The effect of viscous heating on the stability of Taylor–Couette flow

Abstract

The influence of viscous heating on the stability of Taylor–Couette flow is investigated theoretically. Based on a linear stability analysis it is shown that viscous heating leads to significant destabilization of the Taylor–Couette flow. Specifically, it is shown that in the presence of viscous dissipation the most dangerous disturbances are axisymmetric and that the temporal characteristic of the secondary flow is very sensitive to the thermal boundary conditions. If the temperature difference between the two cylinders is small, the secondary flow is stationary as in the case of isothermal Taylor–Couette flow. However, when the temperature difference between the two cylinders is large, time-dependent secondary states are predicted. These linear stability predictions are in agreement with the experimental observations of White & Muller (2000) in terms of onset conditions as well as the spatiotemporal characteristics of the secondary flow. Nonlinear stability analysis has revealed that over a broad range of operating conditions, the bifurcation to the time-dependent secondary state is subcritical, while stationary states result as a consequence of supercritical bifurcation. Moreover, the supercritically bifurcated stationary state undergoes a secondary bifurcation to a time-dependent flow. Overall, the structure of the time-dependent state predicted by the analysis compares very well with the experimental observations of White & Muller (2000) that correspond to slowly moving vortices parallel to the cylinder axis. The significant destabilization observed in the presence of viscous heating arises as the result of the coupling of the perturbation velocity and the base-state temperature gradient that gives rise to fluctuations in the radial temperature distribution. Due to the thermal sensitivity of the fluid these fluctuations greatly modify the fluid viscosity and reduce the dissipation of disturbances provided by the viscous stress terms in the momentum equation.

Effect of inertia on thermoelastic flow instability

“…3(b) and (c). This implies that for the lowest (O(1)) We c values seen in the stability boundary, the corresponding Reynolds number Re c = We c /E c , is O (10). Clearly, under these conditions, the creeping flow assumption is not valid.…”

“…Moreover, the thermally induced modifications of the viscosity can significantly alter the local dissipation rate of velocity perturbations. For viscous, thermally sensitive liquids, gradients in viscosity thus created can lead to centrifugal instabilities at Re c values much lower than that for the isothermal flow [10][11][12][13]. For viscoelastic liquids, this can be further aggravated by thermoelastic mechanisms.…”

“…Even in the absence of elastic forces, i.e. for Newtonian liquids, the reduction in local viscosity caused by temperature increase can reduce the critical shear rate dramatically [5,[10][11][12][13], with the critical Reynolds number Re c decreasing with increasing Na. For relatively narrow gap width, a scaling law of the form Re c (δ) 1/2 Na n = K has been shown to exist, where δ is the ratio of the gap width to inner cylinder radius R 1 , and n and K are constants that depend on the temperature difference between the inner and outer cylinders [14].…”

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

“…Similarly, Newton's law of viscosity says that the viscous stress must be linearly proportional to the rate of strain, but the viscosity coefficient can depend non-linearly on the temperature and specific volume or pressure ( [42], pp. [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

“…Then theory is needed to project the information acquired in simple experiments to the complex reality of the process. The need for robust theories is even more pressing when tackling more complex transport phenomena, like coupled flow and heat transfer [17][18][19][20][21][22] or mass transfer [23][24][25][26][27][28][29][30]; experimental evidence on such processes is still limited [31][32][33][34][35][36].…”