Lattice Boltzmann Simulation of Drops in a Shear Flow

In this paper, we present simulation results of twoand three-dimensional motions of drops in a shear flow based on the lattice Boltzmann method (LBM), where a macroscopic fluid flow results from averaging collisions and propagations of mesoscopic particles. The binary fluid model in LBM used here can reproduce two-phase interface in a self-organizing way by repulsive interaction between particles consistent with the van der Waals-Cahn-Hilliard free energy theory. A finite difference scheme is applied to the lattice-Boltzmann equations governing time evolution of velocity distributions of particle number density. When a drop is suspended in an immiscible second liquid with the same mass and viscosity between moving parallel plates, the numerical results of deformation of drop agree with theoretical solutions and previous numerical results obtained by the volume-of-fluid (VOF) method. Breakup motions of drops in LBM are also reasonable in comparison with the critical Reynolds and capillary numbers predicted by the VOF method. In the simulations of two-drop interaction, it is shown that the breakup motion depends on not only number density of drops but also initial positioning of their volumetric center away from a halfway cross section between the plates. INTRODUCTION In recent years, the lattice Boltzmann method, LBM [1][2], has been developed as an alternative approach for simulating incompressible fluid flows and modeling physics of fluids, on statistical-thermodynamic assumptions that a fluid involves mesoscopic particles repeating collisions and propagations, and that the distributions of particles converge to a state of local equilibrium. The main advantages of the LBM, originated from the lattice gas cellular automaton method[3], are relatively easy implementation of boundary condition for solid surface with geometrical complexity, high efficiency on parallel computing, and self-organizing reproduction of interface. These are provided by the kinetic equations of particles that possesses linear convection (or streaming) operator in velocity space, local collision operator and repulsive interaction between particles. The LBM describes macroscopic variables by averaging the motion of particles from which the Navier-Stokes (NS) equations can be recovered through the chapman-Enskog expansion technique. Unlike conventional numerical methods based on the discretizations of the incompressible NS equations, the pressure in the LBM is calculated by using only the equation of sate, without solving the Poisson equation which requires time-consuming treatment. The gas-liquid model [4] has been proposed to simulate isothermal phase changes consistent with the thermodynamic theory of van der Waals-Cahn-Hillard free energy. After that, the free-energy approach was also applied to the binary fluid model [5][6], called BF model hereafter. In an enhanced gasliquid model [7], the Galilean invariance is improved remarkably. A novel thermal model [8][9] can simulate nonideal two-phase fluid flows with phase change and heat transfer. The concept of the free-energy approach in the abovementioned models is the same as that in the second gradient method (SGM) for the direct numerical simulation of liquidvapor flows using the NS equations [10]. In both of the LBM and the SGM, an interface corresponds to a volumetric transient zone across which physical properties of fluid (i.e. density and viscosity) vary continuously, and the surface tension is reproduced from the energy contribution due to density gradient without the continuous surface force model. So far, to examine applicability of the LBM in numerical analysis of two-phase fluid motions under gravity, we have considered the buoyancy effect on characterized with the nondimensional numbers, and developed the three-dimensional BF model [11][12]. The results of two-dimensional bubble motions by the LBM agreed with those by the volume-of-fluid (VOF) method [13], and the surface tension in the 3D model obeyed the Laplace’s law. It has been also verified that the BF model reproduces two-bubble coalescence and the effect of wall boundary on lateral motion of bubble in a vertical tube [14]. In this paper, we present numerical results of 2D and 3D motions of drops in a shear flow between moving parallel plates by using the lattice-Boltzmann BF model, with the immiscible two-phase scheme and the independent parameters for surface tension and interfacial thickness [15]. The purpose is to investigate deformation and rupture of two-phase interface under simple shear stress for generating fine drops. It is