Further results relating to axi-symmetric rotational flows of elastico-viscous fluids

Abstract

Additional results concerning the effect of uniform suction on the axi-symmetric rotational (elasticoviscous) flows caused by an infinite disk (with the fluid at infinity in a state of solid rotation) are presented. 1. In a previous communication of the same main title (Jones and Thomas (1990)) (hereafter referred to as I ) the motion of an elastico-viscous fluid occupying the halfspace z > 0 due to a disk z = 0 rotating steadily about the z-axis with uniform suction applied across its surface was investigated. The results presented in I are now generalized so as to take-in the B6dewadt inverse problem of an infinite (plane) disk at rest in a fluid rotating with uniform angular velocity at large distance z from the disk: in the present regime the disk is supposed to rotate with angular velocity f2 and the fluid at infinity with uniform angular velocity s~2. The purely viscous B6dewadt-type regime has been studied by several authors, notably by Rogers and Lance (1960), and later generalized by Evans (1969) to allow for uniform suction at the disk surface. Their investigations are primarily concerned with the nature of the flow at large negtive values of s (i.e. with the counter-rotation regime) and on how the flow is affected by the suction mechanism.. What is shown is that for sufficiently small (negative) values of s the disk acts as a centrifugal fan, the flow being directed outward away from the disk axis and downward toward the disk (i.e. a flow having positive radial velocity and negative vertical velocity). As s increases negatively some radial flow reversal begins to showup in the far-flow field, whilst near the disk the flow, although radially outward, becomes progressively weaker. Eventually, the centripetal radial pressure gradient appropriate to the far-flow field becomes the dominant flow mechanism and results in a radial inflow near the disk (i.e. a flow with negative radial velocity and positive vertical velocity): above a critical negative value s = s o the high shear boundary layer that is adjacent to the disk disattaches itself from the disk. The effect of suction applied at the surface of the disk is found to counterbalance somewhat the radially induced pressure gradients, reducing substantially the radial and circumferential velocity components and postponing boundary-layer disattachment to higher (negative) values of s. The purpose of the present note is to ascertain how the elasticity of the fluid, as characterized by the Oldroyd number, affects this general flow structure. It is no more than an exploratory analysis, and may, possibly, make a case for a more detailed numerical study of the problem. 2. The analysis follows closely that of L Again the working elastico-viscous prototype the generalized Maxwell fluid is of the class for which the yon Kfirm~m rotational similarity-type flows are possible, the governing equations reducing in this case to a set of ordinary, non-linear differential equations*). The appropriate kinematic and dynamic variables are non-dimensionalized as in I(6)**), the angular velocity f2 of the disk being chosen as the characteristic velocity. It then follows that the equation of continuity I (7) together with the rheological equations of state I (8) remain unaltered. But to accommodate the changed conditions in the far-flow field, the equations of motion and the associated boundary conditions, viz. Eqs. I(9), I(10), are to replaced by