Gaskell, P.H., Savage, M.D. and Thompson, H.M. (1998) Stagnation–saddle points and flow patterns in Stokes flow between contrarotating cylinders. Journal of Fluid Mechanics, 370. pp. 221247. ISSN 14697645

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Abstract
The steady flow is considered of a Newtonian fluid, of viscosity mu, between contrarotating cylinders with peripheral speeds U1 and U2 The twodimensional velocity field is determined correct to O(H0/2R)(1/2), where 2H(0) is the minimum separation of the cylinders and R an 'averaged' cylinder radius. For flooded/moderately starved inlets there are two stagnationsaddle points, located symmetrically about the nip, and separated by quasiunidirectional flow. These stagnationsaddle points are shown to divide the gap in the ratio U1 : U2 and arise at \X\ = A where the semigap thickness is H(A) and the streamwise pressure gradient is given by dP/dX = mu(Ulf U2)/H2(A). Several additional results then follow.
(i) The effect of nondimensional flow rate, lambda: A(2) = 2RH(0)(3 lambda  1) and so the stagnationsaddle points are absent for lambda < 1/3, coincident for lambda = 1/3 and separated for lambda > 1/3.
(ii) The effect of speed ratio, S = U1/U2: stagnationsaddle points are located on the boundary of recirculating flow and are coincident with its leading edge only for symmetric flows (S = i). 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 not equal 1) flow with two asymmetric effects in balance. Film splitting at the downstream stagnationsaddle point produces uniform films, attached to the cylinders, of thickness H1 and H2, where
H1/H2 = S(S + 3)/3S + 1,
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 stagnationsaddle point either attaches itself to the meniscus or disappears via a saddlenode annihilation according to the flow topology.
Theoretical predictions are supported by experimental data and finite element computations.
Item Type:  Article 

Copyright, Publisher and Additional Information:  Copyright © 1998 Cambridge University Press. 
Institution:  The University of Leeds 
Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Physics and Astronomy (Leeds) The University of Leeds > Faculty of Engineering (Leeds) > School of Mechanical Engineering (Leeds) > Institute of Engineering Thermofluids, Surfaces & Interfaces (iETSI) (Leeds) 
Depositing User:  Repository Officer 
Date Deposited:  15 May 2006 
Last Modified:  06 Jun 2014 12:58 
Published Version:  http://journals.cambridge.org/action/displayIssue?... 
Status:  Published 
Publisher:  Cambridge University Press 
Refereed:  Yes 
URI:  http://eprints.whiterose.ac.uk/id/eprint/1201 