Thermosolutal Convection with a Navier–Stokes–Voigt Fluid

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.

Competitive Double Diffusive Convection in a Kelvin–Voigt Fluid of Order One

“…The two-dimensional surface of instability for oscillatory convection is found to not be a plane in (Ra, Rs, ) space, for fixed γ . When = 0 oscillatory convection may occur as shown in Straughan [47] and then the oscillatory convection branches appear to be straight lines, see Straughan [47], Fig. 1, with positions depending on λ.…”

“…We now denote by v(x, t), T (x, t), C(x, t), p(x, t) the velocity, temperature, concentration of a dissolved solute, and pressure at position x and time t of a body of fluid. When this fluid is of Kelvin-Voigt order one then the governing equations may be taken to be, Sukacheva and Matveeva [27], Straughan [47],…”

“…It is noteworthy that the Kelvin-Voigt order one theory contains two extra parameters ξ 1 andγ which are not present in the Kelvin-Voigt order zero theory (also known as Navier-Stokes-Voigt theory), cf. Straughan [47], and these extra parameters yield a more accurate fit to experiments and provide a more refined fit to the fading memory behaviour, see e.g. Greco and Marano [50].…”

“…Likewise, without the term, this salt effect leads to an increase in the Rayleigh number stability threshold as C increases. It can also lead to oscillatory convection, see Straughan [47], figure 1. Oscillatory convection is increasingly important in multicomponent hydrodynamic systems as is seen in the recent work of Rionero [65], see also Pearlstein [60], Pearlstein et al [61], Straughan and Walker [62], Falsaperla et al [63].…”

“…When the convection Rayleigh number is increased the instability Rayleigh number boundary also increases, see e.g. Joseph [46], Mulone [39], Straughan [47]. However, for a Kelvin-Voigt fluid of order one the viscoelastic term in the momentum equation also has the effect of increasing the critical instability Rayleigh number as the appropriate viscoelastic coefficient is increased.…”