Abstract
Stability of steady, two-dimensional, slide coating flow of Newtonian liquid to small, two-dimensional disturbances is analyzed by means of Galerkin's method and finite element basis functions. The resulting sequence of computational problems consists of large, sparse, asymmetric generalized eigenproblemsJx=λ Mx in whichJ andM depend on system parameters andM is singular. These are solved for the leading modes—eigenvalues of algebraically largest real part, and their eigenvectors—by a flexible method assembled from the iterative Arnoldi algorithm with Schur-Wielandt deflation developed by Saad for the asymmetric eigenproblem; initialization that takes advantage of continuation and can incorporate rational acceleration; complex or real shift of eigenvalue, as appropriate; and—a key ingredient-approximately exponential preconditioning by rational transformation suitable to the singular behavior. The results include leading modes of complicated structure, examples of mode overtaking, turning points beyond which the steady flows do not exist, and Hopf points that mark onset of deleterious, spontaneous oscillations of the flow.
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References
Abbott, J. P. (1978). An efficient algorithm for the determination of certain bifurcation points,J. Comput. Appl. Math. 4, 19–27.
Arnoldi, W. E. (1951). The principle of minimized iteration in the solution of the matrix eigenvalue problem,Q. Appl. Math. 9, 17–29.
Bathe, K.-J. (1982).Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, New Jersey.
Bauer, F. L. (1957). Das Verfahren der Treppeniteration und verwandte Verfahren zur Losung algebraischer Eigenwertprobleme,Z. angew. Math. Phys. 8, 214–235.
Bixler, N. E. (1982). Stability of coating flow, Ph.D. thesis, University of Minnesota, Minneapolis.
Christodoulou, K. N., and Scriven, L. E. (1989a). The fluid mechanics of slide coating,J Fluid Mech., in press.
Christodoulou, K. N., and Scriven, L. E. (1989b). Discretization of free surface flows and other free boundary problems,J. Comput. Phys., to be published.
Christodoulou, K. N., and Scriven, L. E. (1989c). Operability limits of flow systemsx with free boundaries by solving nonlinear eigenvalue problems,Int. J. Num. Meth. Fluids, to be published.
Christodoulou, K. N., and Scriven, L. E. (1989d). Stability of two-dimensional viscous free surface flows to small three-dimensional disturbances, in preparation.
Coyle, D. J. (1984). The fluid mechanics of roll coating: Steady flows, stability, and rheology, Ph.D. thesis, University of Minnesota, Minneapolis.
Daniel, G. W., Gragg, W. B., Kauffmann, L., and Stewart, G. W. (1976). Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization,Math. Comput. 30, 772–795.
Eriksson, L. E., and Rizzi, A. (1985). Computer-aided analysis of the convergence to steady state of discrete approximations to the Euler equations,J. Comput. Phys. 57, 90–128.
Franklin, J. N. (1968).Matrix theory, Prentice-Hall, Engelwood Cliffs, New Jersey.
Goldhirsch, I., Orszag, S. A., and Maulik, B. K. (1987). An efficient method for computing leading eigenvalues of large asymmetric matrices,J. Sci. Comput. 2, 33–58.
Golub, G. H., and Van Loan, C. F. (1983).Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland.
Higgins, B. G. (1980). Capillary hydrodynamics and coating beads, Ph.D. thesis, University of Minnesota, Minneapolis.
Hood, P. (1976). Frontal solution program for unsymmetric matrices,Int. J. Num. Meth. Eng. 10, 379–399.
Hood, P. (1977). Correction,Int. J. Num. Meth. Eng. 11, 1055.
Jennings, A., and Stewart, W. (1980). Simultaneous iteration for partial eigensolution of real matrices,J. Math. Inst. Appl. 15, 351–361.
Jepson, A. D. (1981). Numerical Hopf bifurcation, thesis, Part II, California Institute of Technology, Pasadena.
Katagiri, Y., and Scriven, L. E. (1989). Analysis of transient response of a coating operation, in preparation.
Keller, H. B. (1977). Numerical solution of bifurcation and nonlinear eigenvalue problems. In Rabinowitz, P. H. (ed.),Applications of Bifurcation Theory, Dekker, New York, pp. 45–52.
Kheshgi, H. S., and Scriven, L. E. (1982). Penalty finite element analysis of time-dependent two-dimensional film flows, in Kawai, T. (ed.),Finite Element Flow Analysis, University of Tokyo Press, Tokyo, pp. 113–120.
Lamb, H. (1945).Hydrodynamics, 6th ed., Dover Publications, New York.
Lanczos, C. (1950). An iteration method for the solution of the eigenvalue problem of linear differential and integral operator,J. Res. Natl. Bur. Stand. 45, 255–282.
Parlett, B. (1980).The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, New Jersey.
Parlett, B., and Saad, Y. (1985). Complex shift and invert strategies for real matrices, Technical Report No. YALEU/DCS-RR-424, Yale University, Computer Science Department, New Haven, Connecticut.
Parlett, B. N., Taylor, D. T., and Liu, Z. S. (1985). A look-ahead Lanczos algorithm for unsymmetric matrices,Math. Comput. 44, 105–124.
Petzold, L., and Loetstedt, P. (1986). Numerical solultion of nonlinear differential equations with algebraic constraints II: Practical implications.SIAM J. Sci. Stat. Comput. 7, 720–733.
Pritchard, W. G. (1986). Instability and chaotic behaviour in a free-surface flow,J. Fluid Mech. 165, 1–60.
Reusch, M. F., Ratzan, L., Pomphrey, N., and Park, W. (1988). Diagonal Padé approximations for initial value problems,SIAM J. Sci. Stat. Comput. 9, 829–838.
Ruschak, K. J. (1980). A method for incorporating free boundaries with surface tension in finite element fluid flow simulator,Int. J. Num. Meth. Eng. 15, 639–1648.
Ruschak, K. J. (1983). A three-dimensional linear stability analysis for two-dimensional free boundary flows by the finite element method,Comput. Fluids 11, 391–401.
Saad, Y. (1980). Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices,Lin. Alg. Appl. 34, 269–295.
Saad, Y. (1982). Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Technical Report No. 255, Department of Computer Science, Yale University, New Haven, Connecticut.
Saad, Y. (1989). Numerical solution of large nonsymmetric eigenvalue problems, in Proceedings of the Workshop on Practical Iterative Methods for Large Scale Computations. University of Minnesota Supercomputer Institute, Minneapolis, October 23–25, 1988. Elsevier Science Publishers B.V., Amsterdam.
Saff, E. B., Schoenhage, A., and Varge, R. S. (1976). Geometric convergence toe −z by rational functions with real poles,Numer. Math. 25, 307–322.
Siscovec, R. F., Dembart, B., Epton, M. A., Erisman, A. M., Manke, J. W., and Yip, E. L. (1979). Solvability of large scale descriptor systems, Report, DOE Contract No. ET-78-C-01-2876, Boeing Computer Services Company, Seattle, Washington.
Stewart, G. W. (1973).Introduction to Matrix Computations, Academic Press, New York.
Stewart, G. W. (1976). Simultaneous iteration for computing invariant subspaces of nonHermitian matrices,Numer. Math. 25, 123–136.
Stewart, G. W. (1978). SRRIT-A Fortran subroutine to calculate the dominant invariant subspaces of a real matrix, Technical Report TR-514, University of Maryland, College Park.
Storer, R. G. (1986). Numerical studies of toroidal resistive magnetohydrodynamic instabilities,J. Comput. Phys. 66, 294–323.
Strang, G., and Fix, G. (1973).An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, New Jersey.
Wilkinson, J. H. (1965).The Algebraic Eigenvalue Problem, Clarendon Press, Oxford.
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Christodoulou, K.N., Scriven, L.E. Finding leading modes of a viscous free surface flow: An asymmetric generalized eigenproblem. J Sci Comput 3, 355–406 (1988). https://doi.org/10.1007/BF01065178
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DOI: https://doi.org/10.1007/BF01065178