Stationary shear flow around fixed and free bodies at finite Reynolds number

Abstract

Numerical solutions of stationary flow resulting from immersion of a single body in simple shear flow are reported for a range of Reynolds numbers. Flows are computed using finite-element methods. Comparisons to results of asymptotic low-Reynolds-number theory, experimental study, and other numerical techniques are provided. Results are presented primarily for isotropic bodies, i.e. the circular cylinder and sphere, for both of which the two conditions of a torque-free (freely-rotating) and fixed body are investigated. Conditions studied for the sphere are $0 \,{<}\, \hbox{\it Re} \,{\le}\, 100$, and for the circular cylinder $0 \,{<}\, \hbox{\it Re} \,{\le}\, 500$, with the shear-flow Reynolds number defined as $\hbox{\it Re}\,{=}\, \gammadot a^2/\nu$; $\gammadot$ is the shear rate of the Cartesian simple shear flow $\bu \,{=}\, (\gammadot y, 0, 0)$, $a$ is the cylinder or sphere radius, and $\nu$ is the kinematic viscosity of the fluid. In the torque-free case, the rotation rate of the body decreases with increasing $\hbox{\it Re}$. Qualitative dependence, seen in the $\hbox{\it Re} \,{=}\,0$ fluid flow field, upon whether the body is fixed against rotation or torque-free vanishes as $\hbox{\it Re}$ increases and the fluid flow is more similar to that around the $\hbox{\it Re}\,{=}\,0$ fixed body: the influence of rotation of the body and the associated closed streamlines are confined to a narrow layer about the body for $\hbox{\it Re}\,{>}\,O(1)$. Separation of the boundary layer is observed in the case of a fixed cylinder at $\hbox{\it Re} \,{\approx}\, 85$, and for a fixed sphere at $\hbox{\it Re} \,{\approx}\, 100$; similar separation phenomena are observed for a freely rotating cylinder. The surface stress and its symmetric first moment (the stresslet) are presented, with the latter providing information on the particle contribution to the mixture rheology at finite $\hbox{\it Re}$. Stationary flow results are also presented for elliptical cylinders and oblate spheroids, with observation of zero-torque inclinations relative to the flow direction which depend upon the aspect ratio, confirming and extending prior findings.