LATERAL INERTIAL MIGRATION OF A SMALL SPHERE IN FAST LAMINAR-FLOW THROUGH A MEMBRANE DUCT

The hydrodynamic response of a small suspended sphere to laminar flow is studied for relatively fast flow rates in a duct with porous walls. There is a lift effect on the sphere which competes with a wall suction effect. The details of this balance determine whether the sphere will reach the membrane. An expression is developed, from first principles, to predict conditions under which a sufficiently long cross-flow plate and frame membrane module exposed to dilute suspensions of essentially spherical particles will not foul, assuming nonhydrodynamic attractions between a particle and the membrane are negligible. This is the first expression which might be properly tested experimentally to determine if the elevated permeation rates observed with colloidal suspensions (Porter, 1972) are due to particle lift or to some other phenomena such as a flowing cake. The strongest particle lift effect occurs when the sphere is much closer to one wall than the other and is due to convective interaction of the disturbance caused by the sphere and the undisturbed flow, in the presence of the nearby wall. The lift velocity has the same dependence on parameters as the previous results derived for slow laminar flow (Cox and Brenner, 1968; Ho and Leal, 1974). However, the maximum lift velocity is smaller than that found by Vasseur and Cox (1976) by roughly a factor of 2.6. These results agree with those of Schonberg and Hinch (1989) in the following features: the equilibrium position moves with increasing Reynolds number towards the wall, the magnitude of the velocity near the wall is in good agreement, and our results extend to higher Reynolds number. Also, the results are compared with the experiments of Segre and Silberberg (1962a, b).