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, where

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provided 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.