Shears and vortices of rotational couette flow in a cylindrical gap with radial injection and suction

Abstract

We examine the shear states and vortices of rotational Couette flows with radial injection and suction. The gap of finite radius and infinite axial length lying between two concentric cylinders is assumed to be filled with incompressible Newtonian viscous fluids. To this goal, a rectilinear injected Couette flow is briefly reviewed as a preliminary to a rotational injected Couette flow. After making clear the differences among a simple shear, a pure shear, and generic shears, we formulate a single normalized shear parameter along any curve on a two-dimensional plane. By this way, the relative roles of stretch, shear, and vorticity are clarified through a suitable three-part decomposition of a velocity gradient tensor. For both rectilinear and rotating Couette flows, a certain material fluid particle spends a finite time while traveling from one boundary of injection to the other boundary of suction. We have solved material dynamics for both types of Couette flows to show how a certain fluid particle undergoes various shear states along a curved material trajectory. Such varied shear states are then compared with the shear states found from conventional spatial dynamics. In addition, the roles of enstropy and energy-density gradient are freshly examined from the viewpoints of shear states. The resulting relationships among stretch, shear, and vorticity for the injected rotational Couette flow turn out to shed light on what are going on in realistic flow configurations such as fountain flows, sink-hole flows, tokamak plasma, and tumor-cell flows, to name a few.