Stagnation–saddle points and flow patterns in Stokes flow between contra-rotating cylinders

Abstract

The steady flow is considered of a Newtonian fluid, of viscosity μ, between contra-rotating cylinders with peripheral speeds U1 and U2. The two-dimensional velocity field is determined correct to O(H0/2R)1/2, where 2H0 is the minimum separation of the cylinders and R an ‘averaged’ cylinder radius. For flooded/moderately starved inlets there are two stagnation–saddle points, located symmetrically about the nip, and separated by quasi-unidirectional flow. These stagnation–saddle points are shown to divide the gap in the ratio U1[ratio ]U2 and arise at [mid ]X[mid ]=A where the semi-gap thickness is H(A) and the streamwise pressure gradient is given by dP/dX =μ(U1+U2)/H2(A). Several additional results then follow.(i) The effect of non-dimensional flow rate, λ: A2=2RH0(3λ−1) and so the stagnation–saddle points are absent for λ<1/3, coincident for λ=1/3 and separated for λ>1/3.(ii) The effect of speed ratio, S=U1/U2: stagnation–saddle points are located on the boundary of recirculating flow and are coincident with its leading edge only for symmetric flows (S=1). The effect of unequal cylinder speeds is to introduce a displacement that increases to a maximum of O(RH0)1/2 as S→0.Five distinct flow patterns are identified between the nip and the downstream meniscus. Three are asymmetric flows with a transfer jet conveying fluid across the recirculation region and arising due to unequal cylinder speeds, unequal cylinder radii, gravity or a combination of these. Two others exhibit no transfer jet and correspond to symmetric (S=1) or asymmetric (S≠1) flow with two asymmetric effects in balance. Film splitting at the downstream stagnation–saddle point produces uniform films, attached to the cylinders, of thickness H1 and H2, whereformula hereprovided the flux in the transfer jet is assumed to be negligible.(iii) The effect of capillary number, Ca: as Ca is increased the downstream meniscus advances towards the nip and the stagnation–saddle point either attaches itself to the meniscus or disappears via a saddle–node annihilation according to the flow topology.Theoretical predictions are supported by experimental data and finite element computations.