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

Gaskell, P.H., Savage, M.D. and Thompson, H.M. (1998) Stagnation–saddle points and flow patterns in Stokes flow between contra-rotating cylinders. Journal of Fluid Mechanics, 370. pp. 221-247. ISSN 1469-7645

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The steady flow is considered of a Newtonian fluid, of viscosity mu, between contra-rotating cylinders with peripheral speeds U-1 and U-2 The 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 '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 U-1 : U-2 and arise at \X\ = A where the semi-gap thickness is H(A) and the streamwise pressure 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, lambda: A(2) = 2RH(0)(3 lambda - 1) and so the stagnation-saddle 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 = U-1/U-2: stagnation-saddle 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 stagnation-saddle point produces uniform 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 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.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 1998 Cambridge University Press.
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: 08 Feb 2013 17:02
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

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