STAGNATION-SADDLE POINTS AND FLOW PATTERNS IN STOKES-FLOW BETWEEN CONTRA-ROTATING CYLINDERS

The steady flow is considered of a Newtonian fluid, of viscosity mu, b etween contra-rotating cylinders with peripheral speeds U-1 and U-2 Th e two-dimensional velocity field is determined correct to O(H-0/2R)(1/ 2), where 2H(0) is the minimum separation of the cylinders and R an 'a veraged' 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 p oints are shown to divide the gap in the ratio U-1 : U-2 and arise at \X\ = A where the semi-gap thickness is H(A) and the streamwise pressu re gradient is given by dP/dX = mu(Ulf U-2)/H-2(A). Several additional results then follow. (i) The effect of non-dimensional flow rate, lam bda: A(2) = 2RH(0)(3 lambda - 1) and so the stagnation-saddle points a re absent for lambda < 1/3, coincident for lambda = 1/3 and separated for lambda > 1/3. (ii) The effect of speed ratio, S = U-1/U-2: stagnat ion-saddle points are located on the boundary of recirculating flow an d are coincident with its leading edge only for symmetric flows (S = i ). The effect of unequal cylinder speeds is to introduce a displacemen t 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 menis cus. Three are asymmetric flows with a transfer jet conveying fluid ac ross the recirculation region and arising due to unequal cylinder spee ds, unequal cylinder radii, gravity or a combination of these. Two oth ers exhibit no transfer jet and correspond to symmetric (S = 1) or asy mmetric (S not equal 1) flow with two asymmetric effects in balance. F ilm splitting at the downstream stagnation-saddle point produces unifo rm films, attached to the cylinders, of thickness H-1 and H-2, where H -1/H-2 = S(S + 3)/3S + 1, provided the flux in the transfer jet is ass umed 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 dis appears via a saddle-node annihilation according to the flow topology. Theoretical predictions are supported by experimental data and finite element computations.