Flow induced by the rotation of two circular cylinders in a viscous fluid
Résumé
The Stokes flow driven by the rotation of two parallel cylinders of equal unit radius is determined by both numerical and analytical techniques. A numerical solution is obtained by enclosing the system in an outer cylinder of radius R_0>>1, on which different boundary conditions can be imposed. With a gap 2epsilon between the inner cylinders, attention is focused on the small gap situation epsilon<<1 when lubrication theory becomes applicable. Good agreement with the numerical solution is obtained for both counter-rotating and co-rotating cases. In the counter-rotating situation, the total force F acting on the cylinders is determined. Numerical evidence is presented for the conclusion that F~log(R_0)^-1 as R_0->infty. An exact analytic solution is obtained in the contact limit epsilon=0, and a contact force in F_c is identified, which contributes to the torque that each cylinder experiences about its axis. The far-field torque doublet is also identified. The manner in which the flow topology adapts to the change in topology of the fluid domain when the cylinders are brought into contact is noted. The sliced-cylinder situation when epsilon<0 is also considered, and in this case a distributed contact force is identified, and a similarity solution is found that describes the flow near the corner singularities. In the case of co-rotating cylinders, the theory of Watson is elucidated and shown to agree well with the numerical solution and with lubrication theory when epsilon<0.01. The torque generated by the co-rotation of the cylinders is determined, with asymptotic value ~17.25 as epsilon->0. An alternative exact analytic solution in the contact limit is obtained, for which the torque is zero and the far-field flow is one of uniform rotation; in a rotating frame of reference in which the fluid at infinity is at rest, the relative flow in this case is identified as a `radial quadrupole'.