Jeffery’s paradox for the rotation of a single ‘stick–slip’ cylinder

Abstract

We consider the problem of determining the two-dimensional fluid velocity due to the rotation of an infinitely-long ‘stick–slip’ cylinder in an otherwise quiescent Stokes flow. Stick–slip boundary conditions are introduced as a model of a rough superhydrophobic surface, via a distribution of alternating solid–liquid (stick) and gas–liquid (slip) interfaces. This leads to a mixed boundary-value problem for Stokes flow. Complex variable techniques are employed to transform the flow problem into a Hilbert problem, which involves finding a function analytic in a plane region assuming that on some portions of the boundary its real part is known, while on others its imaginary part is given. We solve the Hilbert problem to obtain semi-analytic expressions for all the pertinent fluid-dynamic quantities. We find that in the general aperiodic case there is no solution in which the velocity of the fluid vanishes at infinity. This is a form of Jeffery’s paradox, typically associated with viscous flow due to the counter-rotation of two equal rigid cylinders. Our work provides the first example of Jeffery’s paradox due to the rotation of a single cylinder.