Three-dimensional flow in a cylinder with a stress-free sidewall

Abstract

Stokes flow in a cylindrical column of fluid, with a stress-free cylindrical sidewall, is considered. The motion is assumed to be generated by the linear, uniform motion of either or both of the flat endwalls. The field is obtained by a vector eigenfunction expansion procedure. If the field is assumed to have a θ- and z-dependence of the type exp(iθ+kz), k has to satisfy the equation (1-k²)J³₁ + ĵ₁[2J²₁ - ĵ₁[2ĵ₁ + (1 + k²J₁}]= 0, where ĵ₁=kJ₀-J₁ and the argument of each Bessel function is k. This equation admits, unlike in the plane case and with important consequences, not just a real sequence {˜λ_n} of eigenvalues but also a complex one {˜μ_n}. Using a least squares procedure to satisfy the boundary conditions on the top and bottom walls, the three-dimensional velocity field in the column is determined for various values of column height h and wall speed ratio S. Detailed computations show that there are strong effects of both the stress-free boundary and three-dimensionality. The principal effect of the former is to permit motion on that boundary leading to large azimuthal motions and of the latter, unlike in the plane flow, to multiple primary eddies when h is sufficiently large. A number of new eddy structures are also found, which demonstrate that three-dimensionality often leads to the elimination of compactness found in plane flows. It is finally shown that the flow fields exhibit interesting bifurcations as S and h are varied.