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
field is determined correct to
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
Several additional results then follow.
(i) The effect of non-dimensional flow rate,
and so the stagnation–saddle points are absent for λ<1/3,
λ=1/3 and separated for λ>1/3.
(ii) The effect of speed ratio,
stagnation–saddle points are located
on the boundary of recirculating flow and are coincident with its leading
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
recirculation region and arising due to unequal cylinder speeds, unequal
gravity or a combination of these. Two others exhibit no transfer jet and
to symmetric (S=1) or asymmetric (S≠1)
flow with two asymmetric effects in
balance. Film splitting at the downstream stagnation–saddle point
films, attached to the cylinders, of thickness H1
and H2, where
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