Abstract
Two industrially important free surface flows arising in polymer processing and thin film coating applications are modelled as lid-driven cavity problems to which a creeping flow analysis is applied. Each is formulated as a biharmonic boundary-value problem and solved both analytically and numerically. The analytical solutions take the form of a truncated biharmonic series of eigenfunctions for the streamfunction, while numerical results are obtained using a linear, finite-element formulation of the governing equations written in terms of both the streamfunction and vorticity. A key feature of the latter is that problems associated with singularities are alleviated by expanding the solution there in a series of separated eigenfunctions. Both sets of results are found to be in extremely good agreement and reveal distinctive flow transformations that occur as the operating parameters are varied. They also compare well with other published work and experimental observation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burggraf, O.R. (1966). Analytical and numerical studies of steady separated flows. J. Fluid Mech., Vol. 24, p. 133.
Canedo, E.L., Denson, C.D. (1989). Flow in driven cavities with a free surface. AIChE J., Vol. 35, pp. 129–138.
Chung, T.J. (1978). Finite Element Analysis of Fluid Dynamics. McGraw-Hill, New York.
Cox, R.G. (1986). The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid. Mech., Vol. 168, pp. 169–194.
Coyle, D.J., Macosko, C.W., Scriven, L.E. (1990). The fluid dynamics of reverse roll coating. AIChE J., Vol. 36.
Dandy, D.S., Leal, L.G. (1989). A newton's method scheme for solving free surface flow problems. Internat. J. Numer. Methods Fluids, Vol. 9, pp. 1469–1486.
Downson, D., Taylor, C.M. (1979). Cavitation in bearings. Ann. Rev. Fluid. Mech., Vol. 11.
Gaskell, P.H., Mobbs, S.D. (1985). An efficient vorticity-streamfunction finite element method for viscous flow. 23rd Brit. Theo. Mech. Coll. Leeds.
Gaskell, P.H., Savage, M.D. (1996). Meniscus roll coating. In Liquid Film Coating (S.F. Kistler and P.M. Schweizer, eds). Chapman & Hall, New York, pp. 573–597.
Gaskell, P.H., Savage, M.D., Summers, J.L., Thompson, H.M. (1995). Modelling and analysis of meniscus roll coating. J. Fluid Mech., Vol. 298, pp. 113–137.
Gaskell, P.H., Savage, M.D., Wilson, M. (1996a). Flow structures in a half-filled annulus between rotating co-axial cylinders. Submitted to J. Fluid. Mech.
Gaskell, P.H., Innes, G.E., Savage, M.D. (1996b). An experimental investigation of meniscus roll coating. J. Fluid. Mech., in press.
Gunzburger, M.D. (1989). Finite Element Methods for Viscous Incompressible Flows. Computer Science and Scientific Computing. Academic Press, New York.
Gurcan, F. (1996). Flow structures in rectangular, driven cavities. Ph.D. Thesis, University of Leeds.
Harper, J.F., Wake, G.C. (1983). Stokes flow between parallel plates due to a transversely moving end wall. I.M.A. J. Appl. Math., Vol. 30, pp. 141–149.
Heubner, K.H. (1975). The Finite Element Method for Engineers. Wiley, New York.
Hillman, A.P., Salzer, H.E. (1943). Roots of sin z=z. Phil. Mag., Vol. 34.
Hood, P. (1976). Frontal solution program for unsymmetric matrices. Internat. J. Numer. Methods Engng., Vol. 10, pp. 379–399.
Jana, S.C., Metcalfe, G., Ottino, J.M. (1994). Experimental and computational studies of mixing in complex stokes flows: the vortex mixing flow and multicellular cavity flows. J. Fluid Mech., Vol. 269, pp. 199–246.
Joseph, D.D., Sturges, L. (1978). The convergence of biorthogonal series for biharmonic and Stokes flow edge problems: Part II. SIAM J. Appl. Math., Vol. 34, pp. 7–26.
Kelmanson, M.A. (1983). Boundary integral equation solution of viscous flows with free surfaces. J. Engng. Math., Vol. 17, pp. 329–343.
Malone, B. (1992). An experimental investigation of roll coating phenomena. Ph.D. Thesis, University of Leeds.
Moffatt, H.K. (1964). Viscous and resistive eddies near a sharp corner. J. Fluid Mech., Vol. 18, pp. 1–18.
Pan, F., Acrivos, A. (1967). Steady flows in rectangular cavities. J. Fluid Mech., Vol. 28, No. 4, pp. 643–655.
Peeters, M.F., Habashi, W.G., Dueck, E.G. (1987). Finite Element streamfunction-vorticity solutions of the incompressible Navier-Stokes equations. Internat. J. Numer. Methods in Fluids, Vol. 7, pp. 17–27.
Shankar, P.N. (1993). The eddy structure in Stokes flow in a cavity. J. Fluid Mech., Vol. 28, No. 4, pp. 643–655.
Shikhmurzaev, Y.D. (1993). The moving contact line on a smooth solid surface. Internat. J. Mult. Flow, Vol. 19, No. 4, pp. 589–610.
Smith, R.C.T. (1952). The bending of a semi-infinite strip. Austral. J. Sci. Res., Vol. 5, pp. 227–237.
Srinivasan, R. (1995). Accurate solutions for steady plane flow in the driven cavity. I. Stokes flow. Z. Angew. Math. Phys., Vol. 46, pp. 524–545.
Sturges, L.D. (1986). Stokes flow in a two-dimensional cavity with moving end walls. Phys. Fluids, Vol. 29, pp. 1731–1733.
Thompson, H.M. (1992). A theoretical investigation of roll coating phenomena. Ph.D. Thesis, University of Leeds.
Author information
Authors and Affiliations
Additional information
Communicated by Philip Hall
Rights and permissions
About this article
Cite this article
Gaskell, P.H., Summers, J.L., Thompson, H.M. et al. Creeping flow analyses of free surface cavity flows. Theoret. Comput. Fluid Dynamics 8, 415–433 (1996). https://doi.org/10.1007/BF00455993
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00455993