Small Reynolds number flow between eccentric rotating cylinders with a permeable sleeve

Two eccentric rotating cylinders together with a permeable membrane surrounding the inner cylinder are used to model the flow around a modified viscometer. A perturbation method is used to solve for the flow between the membrane and the outer cylinder; the how between the inner rotor and the membrane is assumed to be governed by Stoke's equation, and the two flow regimes are coupled by the through-flow across the membrane. For moderate values of Reynolds number and eccentricity, the permeability of the membrane plays a negligible role, and the flow through the membrane is found to be eccentricity dependent. High eccentricities result in the formation of eddies which, upon increasing the Reynolds number, move in a direction opposite to that of the rotation of the outer bowl.