On the linear growth rate of disturbance in double-diffusive Navier–Stokes–Voigt fluid with uniform rotation

linear growth rate of disturbance in double-diffusive Navier–Stokes–Voigt fluid

Abstract

In the present paper linear mathematical analysis is performed for a rotatory incompressible Navier–Stokes–Voigt (NSV) fluid. It is analytically proved that the principle of the exchange of stabilities in a rotatory incompressible Navier–Stokes–Voigt fluid is valid in the regime $ \frac{{{R}_{s}}\Pr Le}{2{{\pi }^{4}}}+\frac{{{T}_{a}}}{{{\pi }^{2}}}\left( \frac{1}{{{\pi }^{2}}}+\lambda \right)\le 1.$ Further, upper bounds for the linear growth rate of disturbance are also obtained. It is mathematically established that upper bounds for the linear growth rate $\sigma ={{\sigma }_{r}}+i{{\sigma }_{i}} $ (${{\sigma }_{r}}$ and ${{\sigma }_{i}}$are the real and imaginary parts of $\sigma $ , respectively) of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude, lies within a semicircle in the right half of the ${{\sigma }_{r}}{{\sigma }_{i}}-$ plane, whose centre is at the origin and radius $=max\left( \sqrt{\frac{{{R}_{s}}}{\Pr Le(1+2{{\pi }^{2}}\lambda )}} \right.,\left. \sqrt{\frac{{{T}_{a}}}{\left( 1+{{\pi }^{2}}\lambda \right)}} \right)$, where ${{R}_{s}}$ is concentration Rayleigh number, and and $Le$ is the Lewis number, $Pr $ is the thermal Prandtl number, ${{T}_{a}}$ is the Taylor number and $\lambda $ is the Navier-Stokes-Voigt parameter. The results derived herein are uniformly valid for any combination of rigid and free boundaries.