Effect of Temperature Upon Double Diffusive Instability in Navier–Stokes–Voigt Models with Kazhikhov–Smagulov and Korteweg Terms

Abstract

We present models for convection in a mixture of viscous fluids when the layer is heated from below and simultaneously the pointwise volume concentration of one of the fluids is heavier below. This configuration produces a problem of competitive double diffusion since heating from below promotes instability, but the greater density of fluid below is stabilizing. The fluids are of linear viscous type which may contain Kelvin–Voigt terms, but density gradients due to the mixture appear strongly in the governing equations. The density gradients give rise to Korteweg stresses, but may also be described by theory due to Kazhikhov and Smagulov. The systems of equations which appear are thus highly nonlinear. The instability surface threshold is calculated and this is found to have a complex nonlinear shape, very different from the linear ones found in classical thermohaline convection in a Navier–Stokes fluid. It is shown that the Kazhikhov–Smagulov terms, Korteweg terms and Kelvin–Voigt term play a key role in acting as stabilizing agents but the associated effect is very nonlinear. Quantitative values of the instability surface are displayed showing the effect Korteweg terms, Kazhikhov–Smagulov terms, and the Kelvin Voigt term have. The nonlinear stability problem is addressed by means of a generalized energy theory deriving different results depending on which underlying theory is employed.