Abstract.
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Pozrikidis, Instability of multi-layer channel and film flows. Adv. Appl. Mech. 40 (2004) In press.
C.S. Yih (1967) ArticleTitleInstability due to viscosity stratification J.Fluid Mech. 27 337–352 Occurrence Handle1967JFM....27..337Y Occurrence Handle0144.47102
A.L. Frenkel D. Halpern (2002) ArticleTitleStokes-flow instability due to interfacial surfactant Phys. Fluids. 14 45–48 Occurrence Handle10.1063/1.1483838 Occurrence Handle2002PhFl...14L..45F
D. Halpern A.L. Frenkel (2003) ArticleTitleDestabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers J. Fluid Mech. 485 191–220 Occurrence Handle10.1017/S0022112003004476 Occurrence Handle2003JFM...485..191H Occurrence Handle1052.76022
M. Blyth C. Pozrikidis (2004) ArticleTitleEffect of surfactants on the stability of two-layer channel flow J. Fluid Mech. 505 59–86 Occurrence Handle10.1017/S0022112003007821 Occurrence Handle2004JFM...505...59B Occurrence Handle1064.76043 Occurrence Handle2258992
M.G. Blyth C. Pozrikidis (2004) ArticleTitleEffect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis J. Engng. Math. 50 329–341 Occurrence Handle10.1007/s10665-004-3691-z Occurrence Handle2116256 Occurrence Handle1073.76036
C. Pozrikidis (1997) ArticleTitleInstability of two-layer creeping flow in a channel with parallel-sided walls J. Fluid Mech. 351 139–165 Occurrence Handle1997JFM...351..139P Occurrence Handle0918.76024 Occurrence Handle1488832
C. Pozrikidis (1998) ArticleTitleGravity-driven creeping flow of two adjacent layers or superimposed films through a channel and down a plane wall J. Fluid Mech. 371 345–376 Occurrence Handle10.1017/S0022112098002213 Occurrence Handle1998JFM...371..345P Occurrence Handle0946.76027 Occurrence Handle1650181
A.V. Coward Y.Y. Renardy M. Renardy J.R. Richards (1997) ArticleTitleTemporal evolution of periodic disturbances in two-layer Couette flow J. Comp. Phys. 132 346–361 Occurrence Handle1445002 Occurrence Handle0880.76055 Occurrence Handle1997JCoPh.132..346C
J. Li Y.Y. Renardy A Renardy M. (1998) ArticleTitlenumerical study of periodic disturbances in two-layer Couette flow Phys. Fluids. 10 3056–3071 Occurrence Handle10.1063/1.869834 Occurrence Handle1998PhFl...10.3056L
R.R. Yiu K.P. Chen (1996) Numerical experiments on disturbed two-layer flows in a channel Y. Renardy A.V. Coward D. Papageorgiou S.M. Sun (Eds) Advances in Multi-Fluid Flows SIAM Philadelphia 368–382
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).
P. Yecko S. Zaleski (2000) ArticleTitleTwo-phase shear instability: waves, fingers, and drops Ann. NY Acad. Sci. 898 127–143 Occurrence Handle2000NYASA.898..127Y Occurrence Handle10.1111/j.1749-6632.2000.tb06169.x
G. Tryggvason S.O. Unverdi (1999) The shear breakup of an immiscible fluid interface W. Shyy R. Narayanan (Eds) Fluid Dynamics at Interfaces Cambridge University Press New York 142–155
J. Zhang M.J. Miksis S.G. Bankoff G. Tryggvason (2002) ArticleTitleNon-linear dynamics of an interface in an inclined channel Phys. Fluids. 14 1877–1885 Occurrence Handle10.1063/1.1433495 Occurrence Handle2002PhFl...14.1877Z Occurrence Handle1906223
C.S. Peskin (2002) ArticleTitleThe immersed boundary method Acta Numer. 11 479–517 Occurrence Handle2009378 Occurrence Handle1123.74309
K. Sheth C. Pozrikidis (1995) ArticleTitleEffects of inertia on the deformation of liquid drops in simple shear flow Comput. Fluids. 24 101–119 Occurrence Handle10.1016/0045-7930(94)00025-T Occurrence Handle0826.76054
C. Pozrikidis (1997) Introduction to Theoretical and Computational Fluid Dynamics Oxford University Press New York 675 Occurrence Handle0886.76002
X. Li C. Pozrikidis (1997) ArticleTitleThe effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow J. Fluid Mech. 341 165–194 Occurrence Handle1997JFM...341..165L Occurrence Handle1457708 Occurrence Handle0892.76013
S. Yon C. Pozrikidis (1998) ArticleTitleA finite-volume/boundary-element method for flow past interfaces in the presence of surfactants, with application to shear flow past a viscous drop Comput. Fluids. 27 879–902 Occurrence Handle10.1016/S0045-7930(98)00013-9 Occurrence Handle0978.76058
A.W. Adamson (1990) Physical Chemistry of Surfaces EditionNumber5 Wiley New York
C. Pozrikidis (1998) Numerical Computation in Science and Engineering Oxford Univ. Press New York 627 Occurrence Handle0971.65001
C. Pozrikidis (2003) ArticleTitleOn the relationship between the pressure and the projection function for the numerical computation of incompressible flow Euro. J. Mech. B/Fluids. 22 105–121 Occurrence Handle1051.76595 Occurrence Handle1980462 Occurrence Handle2003EJMF...22..105P
C.S. Peskin (1977) ArticleTitleNumerical analysis of blood flow in the heart J. Comp. Phys. 25 220–252 Occurrence Handle0403.76100 Occurrence Handle490027 Occurrence Handle1977JCoPh..25..220P
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pozrikidis, C. Effect of inertia on the Marangoni instability of two-layer channel flow, Part I: numerical simulations. J Eng Math 50, 311–327 (2004). https://doi.org/10.1007/s10665-004-3690-0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10665-004-3690-0