Journal of Engineering Mathematics

, Volume 50, Issue 2–3, pp 311–327

Effect of inertia on the Marangoni instability of two-layer channel flow, Part I: numerical simulations



A numerical method is developed for simulating the flow of two superposed liquid layers in a two-dimensional channel confined between two parallel plane walls, in the presence of an insoluble surfactant. The algorithm combines Peskin’s immersed-interface method with the diffuse-interface approximation, wherein the step discontinuity in the fluid properties is replaced by a transition zone defined in terms of a mollifying function. A finite-difference method is implemented for integrating the generalized Navier–Stokes equation incorporating the jump in the interfacial traction, and a finite-volume method is implemented for solving the surfactant transport equation over the evolving interface. The accuracy of the overall scheme is confirmed by successfully comparing the numerical results with the predictions of linear stability analysis and numerical simulations based on a boundary-element method for Stokes flow. Results for selected case studies suggest that inertial effects have a mild effect on the growth rate of the surfactant-induced Marangoni instability.


channel flow immersed-interface method liquid films Marangoni instability surfactants 


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  1. C. Pozrikidis, Instability of multi-layer channel and film flows. Adv. Appl. Mech. 40 (2004) In press.Google Scholar
  2. Yih, C.S. 1967Instability due to viscosity stratificationJ.Fluid Mech.27337352ADSMATHGoogle Scholar
  3. Frenkel, A.L., Halpern, D. 2002Stokes-flow instability due to interfacial surfactantPhys. Fluids.144548CrossRefADSGoogle Scholar
  4. Halpern, D., Frenkel, A.L. 2003Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbersJ. Fluid Mech.485191220CrossRefADSMATHGoogle Scholar
  5. Blyth, M., Pozrikidis, C. 2004Effect of surfactants on the stability of two-layer channel flowJ. Fluid Mech.5055986CrossRefADSMATHMathSciNetGoogle Scholar
  6. Blyth, M.G., Pozrikidis, C. 2004Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysisJ. Engng. Math.50329341CrossRefMathSciNetMATHGoogle Scholar
  7. Pozrikidis, C. 1997Instability of two-layer creeping flow in a channel with parallel-sided wallsJ. Fluid Mech.351139165ADSMATHMathSciNetGoogle Scholar
  8. Pozrikidis, C. 1998Gravity-driven creeping flow of two adjacent layers or superimposed films through a channel and down a plane wallJ. Fluid Mech.371345376CrossRefADSMATHMathSciNetGoogle Scholar
  9. Coward, A.V., Renardy, Y.Y., Renardy, M., Richards, J.R. 1997Temporal evolution of periodic disturbances in two-layer Couette flowJ. Comp. Phys.132346361MathSciNetMATHADSGoogle Scholar
  10. Li, J., Renardy, Y.Y., Renardy M., A 1998numerical study of periodic disturbances in two-layer Couette flowPhys. Fluids.1030563071CrossRefADSGoogle Scholar
  11. Yiu, R.R., Chen, K.P. 1996

    Numerical experiments on disturbed two-layer flows in a channel

    Renardy, Y.Coward, A.V.Papageorgiou, D.Sun, S.M. eds. Advances in Multi-Fluid FlowsSIAMPhiladelphia368382
    Google Scholar
  12. S. Zaleski, J. Li, R. Scardovelli and G. Zanetti, Direct simulation of multiphase flows with density variations. In: L. Fulachier, J.L. Lumley, and F. Anselmet (eds.), IUTAM Symposium on Variable Density Low Speed Turbulent Flows. Proc. IUTAM Symposium, Marseille, France, 1996. Dordrecht: Kluwer (1997).Google Scholar
  13. Yecko, P., Zaleski, S. 2000Two-phase shear instability: waves, fingers, and dropsAnn. NY Acad. Sci.898127143ADSCrossRefGoogle Scholar
  14. Tryggvason, G., Unverdi, S.O. 1999

    The shear breakup of an immiscible fluid interface

    Shyy, W.Narayanan, R. eds. Fluid Dynamics at InterfacesCambridge University PressNew York142155
    Google Scholar
  15. Zhang, J., Miksis, M.J., Bankoff, S.G., Tryggvason, G. 2002Non-linear dynamics of an interface in an inclined channelPhys. Fluids.1418771885CrossRefADSMathSciNetGoogle Scholar
  16. Peskin, C.S. 2002The immersed boundary methodActa Numer.11479517MathSciNetMATHGoogle Scholar
  17. Sheth, K., Pozrikidis, C. 1995Effects of inertia on the deformation of liquid drops in simple shear flowComput. Fluids.24101119CrossRefMATHGoogle Scholar
  18. Pozrikidis, C. 1997Introduction to Theoretical and Computational Fluid DynamicsOxford University PressNew York675MATHGoogle Scholar
  19. Li, X., Pozrikidis, C. 1997The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flowJ. Fluid Mech.341165194ADSMathSciNetMATHGoogle Scholar
  20. Yon, S., Pozrikidis, C. 1998A finite-volume/boundary-element method for flow past interfaces in the presence of surfactants, with application to shear flow past a viscous dropComput. Fluids.27879902CrossRefMATHGoogle Scholar
  21. Adamson, A.W. 1990Physical Chemistry of Surfaces5WileyNew YorkGoogle Scholar
  22. Pozrikidis, C. 1998Numerical Computation in Science and EngineeringOxford Univ. PressNew York627MATHGoogle Scholar
  23. Pozrikidis, C. 2003On the relationship between the pressure and the projection function for the numerical computation of incompressible flowEuro. J. Mech. B/Fluids.22105121MATHMathSciNetADSGoogle Scholar
  24. Peskin, C.S. 1977Numerical analysis of blood flow in the heartJ. Comp. Phys.25220252MATHMathSciNetADSGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan Diego, La JollaUSA

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