Effect of partial slip on the oscillatory slow viscous flow around a cylindrical obstacle

Two-dimensional transient low-Reynolds number viscous flows past a circular cylinder with Navier slip boundary conditions are examined. The Lagrange stream function method is used to solve the oscillatory Stokes flow problem around a cylinder, yielding analytic solutions in terms of second-kind modified Bessel functions with complex arguments. To separate the real and imaginary parts of the exact solutions, Kelvin functions are introduced. The flow fields around the cylinder contour are significantly impacted by the two key parameters, namely the frequency λ and the slip coefficient ξ. A term consisting of a biharmonic function discovered by Stokes that grows logarithmically dominates the flow in the very low frequency limit. Interesting flow patterns can be seen in local streamlines for short times. As ξ is varied, the attached eddies resulting from flow separation in the no-slip case are either delayed or pushed farther away from the cylinder. Computed asymptotic results reveal inviscid behavior far away from the cylinder in the frequency range 0 < λ < 1. Our analytic solutions show rather fast changes in the flow domain, despite the finite oscillation frequency. Our approach is also extended to surface velocity slip conditions, that is, squirming boundary conditions. The flow fields in the latter case appear to be very sensitive to the amplitude of the surface squirming modes.