SIMULATING THE AXISYMMETRIC INTERFACIAL FLOWS WITH INSOLUBLE SURFACTANT BY IMMERSED BOUNDARY METHOD

In this paper, we extend our previous work on the two-dimensional immersed boundary method for interfacial flows with insoluble surfactant to the case of three-dimensional axisymmetric interfacial flows. Although the key components of the scheme are similar in spirit to the two-dimensional case, there are two differences introduced in the present work. Firstly, the governing equations are written in an immersed boundary formulation using the axisymmetric cylindrical coordinates. Secondly, we introduce an artificial tangential velocity to the Lagrangian markers so that the uniform distribution of markers along the interface can be achieved and a modified surfactant concentration equation is derived as well. The numerical scheme still preserves the total mass of surfactant along the interface. Numerical convergence of the present scheme has been checked, and several tests for a drop in extensional flows have been investigated in detail.

[1]  Metin Muradoglu,et al.  A front-tracking method for computation of interfacial flows with soluble surfactants , 2008, J. Comput. Phys..

[2]  W. Tian,et al.  Introduction to Microfluidics , 2008 .

[3]  Jie Zhang,et al.  A front tracking method for a deformable intravascular bubble in a tube with soluble surfactant transport , 2006, J. Comput. Phys..

[4]  C. Pozrikidis,et al.  A Finite-volume/Boundary-element Method for Flow Past Interfaces in the Presence of Surfactants, with Application to Shear Flow Past a Viscous Drop , 1998 .

[5]  P. Gennes,et al.  Capillarity and Wetting Phenomena , 2004 .

[6]  Héctor D. Ceniceros,et al.  The effects of surfactants on the formation and evolution of capillary waves , 2003 .

[7]  T. Hou,et al.  Removing the stiffness from interfacial flows with surface tension , 1994 .

[8]  D. An,et al.  The effects of surfactants on drop deformation and breakup By , 2005 .

[9]  Jie Shen,et al.  An Efficient Spectral-Projection Method for the Navier–Stokes Equations in Cylindrical Geometries: I. Axisymmetric Cases , 1998 .

[10]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[11]  Jie Shen,et al.  An Efficient Spectral-Projection Method for the Navier-Stokes Equations in Cylindrical Geometries , 2002 .

[12]  Michael Siegel,et al.  Influence of insoluble surfactant on the deformation and breakup of a bubble or thread in a viscous fluid , 2007, Journal of Fluid Mechanics.

[13]  J. Lowengrub,et al.  A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant , 2004 .

[14]  Jie Li.  The effect of an insoluble surfactant on the skin friction of a bubble , 2006 .

[15]  Joel H. Ferziger,et al.  Introduction to Theoretical and Computational Fluid Dynamics , 1996 .

[16]  Ming-Chih Lai,et al.  An immersed boundary method for interfacial flows with insoluble surfactant , 2008, J. Comput. Phys..

[17]  M. Abdelgawad,et al.  Introduction to Microfluidics , 2006 .

[18]  G. Batchelor,et al.  An Introduction to Fluid Dynamics , 1968 .

[19]  Michael Siegel,et al.  Influence of surfactant solubility on the deformation and breakup of a bubble or capillary jet in a viscous fluid , 2009 .

[20]  S. Yariv,et al.  Physical Chemistry of Surfaces , 1979 .

[21]  David S. Rumschitzki,et al.  On the surfactant mass balance at a deforming fluid interface , 1996 .

[22]  H. Stone.  A simple derivation of the time‐dependent convective‐diffusion equation for surfactant transport along a deforming interface , 1990 .

[23]  Weichung Wang,et al.  A fast spectral/difference method without pole conditions for Poisson-type equations in cylindrical and spherical geometries , 2002 .

[24]  Sashikumaar Ganesan,et al.  A coupled arbitrary Lagrangian-Eulerian and Lagrangian method for computation of free surface flows with insoluble surfactants , 2009, J. Comput. Phys..

[25]  K. Stebe,et al.  Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants , 1996 .

[26]  V. Cristini,et al.  Critical behavior of drops in linear flows. I. Phenomenological theory for drop dynamics near critical stationary states , 2002 .

[27]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.