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KELVIN–HELMHOLTZ CREEPING FLOW AT THE INTERFACE BETWEEN TWO VISCOUS FLUIDS

Part of: Incompressible viscous fluids

Published online by Cambridge University Press:  02 July 2015

LAWRENCE K. FORBES ,
RHYS A. PAUL ,
MICHAEL J. CHEN  and
DAVID E. HORSLEY
LAWRENCE K. FORBES*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania, Australia email Larry.Forbes@utas.edu.au, Rhys.Paul@utas.edu.au, David.Horsley@utas.edu.au
RHYS A. PAUL
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania, Australia email Larry.Forbes@utas.edu.au, Rhys.Paul@utas.edu.au, David.Horsley@utas.edu.au
MICHAEL J. CHEN
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania, Australia email Larry.Forbes@utas.edu.au, Rhys.Paul@utas.edu.au, David.Horsley@utas.edu.au Department of Applied Mathematics, University of Adelaide, South Australia 5005, Australia email michael.chen@adelaide.edu.au
DAVID E. HORSLEY
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania, Australia email Larry.Forbes@utas.edu.au, Rhys.Paul@utas.edu.au, David.Horsley@utas.edu.au
*
✉Larry.Forbes@utas.edu.au
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Abstract

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The Kelvin–Helmholtz flow is a shearing instability that occurs at the interface between two fluids moving with different speeds. Here, the two fluids are each of finite depth, but are highly viscous. Consequently, their motion is caused by the horizontal speeds of the two walls above and below each fluid layer. The motion of the fluids is assumed to be governed by the Stokes approximation for slow viscous flow, and the fluid motion is thus responsible for movement of the interface between them. A linearized solution is presented, from which the decay rate and the group speed of the wave system may be obtained. The nonlinear equations are solved using a novel spectral representation for the streamfunctions in each of the two fluid layers, and the exact boundary conditions are applied at the unknown interface location. Results are presented for the wave profiles, and the behaviour of the curvature of the interface is discussed. These results are compared to the Boussinesq–Stokes approximation which is also solved by a novel spectral technique, and agreement between the results supports the numerical calculations.

Keywords

curvature singularityKelvin–Helmholtz instabilityinterfacespectral representationStokes flow

MSC classification

Primary: 76D07: Stokes and related (Oseen, etc.) flows
Secondary: 76D33: Waves
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 4 , April 2015 , pp. 317 - 358
Copyright
© 2015 Australian Mathematical Society 

References

Abramowitz, M. and Stegun, I. A. (eds), Handbook of mathematical functions (Dover, New York, 1972).Google Scholar
Baker, G. R. and Pham, L. D., “A comparison of blob-methods for vortex sheet roll-up”, J. Fluid Mech. 547 (2006) 297–316 doi:10.1017/S0022112005007305.CrossRefGoogle Scholar
Barnea, D. and Taitel, Y., “Kelvin–Helmholtz stability criteria for stratified flow: viscous versus non-viscous (inviscid) approaches”, Int. J. Multiphase Flow 19 (1993) 639–649 doi:10.1016/0301-9322(93)90092-9.CrossRefGoogle Scholar
Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, Cambridge, 1967).Google Scholar
Chandrasekhar, S., Hydrodynamic and hydromagnetic stability (Dover, New York, 1981).Google Scholar
Chen, M. J. and Forbes, L. K., “Accurate methods for computing inviscid and viscous Kelvin–Helmholtz instability”, J. Comput. Phys. 230 (2011) 1499–1515 doi:10.1016/j.jcp.2010.11.017.CrossRefGoogle Scholar
Cowley, S. J., Baker, G. R. and Tanveer, S., “On the formation of Moore curvature singularities in vortex sheets”, J. Fluid Mech. 378 (1999) 233–267 doi:10.1017/S0022112098003334.CrossRefGoogle Scholar
Drazin, P. G. and Reid, W. H., Hydrodynamic stability, 2nd edn (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Eggers, J. and Villermaux, E., “Physics of liquid jets”, Rep. Progr. Phys. 71 (2008) 1–79 doi:10.1088/0034-4885/71/3/036601.CrossRefGoogle Scholar
Faltinsen, O. and Timokha, A., “An adaptive multimodal approach to nonlinear sloshing in a rectangular tank”, J. Fluid Mech. 432 (2001) 167–200http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=78569.CrossRefGoogle Scholar
Forbes, L. K., “The Rayleigh–Taylor instability for inviscid and viscous fluids”, J. Engrg. Math. 65 (2009) 273–290 doi:10.1007/s10665-009-9288-9.CrossRefGoogle Scholar
Forbes, L. K., “How strain and spin may make a star bi-polar”, J. Fluid Mech. 746 (2014) 332–367 doi:10.1017/jfm.2014.130.CrossRefGoogle Scholar
Forbes, L. K., Chen, M. J. and Trenham, C. E., “Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow”, J. Comput. Phys. 221 (2007) 269–287 doi:10.1016/j.jcp.2006.06.010.CrossRefGoogle Scholar
Forbes, L. K. and Cosgrove, J. M., “A line vortex in a two-fluid system”, J. Engrg. Math. 84 (2014) 181–199 doi:10.1007/s10665-012-9606-5.CrossRefGoogle Scholar
Hamming, R. W., Numerical methods for scientists and engineers (McGraw-Hill, New York, 1973).Google Scholar
Horsley, D. E. and Forbes, L. K., “A spectral method for Faraday waves in rectangular tanks”, J. Engrg. Math. 79 (2013) 13–35 doi:10.1007/s10665-012-9562-0.CrossRefGoogle Scholar
Krasny, R., “Desingularization of periodic vortex sheet roll-up”, J. Comput. Phys. 65 (1986) 292–313 doi:10.1016/0021-9991(86)90210-X.CrossRefGoogle Scholar
Kreyszig, E., Advanced engineering mathematics, 9th edn (Wiley, New York, 2006).Google Scholar
Li, J., Renardy, Y. Y. and Renardy, M., “A numerical study of periodic disturbances on two-layer Couette flow”, Phys. Fluids 10 (1998) 3056–3071 doi:10.1063/1.869834.CrossRefGoogle Scholar
Moore, D. W., “Spontaneous appearance of a singularity in the shape of an evolving vortex sheet”, Proc. R. Soc. Lond. A 365 (1979) 105–119 doi:10.1098/rspa.1979.0009.Google Scholar
Ockendon, H. and Ockendon, J. R., Viscous flow (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Pozrikidis, C., “Instability of two-layer creeping flow in a channel with parallel-sided walls”, J. Fluid Mech. 351 (1997) 139–165 doi:10.1017/S0022112097007052.CrossRefGoogle Scholar
Shadloo, M. S. and Yildiz, M., “Numerical modeling of Kelvin–Helmholtz instability using smoothed particle hydrodynamics”, Internat. J. Numer. Methods Engrg. 87 (2011) 988–1006 doi:10.1002/nme.3149.CrossRefGoogle Scholar
Siegel, M., “Cusp formation for time-evolving bubbles in two-dimensional Stokes flow”, J. Fluid Mech. 412 (2000) 227–257 doi:10.1017/S002211200000834X.CrossRefGoogle Scholar
Tauber, W., Unverdi, S. O. and Tryggvason, G., “The nonlinear behavior of a sheared immiscible fluid interface”, Phys. Fluids 14 (2002) 2871–2885 doi:10.1063/1.1485763.CrossRefGoogle Scholar
Tryggvason, G., Dahm, W. J. A. and Sbeih, K., “Fine structure of vortex sheet rollup by viscous and inviscid simulation”, J. Fluids Engrg. 113 (1991) 31–36 doi:10.1115/1.2926492.CrossRefGoogle Scholar
Van Dyke, M., An album of fluid motion (Parabolic Press, Stanford, CA, 1982).CrossRefGoogle Scholar
von Winckel, G., lgwt.m, at: MATLAB file exchange website (2004), http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=4540&objectType=file.Google Scholar