Oscillatory Stokes flow in a cylindrical container

We study the three-dimensional viscous incompressible flow generated in a cylindrical container, of radius R, by the sinusoidally oscillatory motion of the top lid, of frequency Ω. This study generalises the one in Shankar [1997a. Three-dimensional eddy structure in a cylindrical container, J. Fluid Mech. 342, 97-118] where inertial effects were not included. For sufficiently small velocities, the motion will be governed by the linearised Navier-Stokes equation. The solution is constructed as a sum of vector eigenfunctions that satisfy the no-slip conditions on the lateral wall. The eigenvalues are found as the roots of the relation, 4k2Jm-1(α)Jm(k)Jm+1(α)+kαJm(α)(Jm-1(k)-Jm+1(k))(Jm-1(α)-Jm+1(α))-4m2Jm(k)Jm2(α)=0 where m and k are the azimuthal and vertical mode numbers and α=k2-iRe with Re being the oscillatory Reynolds number ΩR2/ν. Jm is the Bessel function of the first kind of order m. For the flow generated by the sinusoidal motion of the lid, only the first non-axisymmetric mode m=1 is excited. Roots of the above equation with m=1 are calculated and the coefficients in the finite sum representations for the velocity fields are determined so as to satisfy the no-slip conditions on the top and bottom lids in a least squares sense. Detailed studies for a cylinder depth of 1 and an Re of 10 are made and these reveal interesting streamline patterns that include heteroclinic cycle type structures at the top lid (apart from the usual one at the bottom), saddle and closed stagnation lines, especially when the lid reverses its direction of motion.