Short-wavelength local instabilities of a circular Couette flow with radial temperature gradient

We perform a linearized local stability analysis for short-wavelength perturbations of a circular Couette flow with a radial temperature gradient. Axisymmetric and non-axisymmetric perturbations are considered and both the thermal diffusivity and the kinematic viscosity of the fluid are taken into account. The effect of asymmetry of the heating both on centrifugally unstable flows and on the onset of instabilities of centrifugally stable flows, including flows with a Keplerian shear profile, is thoroughly investigated. It is found that an inward temperature gradient destabilizes the Rayleigh-stable flow either via Hopf bifurcation if the liquid is a very good heat conductor or via steady state bifurcation if viscosity prevails over the thermal conductance.

Determining True Material Constants of Semisolid Slurries from Rotational Rheometer Data

In this work we revisit the issue of obtaining true material constants for semisolid slurries. Therefore, we consider the circular Couette flow of Herschel-Bulkley fluids. We first show how true constants can be obtained using an iterative procedure from experimental data to theory and vice versa. The validity of the assumption that the rate-of-strain distributions across the gap share a common point is also investigated. It is demonstrated that this is true only for fully-yielded Bingham plastics. In other cases, e.g., for partially-yielded Bingham plastics or fully-yielded Herschel-Bulkley materials, the common point for the fully-yielded Bingham case provides a good approximation for determining the material constants only if the gap is sufficiently small. It can thus be used to simplify the iterative procedure in determining the material constants.

Extension to nonlinear stability theory of the circular Couette flow

A nonlinear stability analysis of the viscous circular Couette flow to axisymmetric finite-amplitude perturbations under axial periodic boundary conditions is developed. The analysis is based on investigating the properties of a reduced Arnol’d energy-Casimir function $\mathscr{A}_{rd}$ of Wang (Phys. Fluids, vol. 2, 2009, 084104). A weighted kinetic energy of the perturbation, which has a form of ${\rm\Delta}\mathscr{A}_{rd}$, the difference between the reduced Arnol’d function and its base flow value, is used as a Lyapunov function. We show that all the inviscid flow effects as well as all the viscous-dependent terms that are related to the flow boundaries vanish. The evolution of ${\rm\Delta}\mathscr{A}_{rd}$ depends only on the viscous effects of the perturbation’s dynamics inside the flow domain. The requirement for the temporal decay of ${\rm\Delta}\mathscr{A}_{rd}$ leads to two novel sufficient conditions for the nonlinear stability of the circular Couette flow in response to axisymmetric perturbations. The linearized version of these conditions for infinitesimally small perturbations recovers the recent linear stability results by Kloosterziel (J. Fluid Mech., vol. 652, 2010, pp. 171–193). By examining the nonlinear stability conditions, we establish a definite operational region of the viscous circular Couette flow that is independent of the fluid viscosity. In this region of operation, the flow is nonlinearly stable in response to perturbations of any size, provided that the initial total circulation function is above a minimum level determined by the operational conditions of the base flow. Comparisons with historical studies show that our results shed light on the experimental measurements of Wendt (Ing.-Arch., vol. 4, 1933, pp. 577–595) and extend the classical nonlinear stability results of Serrin (Arch. Rat. Mech. Anal., vol. 3, 1959, pp. 1–13) and Joseph & Hung (Arch. Rat. Mech. Anal., vol. 44, 1971, pp. 1–22). When the flow is nonlinearly stable and evolves axisymmetrically for all time, then it always decays asymptotically in time to the circular Couette flow determined uniquely by the set-up of the rotating cylinders. Finally, we derive upper-bound estimates on the decay rate of finite-amplitude perturbations for the solid-body rotation flow between two coaxial rotating cylinders and for the circular Couette flow. We demonstrate via numerical simulations that the theoretical upper bound is relevant to the dynamics of various axisymmetric perturbations tested, where it is strictly obeyed. This present study provides new physical insights into a classical flow problem that was studied for many decades.