Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-02T15:54:09.345Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Solutions to Free Boundary Problems*

Published online by Cambridge University Press:  07 November 2008

Thomas Y. Hou
Thomas Y. Hou
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125 E-mail: hou@ama.caltech.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Many physically interesting problems involve propagation of free surfaces. Vortex-sheet roll-up in hydrodynamic instability, wave interactions on the ocean's free surface, the solidification problem for crystal growth and Hele-Shaw cells for pattern formation are some of the significant examples. These problems present a great challenge to physicists and applied mathematicians because the underlying problem is very singular. The physical solution is sensitive to small perturbations. Naïve discretisations may lead to numerical instabilities. Other numerical difficulties include singularity formation and possible change of topology in the moving free surfaces, and the severe time-stepping stability constraint due to the stiffness of high-order regularisation effects, such as surface tension.

This paper reviews some of the recent advances in developing stable and efficient numerical algorithms for solving free boundary-value problems arising from fluid dynamics and materials science. In particular, we will consider boundary integral methods and the level-set approach for water waves, general multi-fluid interfaces, Hele–Shaw cells, crystal growth and solidification. We will also consider the stabilising effect of surface tension and curvature regularisation. The issue of numerical stability and convergence will be discussed, and the related theoretical results for the continuum equations will be addressed. This paper is not intended to be a detailed survey and the discussion is limited by both the taste and expertise of the author.

Type
Research Article
Information
Acta Numerica , Volume 4 , January 1995 , pp. 335 - 415
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Almgren, A., Buttke, T. and Colella, P. (1994), ‘A fast adaptive vortex method in 3 dimensions’, J. Comp. Phys., 113, 117–200.Google Scholar
Almgren, R. (1993), ‘Variational algorithms and pattern-formation in dendritic solidification’, J. Comp. Phys., 106, 337–354.Google Scholar
Almgren, R., Dai, W.-S. and Hakim, V. (1993), ‘Scaling behaviour in anisotropic Hele–Shaw flow’, Phys. Rev. Lett., 71, 3461–3464.Google Scholar
Anderson, C. (1985), ‘A vortex method for flows with slight density variations’, J. Comp. Phys., 61, 417–444.CrossRefGoogle Scholar
Anderson, C. (1986), ‘A method of local corrections for computing the velocity due to a distribution of virtex blobs’, J. Comp. Phys., 62, 111–123.CrossRefGoogle Scholar
Anderson, C. and Greengard, C. (1985), ‘On vortex methods’, SIAM J. Numer. Anal, 22, 413–440.CrossRefGoogle Scholar
Ascher, U., Ruuth, S and Wetton, B., ‘Implicit–explicit methods for time-dependent PDE's’, to appear in SIAM J. Numer. Anal.Google Scholar
Baker, G. (1979), ‘The ‘cloud-in-cell’ technique applied to the roll-up of vortex sheets’, J. Comp. Phys., 31, 76–95.Google Scholar
Baker, G. (1983), Generalized vortex methods for free-surface flows, in Waves on Fluid Interfaces, (Meyer, R. E., ed.), Academic Press pp. 53–81.Google Scholar
Baker, G., Caflisch, R. and Siegal, M., ‘Singularity formation during the Rayleigh–Taylor instability’, to appear J. Fluid Mech.Google Scholar
Baker, G., Meiron, D. and Orszag, S. (1982), ‘Generalized vortex methods for free-surface flow problems’, J. Fluid Mech., 123, 477–501.Google Scholar
Baker, G. R. and Shelley, M. J. (1990), ‘On the connection between thin vortex layers and vortex sheets’, J. Fluid Mech., 215, 161–194.Google Scholar
Baker, G. and Nachbin, A., ‘Stable methods for vortex sheet motion in the presence of surface tension’, to appear in J. Comp. Phys.Google Scholar
Beale, J. T. and Majda, A. (1982), ‘Vortex methods, I: Convergence in three dimensions’, Math. Comp., 32, 1–27.Google Scholar
Beale, J. T., Hou, T. Y. and Lowengrub, J. S. (1993a), ‘Growth rates for the linear motion of fluid interfaces far from equilibrium’, Comm. Pure Appl. Math., 46, 1269–1301.Google Scholar
Beale, J. T., Hou, T. Y. and Lowengrub, J. S., ‘Convergence of a boundary integral method for water waves’, to appear in SIAM J. Num. Anal.Google Scholar
Beale, J. T., Hou, T. Y. and Lowengrub, J. S. (1993b), On the well-posedness of two fluid interfacial flows with surface tension, Singularities in Fluids, Plasmas and Optics, (eds. Caflisch, R. C. and Papanicolaou, G. C.), Kluwer Academic, London, pp. 11–38.Google Scholar
Beale, J. T., Hou, T. Y., Lowengrub, J. and Shelley, M., ‘Spatial and temperal stability issues for interfacial flows with surface tension’, to appear in J. Math. Computer Modelling, Vol. 20, No. 10/11, pp. 1–27, 1994.Google Scholar
Beale, J. T., Hou, T. Y. and Lowengrub, J. S., Two Fuid Flows with Surface Tension, Part 1: Growth Rates of the Linearized Motion Far From Equilibrium; Part 2: Convergence of Boundary Integral Methods, in preparation, Applied Math., Caltech.Google Scholar
Bell, J. B., Colella, P. and Glaz, H. M. (1989), ‘A 2nd order projection method for the incompressible Navier–Stokes equations’, J. Comp. Phys., 85, 257–283.CrossRefGoogle Scholar
Bell, J. B. and Marcus, D. L. (1992), ‘A second-order projection method for variable-density flows’, J. Comp. Phys., 101, 334–348CrossRefGoogle Scholar
Benamar, M. and Pomeau, Y. (1986), ‘Theory of dendritic growth in a weakly undercooled melt’, Europhys. Lett., 2, 307–314.Google Scholar
Bertozzi, A. L. and Constantin, P. (1993), ‘Global regularity for vortex patches, Comm. Math. Phys.’, 152, 19–28.CrossRefGoogle Scholar
Birkhoff, G. (1962), ‘Helmholtz and Taylor instability’, in Proc. Symp. Appl. Math., 13, 55–76, Vol. XIII, Publ. American Math. Society, Providence, R.I.Google Scholar
Brackbill, J. U., Kothe, D. B. and Zemach, C. (1992), ‘A continuum method modeling surface tension’, J. Comp. Phys., 100, 335–354.CrossRefGoogle Scholar
Brattkus, K. and Meiron, D. I. (1992), ‘Numerical simulation of unsteady crystal growth’, SIAM J. Appl. Math., 52, 1303–1320.Google Scholar
Buttke, T. F. (1989), ‘The observation of singularities in the boundary of patches of constant vorticity’, Phys. Fluids A, 1, 1283–1285.Google Scholar
Caflisch, R. and Orellana, O. (1988), ‘Long time existence for a slightly perturbed vortex sheet’, Comm. Pure Appl. Math., 39, 807–838.CrossRefGoogle Scholar
Caflisch, R. and Orellana, O. (1989), ‘Singularity solutions and ill-posedness for the evolution of vortex sheets’, SIAM J. Math. Anal., 20, 293–307.Google Scholar
Caflisch, R. and Lowengrub, J. (1989), ‘Convergence of the vortex method for vortex sheets’, SIAM J. Numer. Anal., 26, 1060–1080.CrossRefGoogle Scholar
Caflisch, R., Li, X. and Shelley, M. (1993), ‘The collapse of an axisymmetrical, swirling vortex sheet’, Nonlinearity, 6, 843–867CrossRefGoogle Scholar
Caflisch, R., Ercolani, N., Hou, T. Y. and Landis, Y. (1993), ‘Multi-valued solutions and branch point singularities for nonlinear and elliptic systems’, Comm Pure Appl Math., 46, 453–499.Google Scholar
Caflisch, R., Hou, T. Y. and Lowengrub, J. (1994), Almost Optimal Convergence of the Point Vortex Method for Vortex Sheets Using Numerical Filtering, preprint, Dept of Math., Univ. of Minnesota, submitted to Math. Comput.Google Scholar
Gaginalp, G. and Fife, P. C. (1988), ‘Dynamics of layered interfaces arising from phase boundaries’, SIAM J. Appl. Math., 48, 506–518.CrossRefGoogle Scholar
Chadam, J. and Ortoleva, P. (1983), ‘The stabilizing effect of surface tension on the development of the free-boundary in a planar, one-dimensional, Cauchy-Stefan problem’, IMA J. Appl. Math., 30, 57–66.Google Scholar
Chang, Y. C., Hou, T. Y., Merriman, B. and Osher, S., ‘Eularian capturing methods based on a level set formulation for incompressible fluid interfaces’, to appear in J. Comp. Phys.Google Scholar
Ceniceros, H. and Hou, T. Y., Convergence of a Non-stiff Boundary Integral Method for Interfacial Flows with Surface Tension, preprint, Applied Math., Caltech.Google Scholar
Chemin, J.-Y. (1993), ‘Presistency of geometric structures in bidimensional incompressible fluids’, Ann. Sei. EC, 26, 517–542.Google Scholar
Chopp, D. L. (1993), ‘Computing minimal surfaces via level set curvature flow,’ J. Comp. Phys., 106, 77–91.Google Scholar
Chorin, A. (1968), ‘Numerical solution of the Navier–Stokes equations’, Math. Comput, 22, 745–762.CrossRefGoogle Scholar
Chorin, A. (1973), ‘Numerical study of slightly viscous flow’, J. Fluid Mech., 57, 785–796.Google Scholar
Chorin, A. and Bernard, P. S. (1973), ‘Discretization of a vortex sheet, with an example of roll-up’, J. Comp. Phys., 13, 423–429.Google Scholar
Christiansen, J. P. (1973), ‘Numerical simulation of hydrodynamics by the method of point vortices’, J. Comp. Phys., 13, 363–379.Google Scholar
Constantin, P., Lax, P. D. and Majda, A. (1985), ‘A simple one dimensional model for the three dimensional vorticity equations’, Comm. Pure Appl. Math., 38, 715–724.CrossRefGoogle Scholar
Colella, P. and Woodward, P. R. (1984), ‘The piecewise parabolic method (PPM) for gas dynamical simulations’, J. Comp. Phys., 54, 174–201.CrossRefGoogle Scholar
Cottet, G. H. (1988), ‘On the convergence of vortex methods in two and three dimensions’, Ann. Insti. H. Poincaré, 5, 641–672.Google Scholar
Cottet, G. H., Goodman, J. and Hou, T. Y. (1991), ‘Convergence of the grid free point vortex method for the 3-D Euler equations’, SIAM J. Numer. Anal., 28, 291–307.Google Scholar
Cottet, G. H. (1987), ‘Convergence of a vortex in cell method for the two-dimensional Euler equations’, Math. Comp., 49, 407–425.Google Scholar
Craig, W. (1985), ‘An existence theory for water waves and the Bousinesq and the Kortewegde–de Vries scaling limits’, Comm. Partial Diff. Eqns., 10, 787–1003.Google Scholar
Craig, W. and Sulem, C. (1993), ‘Numerical simulation of Gravity Waves, J. Comp. Phys.’, 108, 78–83.Google Scholar
Dahm, W. J. A., Frieler, C. E. and Tryggvason, G. (1992), ‘Vortex structure and dynamics in near field of a coaxial jet’, J. Fluid Mech., 241, 371–402.Google Scholar
Dai, W.-S. and Shelley, M. J. (1993), ‘A numerical study of the effect of surface tension and noise on an expanding Hele–Shaw bubble’, Phys. Fluids A, 5(9), 2131–2146.CrossRefGoogle Scholar
Delort, J.-M. (1991), ‘Existence de nappes de tourbillon en dimension deux’, J. Amer. Math. Soc, 4, 553–586.Google Scholar
DiPerna, R. and Majda, A. (1987a), ‘Concentration in regularizations for 2-D incompressible flow’, Comm. Pure Appl. Math., 40, 301–345.Google Scholar
DiPerna, R. and Majda, A. (1987b), ‘Oscillations and concentrations in weak solutions of the incompressible fluid equations’, Comm. Math. Phys., 108, 667–689.Google Scholar
Dold, J. W. (1992), ‘An efficient surface-integral algorithm applied to unsteady gravity waves’, J. Comp. Phys., 103, 90–115.CrossRefGoogle Scholar
Dommermuth, D. G. and Yue, D. K. P. (1987), ‘A high order spectral method for the study of nonlinear gravity waves’. J. Fluid Mech., 184, 267–288.CrossRefGoogle Scholar
Draghicescu, C. I. (1994), ‘An efficient implementation of particle methods for the incompressible Euler equations’, SIAM J. Numer. Anal., 31, 1090–1108.Google Scholar
Dritschel, D. G. (1989), ‘Contour dynamics and contour surgery – Numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows’, Comp. Phys. Review, 10, 77–146.Google Scholar
Dritschel, D. G. and McIntyre, M. E. (1990), ‘Does contour dynamics go singular?’ Phys. Fluids A, 2, 748–753.CrossRefGoogle Scholar
Duchon, J. and Robert, R. (1986), ‘Solution globales avec nape tourbillionaire pour les equations d's Euler dans le plan’, C R. Acad. Sc. Paris, 302, Series I. 5, 183–186.Google Scholar
Duchon, J. and Robert, R. (1988), ‘Global vortex sheet solutions of Euler equations in the plane’, J. Diff. Eqn., 73, 215–224.CrossRefGoogle Scholar
Ebin, D. (1988), ‘Ill-posedness of the Rayleigh–Taylor and Helmholtz problems for incompressible fluids’, Comm. PDE, 13, 1265–1295.CrossRefGoogle Scholar
Evans, L. C. and Spruck, J. (1991), ‘Motion of level sets by mean-curvature, I’, J. Diff. Geom., 33, 635–681.Google Scholar
Evans, L. C. and Spruck, J. (1992), ‘Motion of level sets by mean-curvature, II’, Tran. Amer. Math. Soc., 330, 321–332.Google Scholar
Fenton, J. D. and Rienecker, M. M. (1982), ‘A Fourier method for solving nonlinear water-wave problems: Application to solitary-wave interactions’, J. Fluid Mech., 118, 411–443.Google Scholar
Fink, P. T. and Soh, W. K. (1978), ‘A new approach to roll-up calculations of vortex sheets’, Proc Roy. Soc. London A., 362, 195–209.Google Scholar
Glozman, M., Agnon, Y. and Stiassnie, M. (1993), ‘High-order formulation of the water-wave problem’, Physica D, 66, 347–367.CrossRefGoogle Scholar
Goldstein, R., Pesci, A. and Shelley, M. (1993), ‘Topology transitions and singularities in viscous flows’, Phys. Rev. Lett., 70, 3043–3046.CrossRefGoogle ScholarPubMed
Goodman, J., Hou, T. Y. and Lowengrub, J. (1990), ‘The convergence of the point vortex method for the 2-D Euler equations’, Comm. Pure and Appl. Math., 43, 415–430.Google Scholar
Goodman, J., Hou, T. Y. and Tadmor, E. (1994), ‘On the stability of the unsmoothed Fourier method for hyperbolic equations’, Numer. Math., 67, 93–129.Google Scholar
Gray, L. J., Chisholm, M. F. and Kaplan, T. (1993), Morphological stability of thin films, in Boundary Element Technology VIII, (Pina, H. and Brebbia, C. A., eds.), Southampton, UK, Computational Mechanics Publications, pp. 181–190.Google Scholar
Greenbaum, A., Greengard, L. and McFadden, G. B. (1993), ‘Laplace's equation and the Dirichlet–Neumann map in multiply connected domains’, J. Comp. Phys., 105, 267–278.Google Scholar
Greengard, L. and Rokhlin, V. (1987), ‘A fast algorithm for particle summations’, J. Comp. Phys., 73, 325–348.Google Scholar
Greengard, L. and Strain, J. (1990), ‘A fast algorithm for the evaluation of heat potentials’, Coram. Pure Appl. Math., 43, 949–963.Google Scholar
Gresho, P. M. and Sani, R. L. (1987), ‘On pressure boundary conditions for the incompressible Navier–Stokes equations’, Internat. J. Numer. Methods Fluids, 7, 1111–1145.CrossRefGoogle Scholar
Gottlieb, D. and Orszag, S. A. (1977), Numerical Analysis of Spectral Methods: Theory and Applications, CBMS–NSF Regional Conference Series in Applied Mathematics 26, SIAM Publ., Philadelphia.Google Scholar
Gurtin, M. E. (1986), ‘On the 2-phase stefan problem with interfacial energy and entropy’, Arch. Ration. Mech. Anal., 96, 199–241.CrossRefGoogle Scholar
Hald, O. (1979), ‘Convergence of vortex methods II’, SIAM J. Numer. Anal., 16, 726–755.Google Scholar
Hald, O. (1991), ‘Convergence of vortex methods’, in Vortex Methods and Vortex Motion, (Gustafson, and Sethian, , eds.), SIAM publications, Philadelphia, pp. 33–58.Google Scholar
Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. R. (1987), ‘Uniformly high order accurate essentially non-oscillatory schemes, III’, J. Comp. Phys., 71, 231–303.Google Scholar
Higdon, J. J. L. and Pozrikidis, C. (1985), ‘The self-induced motion of vortex sheets’, J. Fluid Mech., 150, 203–231.Google Scholar
Hou, T. Y. and Lowengrub, J. (1990), ‘Convergence of a point vortex method for 3-D Euler equations’, Comm. Pure Appl. Math., 43, 965–981.Google Scholar
Hou, T. Y. and Osher, S. (1994), Computing Topological Singularities Using a Tracking Method: Geometric regularization, preprint, Applied Math., Caltech.Google Scholar
Hou, T. Y., Lowengrub, J. S. and Shelley, M. J. (1994), ‘Removing the stiffness from interfacial flows with surface tension’, J. Comp. Phys., 114, 312–338.Google Scholar
Hou, T. Y., Lowengrub, J. S. and Shelley, M. J., The Roll-up and Self-intersection of Vortex Sheets Under Surface Tension, preprint, Courant Institute.Google Scholar
Hou, T. Y., Lowengrub, J. S. and Krasny, R. (1991), ‘Convergence of a point vortex method for vortex sheets’, SIAM J. Numer. Anal., 28, 308–320.Google Scholar
Kaneda, Y. (1990), ‘A representation of the motion of a vortex sheet in a three dimensional flow’, Phys. Fluids A, 2, 458–461.Google Scholar
Karma, A. (1986), ‘Wavelength selection in dendritic solidification’, Phys. Rev. Lett., 57, 858–861.Google Scholar
Kerr, R. M. (1988), ‘Simulation of Rayleigh–Taylor flows using vortex blobs’, J. Comput. Phys., 76, 48–84.Google Scholar
Kessler, D. A. and Levine, H. (1986), ‘Stability of dendritic crystals’, Phys. Rev. Lett., 57, 3069–3072.Google Scholar
Koumoutsakos, P. (1993), Direct Numerical Simulations of Unsteady Separated Flows Using Vortex Methods, Ph.D. thesis, Graduate School of Aeronautics, Caltech.CrossRefGoogle Scholar
Krasny, R. (1986a), ‘A study of singularity formation in a vortex sheet by the point vortex approximation’, J. Fluid Mech., 167, 65–93.CrossRefGoogle Scholar
Krasny, R. (1986b), ‘Desingularization of periodic vortex sheet roll-up’, J. Comp. Phys., 65, 292–313.CrossRefGoogle Scholar
Krasny, R. (1987), ‘Computation of vortex sheet roll-up in the Trefftz plane’, J. Fluid Mech., 184, 123–155.Google Scholar
Kreiss, H.-O. and Oliger, J. (1979), ‘Stability of the Fourier method’, SIAM J. Num. Anal., 16, 421–433.Google Scholar
Langer, J. S. (1980), ‘Instabilities and pattern formation in crystal growth’, Rev. Modern Phys., 52, 1–28.Google Scholar
Langer, J. S. (1986), ‘Existence of needle crystals in local models of solidification’, Phys. Rev. A, 33, 435–441.CrossRefGoogle ScholarPubMed
Leonard, A. (1980), ‘Vortex methods for flow simulation’, J. Comp. Phys., 37, 289–335.CrossRefGoogle Scholar
Liu, J. G. and Xin, Z. P., Convergence of Vortex Methods for Weak Solutions to the 2-D Euler Equations with Vortex Sheet Data, preprint, Courant Institute.Google Scholar
Longuet-Higgins, M. S. and Cokelet, E. D. (1976), ‘The deformation of steep surface waves on water, I. A numerical method of computation’, Proc. Roy. Soc. London A, 350, 1–26.Google Scholar
Majda, A. (1986), ‘Vorticity and the mathematical theory of incompressible fluid flow’, Comm. Pure Appl. Math., 39, 5187–5220.Google Scholar
Meiron, D. I. (1986), ‘Boundary Integral formulation of the two-dimensional symmetric model of dendritic growth’, Physica-D, 23, 329–339.Google Scholar
Meiron, D. I., Baker, G. R. and Orszag, S. A. (1982), ‘Analytic structure of vortex sheet dynamics, 1. Kelvin–Helmholtz instability’, J. Fluid Mech., 114, 283–298.Google Scholar
Mulder, W., Osher, S. and Sethian, J. (1992), ‘Computing interface motion in compressible gas dynamics’, J. Comp. Phys., 100, 209–228.CrossRefGoogle Scholar
Moore, D. W. (1979), ‘The spontaneous appearance of a singularity in the shape of an evolving vortex sheet’, Proc. R. Soc. Lond. Ser. A, 365, 105–119.Google Scholar
Moore, D. W. (1981), ‘On the point vortex method’. SIAM J. Sei. Stat., 2, 65–84.Google Scholar
Moore, D. W. (1978), ‘The equations of motion of a vortex layer of small thickness’, Stud. Appl. Math., 58, 119–140.Google Scholar
Moore, D. W. (1985), Numerical and analytical aspects of Helmholtz instability. In Theoretical and Applied Mechanics, Proc. XVI IUTAM (eds., Niodsen, and Olhoff, ), pp. 263.Google Scholar
Mullins, W. and Sekerka, R. (1963), ‘Morphological stability of a particle growing by diffusion or heat flow’, J. Appl. Phys., 34, 323–329.Google Scholar
New, A. L., McIver, P. and Peregrine, D. H. (1985), ‘Computations of overturning waves’, J. Fluid Mech., 150, 233–251.Google Scholar
Nitsche, M. and Krasny, R. (1994), A Numerical Study of Vortex Ring Formation at the Edge of a Circular Tube, preprint, Dept. of Math., Univ. of Michigan.Google Scholar
Osher, S. and Sethian, J. (1988), ‘Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations’, J. Comp. Phys., 79, 12–49.Google Scholar
Osher, S. and Shu, C. W. (1991), ‘High order essentially nonoscillatory schemes for Hamilton–Jacobi equations’, J. Comp. Phys., 28, 907–922.Google Scholar
Osher, S., private communication.Google Scholar
Paterson, L. (1981), ‘Radial fingering in a Hele-Shaw cell’, J. Fluid Mech., 113, 513–519.Google Scholar
Paterson, L. (1985), ‘Fingering with miscible fluids in a Hele–Shaw cell’, Phys Fluids., 28, 26–30.Google Scholar
Pelcé, P. (1988), Dynamics of Curved Fronts, Academic Press, Inc., San Diego.Google Scholar
Peregrine, D. H. (1983), ‘Breaking waves on beaches’, Ann. Rev. Fluid Mech., 15, 149–178.Google Scholar
Peskin, C. (1977), ‘Numerical analysis of blood flow in the heart’, J. Comp. Phys., 25, 220–252.Google Scholar
Pozrikidis, C. (1992), Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press.Google Scholar
Pozrikidis, C. and Higdon, J. J. L. (1985), ‘Nonlinear Kelvin–Helmholtz instability of a finite vortex layer’, J. Fluid Mech., 157, 225–263.CrossRefGoogle Scholar
Prosperetti, A., Crum, L. A. and Pumphrey, H. C. (1989), ‘The underwater noise of rain’, J. Geophys. Res., 94, 3255–3259.Google Scholar
Pugh, D. A. (1989), Development of Vortex Sheets in Boussinesq Flows – Formation of Singularities, Ph.D. Thesis, Imperial College, LondonGoogle Scholar
Pullin, D. I. (1979), ‘Vortex ring formation at tube and orifice openings’, Phys. Fluids, 22, 401–403.Google Scholar
Pullin, D. I. (1982), ‘Numerical studies of surface tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability’, J. Fluid Mech., 119, 507–532.Google Scholar
Rangel, R. and Sirignano, W. (1988), ‘Nonlinear growth of the Kelvin–Helmholtz instability: effect of surface tension and density ratio’, Phys. Fluids, 31, 1845–1855.Google Scholar
Rauseo, S. N., Barnes, P. D. Jr and Maher, J. V. (1987), ‘Development of radial fingering patterns’, Phys. Rev. A, 35, 1245–1251.Google Scholar
Roberts, A. J. (1983), ‘A stable and accurate numerical method to calculate the motion of a sharp interface between fluids’, IMA J. Appl. Math., 31, 13–35.Google Scholar
Rosenhead, L. (1932), ‘The point vortex approximation of a vortex sheet’, Proc. Roy. Soc. London Ser. A, 134, 170–192.Google Scholar
Saffman, P. G. and Taylor, G. I. (1958), ‘The penetration of a fluid into a porous medium of Hele–Shaw cell containing a more viscous fluid’, Proc. Roy. Soc. London Ser. A, 245, 312–329.Google Scholar
Schwartz, L. W. and Fenton, J. D. (1982), ‘Strongly nonlinear waves’, Ann. Rev. Fl. Mech., 14, 39–60.Google Scholar
Shelley, M. (1992), ‘A study of singularity formation in vortex sheet motion by a spectrally accurate vortex method’, J. Fluid Mech., 244, 493–526.Google Scholar
Sethian, J. A. (1985), ‘Curvature and the evolution of fronts’, Comm. Math. Phys., 101, 487–499.Google Scholar
Sethian, J. and Strain, J. (1992), ‘Crystal growth and dendritic solidification’, J. Comp. Phys., 98, 231–253.Google Scholar
Sidi, A. and Israeli, M. (1988), ‘Quadrature methods for periodic singular and weakly singular Fredholm integral equations’, J. Sci. Comp., 3, 201–231.Google Scholar
Snyder, C. W., Orr, B. G., Kessler, D. and Sander, L. M. (1991), ‘Effect of strain on surface morphology in highly strained InGaAs’, Phys. Rev. Lett., 66, 3032–3035.Google Scholar
Spencer, B. J., Voorhees, P. W. and Davis, S. H. (1991), ‘Morphological instability in epitaxially dislocation-free solid films’, Phys. Rev. Lett., 67, 3696–3999.CrossRefGoogle ScholarPubMed
Spencer, B. J. and Meiron, D. I. (1994), ‘Nonlinear Evolution of the stress-driven morphological instability in a 2-dimensional semi-infinite solid’, Act. Met. Mat., 42, 3629–3641.Google Scholar
Stiassnie, M. and Shemer, L. (1984), ‘On modifications of the Zakharov equation for surface gravity waves’, J. Fluid Mech., 143, 47–67.Google Scholar
Strain, J. (1989), ‘A boundary integral approach to unstable solidification’, J. Comp. Phys., 85, 342–389.Google Scholar
Sulem, P., Sulem, C., Bardos, C. and Frisch, U. (1981), ‘Finite time analyticity for the two and three dimensional Kelvin–Helmholtz instability’, Comm. Math. Phys., 80, 485–516.Google Scholar
Sussman, M., Smereka, P. and Osher, S., ‘A level set approach for computing solutions to incompressible 2-phase flow’, to appear in J. Comp. Phys.Google Scholar
Tadmor, E. (1987), ‘Stability analysis of finite difference, pseudospectral and Fourier–Galerkin approximations for time dependent problems’, SIAM Review, 29, 525–555.Google Scholar
Taylor, G. I. (1950), ‘The instability of liquid surfaces when accelerated in a direction perpendicular to their planes’, Proc. Roy. Soc. London Ser. A., 201, 192–196.Google Scholar
Tryggvason, G. and Aref, H. (1983), ‘Numerical experiments on Hele–Shaw flow with a sharp interface’, J. Fluid Mech., 136, 1–30.CrossRefGoogle Scholar
Tryggvason, G. (1988), ‘Numerical simulations of the Rayleigh–Taylor instabilities’, J. Comp. Phys., 75, 253–282.Google Scholar
Tryggvason, G. (1989), ‘Simulations of vortex sheet roll-up by vortex methods’, J. Comp. Phys., 80, 1–16.Google Scholar
Tryggvason, G., Dahm, W. J. A. and Sbeih, K. (1991), ‘Fine structure of vortex sheet roll-up by viscous and inviscid simulation’, J. Fluid Eng., 113, 31–36.Google Scholar
Unverdi, S. O. and Tryggvason, G. (1992), ‘A front-tracking method for viscous, incompressible, multi-fluid flows’, J. Comp. Phys., 100, 25–37.Google Scholar
van de Vooren, A. (1980), ‘A numerical investigation of the rolling-up of vortex sheets’, Proc. Roy. Soc. London Ser. A, 373, 67–91.Google Scholar
van Dommelen, L. and Rundensteiner, E. A. (1989), ‘Fast, adaptive summation of point forces in the two-dimensional Poisson equation’, J. Comp. Phys., 83, 126–147.Google Scholar
Vinje, T. and Brevig, P. (1981), ‘Numerical simulation of breaking waves’, Adv. Water Resources, 4, 77–82.Google Scholar
Voorhees, P. W., McFadden, G. B., Boisvert, R. F. and Meiron, D. I. (1988), ‘Numerical simulation of morphological development during Ostwald ripening’, Acta Metall., 36, 207–222.Google Scholar
Voorhees, P. W. (1992), ‘Ostwald ripening of 2-phase mixtures’, Annual review of Material Sci., 22, 197–215.Google Scholar
West, B. J., Brueckner, K. A. and Janda, R. S. (1987), ‘A new numerical method for surface hydrodynamics’, J. Geophys. Res. C, 92, 11803–24.Google Scholar
Yeung, R. W. (1982), ‘Numerical methods in free-surface flows’, Ann. Rev. Fl. Mech., 14, 395–442.Google Scholar
Yudovich, V. I. (1963), ‘Non-stationary flow of an ideal incompressible liquid’, Zh. Vych. Mat., 3, 1032–1066 (in Russian)Google Scholar
Zabusky, N. J., Hughes, M. H. and Robert, K. V. (1979), ‘Contour dynamics for the Euler equations in two dimensions’, J. Comp. Phys., 30, 96–106.Google Scholar
Zabunksy, N. J. and Overman, E. A. (1983), ‘Regularization of contour dynamics algorithms, 1 Tangential regularization’, J. Comp. Phys., 52, 351–373.Google Scholar