On the effects of rotation on fluid motions in cylindrical containers of various shapes and topological characteristics

Abstract

Abstract Previous theoretical and laboratory studies of mechanically driven fluids in general rotation relative to an inertial frame have shown that there is a special class of flows for which the (Eulerian) flow field u ( r , t ) relative to the rotating frame of reference is unaffected by gyroscopic (Coriolis) forces, and therefore remains the same for all values of the rotation vector Ω . (Here t denotes time and r the position of a general point R in a reference frame attached to the rotating apparatus.) Such flows occur when (a) Ω is independent of time t ; (b) u ( r , t ) is independent of the coordinate z (say) parallel to Ω , (c) the fluid has constant density and is therefore ‘barotropic’ (i.e. no density variations on horizontal surfaces) and (d) the topology of the cross-section of the (cylindrical) container, in planes z = constant, is such that the bounding surfaces can support the concomitant field of (kinematic) pressure P 1 satisfying ▿ P 1 + 2 Ω × u = 0 Condition (d) is equivalent to the requirement that any fluid sources or siks within the system be multipole in character, but not monopole. In the present study the ‘baroclinic’ case is treated, where buoyancy forces due to the action of gravity (and centripetal forces) on horizontal density variations have to be taken into account. These include investigations of flows due entirely to buoyancy forces, such as thermal convection in fluids in rotating cylindrical containers of various shapes and topological characteristics subject to horizontal temperature gradients. The implications for the impressed temperature field of the mathematical requirements that the fields of kinematic pressure P 1 and density ϱ ϑ (where ϱ denotes the mean density) be everywhere single-valued are guiding such investigations and facilitating the interpretation of their findings. The investigations include laboratory studies, reported elsewhere, of convection in a rotating fluid annulus with a circular cross-section blocked by a radial barrier, where it is found inter alia that advective heat transfer is virtually independent of | Ω | over a wide range of conditions. They also include (as yet unpublished) studies of thermal convection in rotating systems with topologically triply connected cross-sections which can be rendered doubly or simply connected by the insertation of suitable barriers.