics & Optimization (2021) 84:2587–2599
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