Material lines in viscous and elastic flow between eccentric co-rotating cylinders

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Abstract
The two-dimensional geometry of two eccentric cylinders is fixed by the ratio of radii and the eccentricity; the relation of these to coaxial (bipolar) coordinates is illustrated. When a Newtonian fluid is between the cylinders and they rotate with equal angular velocity in the same direction, the fluid in the middle of the gap lags behind the walls, not returning to its original position after one revolution. Calculated material lines at successive times, streamlines, and pressure and vorticity contours are illustrated for four configurations. Flow is shearing rather than extensional. Accurate pressures agree with crude estimates based on drag and pressure flows within a factor of four. An eddy circulates in the wide part of the gap when the eccentricity is greater than a critical value dependent on the ratio of radii. When the material in the gap is purely elastic, adhering to the walls, the stress oscillates with time unless the stress-free state is concentric. Infinitesimal elasticity theory is used to calculate steady stress and displacements. The stress agrees exactly with that deriving from Abbott and Walters' theory.

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