Stability and instability of Taylor–Couette flows of a Bingham fluid

We consider in detail the Taylor–Couette problem for a Bingham fluid, presenting a range of analytical and computational results. First, for co-rotating cylinders it is known that the critical inner cylinder Reynolds number $\hbox{\it Re}_{1,c}$, does not increase monotonically with the Bingham number $B$, over a range of small to moderate $B$. It is the only situation that we know of where a yield stress fluid flow is less stable that the corresponding Newtonian fluid flow. This effect was discovered independently by Landry (2003, MSc thesis) and Peng & Zhu (J. Fluid Mech. vol. 512, 2004, p. 21), but the mechanism has not been explained. Here we show that the decrease in critical Reynolds number is due to an increase (at small $B$) in the rate of strain of the basic flow, which amplifies the transfer of energy from the basic flow to the perturbation, via the inertial terms in the energy equation. At larger $B$, the yielded region contracts and the inertial energy transfer is bounded by the yield stress dissipation. We next consider the effects of large $B$. For fixed radius and Reynolds number ratios, we show that for sufficiently large $B$ all basic flows have an unyielded fluid layer attached to the outer wall. For these flows we show that there is a similarity mapping that maps both the basic solution and the linear stability problem onto the stability problem for an outer cylinder of radius equal to the yield surface radius. The Reynolds and Bingham numbers of the transformed problems are smaller than that of the original problem, as is the wavenumber $k$. As $B \,{\to}\, \infty$, the yield surface approaches the inner cylinder, defining a narrow gap limiting problem that differs from the classical narrow gap limit. Via the transformed problem we derive an energy estimate for stability: $\hbox{\it Re}_{1,c} k_c \,{\sim}\, B^{1.5}$ as $B \,{\to}\, \infty$, which compares well with our computed results for a stationary outer cylinder: $\hbox{\it Re}_{1,c} \,{\sim}\, B^{1.25}$ and $k_c \,{\sim}\, B^{0.375}$. We also show how $\hbox{\it Re}_{1,c} \,{\sim}\, B^{1.25}$ can be deduced from a simple order of magnitude analysis, for a stationary outer cylinder. Finally, we consider the second (classical) narrow gap limit in which the radius ratio $\eta$, approaches unity, for fixed $B$ and Reynolds number ratio. We show that $\hbox{\it Re}_{1,c} \,{\gtrsim}\, (k^2[1 + O(B)] + \pi^2)/(1-\eta)^{1/2}$ in this limit.