Thermo-mechanical instabilities in Dean and Taylor–Couette flows: mechanisms and scaling laws

Abstract

The influence of fluid thermal sensitivity on the centrifugal flow instabilities in pressure-driven (Dean) and drag-driven (Taylor–Couette) Newtonian shear flows is investigated. Thermal effects are caused by viscous heating or an externally imposed temperature difference between the outer and inner cylinders, $\uDelta T^{ \ast }$, or a combination of both. In all cases considered, the maximum temperature difference within the gap is small enough such that the base-state velocity profile and consequently the distribution of angular momentum are practically unchanged from those in the isothermal flow. The base-state temperature gradient can be approximated as a linear superposition of $\uDelta T^{ \ast }/d$, where $d$ is the gap width, and that caused by viscous heating. Numerical linear stability analysis shows that when $\uDelta T^{ \ast }\,{=}\, 0$, viscous heating causes the critical Reynolds number, Re$_{c}$, to be greatly reduced when the Nahme number, defined as the product of the Brinkman number, Br, and the dimensionless activation energy associated with the fluid viscosity, $\varepsilon $, is $O(\alpha^{2}/\hbox{\it Pr})$ where $\alpha $ and Pr denote the dimensionless critical axial wavenumber and Prandtl number respectively. Since $\alpha^{2}$ is $O(10)$ and typical Pr values for thermal sensitive liquids could be $O(10^{4})$, appreciable flow destabilization occurs even when Na is $O(10^{ - 3})$. In the absence of viscous heating, an externally imposed temperature gradient can lead to significant reduction in Re$_{c}$ when $S \,{ \equiv}\,(\varepsilon\uDelta T^{ \ast })/T_{1}^{\ast }\,{ <}\, 0$ and $\vert S\vert $ is $O(\alpha^{2}/\hbox{\it Pr})$, where $T_{1}^{\ast }$ denotes the temperature of the inner cylinder. The numerical linear stability analysis results are explained based on a simplified model derived from the linearized governing equations by invoking the narrow-gap approximation. This model shows that the thermo-mechanical coupling, arising from the convection of the base-state temperature gradient by radial velocity perturbation, amplifies the temperature fluctuations within the flow by a factor proportional to Pe/$\alpha^{2}$ where Pe denotes the Péclet number. This results in the reduction of local viscosity. Hence, the rate of dissipation of the velocity perturbations decreases causing the centrifugal instability to occur at lower values of the Reynolds number compared to the isothermal flow. Thermo-mechanical destabilization caused by vis- cous heating for $\uDelta T^{ \ast }\,{=}\, 0$ can be quantified by a scaling law of the form $\Lambda \,{=} [1+ \hbox{\it Pr}\,c_{1}$Na/$\alpha^{2}]^{ - 1 / 2}$ where $\Lambda $ is the ratio of the critical Reynolds number of the non-isothermal flow to that of the isothermal one and $c_{1}$ is a flow-dependent constant. Similarly, in the absence of viscous heating and for $\uDelta T^{ \ast }\,{<}\, 0$, $\Lambda\,{=}\, [1\,{+}\,\hbox{\it Pr}c_{2} S/\alpha^{2}]^{ - 1 / 2}$, where $c_{2}$ is a flow-dependent constant. When $\uDelta T^{ \ast }\,{>}\, 0$ and viscous heating are present, a numerical linear stability analysis shows that $\Lambda\,{\propto}\,\hbox{\it Na}^{k}$ where $k \,{<}\,0$ and it is dependent on $\uDelta T^{ \ast }$ and the flow type. Finally, we perform a nonlinear stability analysis for the Dean flow which shows that the bifurcation is supercritical for both stationary and time-dependent modes of instability.