Effects of inertia in forced corner flows

Abstract

When viscous fluid is contained in the corner between two planes intersecting at an angle α, a flow may be ‘forced’ either by relative motion of the two planes keeping α constant (the ‘paint-scraper’ problem – Taylor 1960) or by relative rotation of the planes about their line of intersection (the hinged-plate problem – Moffatt 1964). In either case, a similarity solution is available describing the flow sufficiently near the corner, where inertia forces are negligible. In this paper, we investigate the effects of inertia forces, by constructing regular perturbation series for the stream function, of which the leading term is the known similarity solution. The first-order inertial effect is obtained analytically, and, for the Taylor problem with α = ½π, 25 terms of the perturbation series for the wall stress are generated numerically. Analysis of the coefficients suggests that the radius of convergence of the series is given by [rscr ] where r is distance from the corner, U is the relative speed of the planes, and v is the kinematic viscosity of the fluid. For the hinged-plate problem, discussed in § 5, the unsteadiness of the flow contributes to an inertial effect which is explicitly incorporated in the analysis. For both problems, streamline plots are presented which indicate the first influence of inertia forces at distances from the corner at which these become significant.