In the present study, the numerical simulation of lateral migration of a three-dimensional deformable bubble in a compound laminar Couette and Poiseuille flow is studied at finite Reynolds numbers. The Navier-Stokes equations are solved for incompressible fluids using a finite-difference method on a regular, fixed, and staggered grid. Interface is tracked explicitly by connecting marker points through a front-tracking method on a triangular moving grid. The effects of surface tension are also accounted for by adding an appropriate source term to the governing equations. The results show that a bubble, regardless of its original position, will be fixed in an equilibrium position between the wall and the centerline of channel. It is observed that by increase of the bubble radius, the bubble migrates to an equilibrium position closer to the centerline. Negative pressure gradient causes that the deformation of bubble increases, so it reaches a steady-state position closer to the center line.
migration, finite-difference/front-tracking method, pressure gradient, combined couette-poiseuille flow
 McCormik ME, Bhattacharyya R. (1973). Drag reduction of a submersible hull by electrolysis. Naval Engineers Journal 85: 11-16.
 Esmaeeli A, Tryggvason G. (1998). Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays. Journal of Fluid Mechanics 377: 313-345.
 Segre G, Silberberg A. (1962). Behavior of macroscopic rigid spheres in Poiseuille flow. Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams. J. Fluid Mech. 14: 115-135.
 Segre G, Silberberg A. (1962). Behavior of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation. J. Fluid Mech. 14: 136-157.
 Matas JP, Morris JF, Guazzelli E. (2004). Inertial migration of rigid spherical particles in Poiseuille flow. J. Fluid Mech. 515: 171.
 Griggs AJ, Zinchenko AZ, Davis RH. (2008). Gravity-driven motion of a deformable drop or bubble near an inclined plane at low Reynolds number. Int. J. Multiphase Flow 34: 408-418.
 Li X, Pozrikidis C. (2000). Wall-bounded shear flow and channel flow of suspensions of liquid drops. Int. J. Multiphase Flow 26: 1247.
 Nourbakhsh A, Mortazavi S. (2010). A three-dimensional study of the motion of a drop in plane Poiseuille flow at finite Reynolds numbers. Iranian J Sci Technol Trans B Eng. 34: 179-196.
 Tsai TM, Miksis MJ. (1994). Dynamics of a drop in a constricted capillary tube. J. Fluid Mech. 274: 197-17.
 Mortazavi S, Tafreshi MM. (2013). On the behavior of suspension of drops on an inclined surface. Physica A 392: 58–71.
 Bayareh M, Mortazavi S. (2011). Three-dimensional numerical simulation of drops suspended in simple shear flow at finite Reynolds numbers. International Journal of Multiphase Flow 37: 1315–1330.
 Bayareh M, Mortazavi S. (2011). Binary collision of drops in simple shear flow at finite Reynolds numbers: Geometry and viscosity ratio effects. Advances in Engineering Software 42: 604–611.
 Amiri M, Mortazavi S. (2013). Three-dimensional numerical simulation of sedimenting drops inside a vertical channel. International Journal of Multiphase Flow 56: 40–53.
 Mortazavi S, Tryggvason GA. (2011). Numerical study of the motion of drops in Poiseuille flow. Part 1. Lateral migration of one drop. Journal of Fluid Mechanics 411: 325-350.
 Unverdi SO, Tryggvason GA. (1992). Front-tracking method for viscous incompressible multi-fluid flows. J. Comput. Phys. 100: 25-82.
 Unverdi SO, Tryggvason G. (1992). Computations of multi-fluid flows. Physics 60: 70-83.
 Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N, Tauber W, Jan YJ. (2001). A front tracking method for the computations of multiphase flow. Journal of Computational Physics. 169: 708-759.
 Adams J. (1989). Mudpack: Multigrid FORTRAN software for the efficient solution of linear elliptic partial differential equations. Appl. Math. Comput. 34: 113-146.
 Feng J, Hu HH, Joseph DD. (1994). Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261: 95–134.
 Feng J, Hu HH, Joseph DD. (1994). Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille. J. Fluid Mech. 277: 271–301.
 Taylor GI. (1934). The formation of emulsions in definable fields of flow. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, pp. 501-523.
 Schleizer AD. Bonnecaze RT. (1999). Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows. J. Fluid Mech. 383: 29-54.
 Doddi SK, Bagchi P. (2008). Lateral migration of a capsule in a plane Poiseuille flow in a channel. Int. J. Multiphase Flow. 34: 966-986.
 Martinez MJ, Udell KS. (1990). Axisymmetric creeping motion of drops though circular tubes. J. Fluid Mech. 210: 565-591.