Lattice Boltzmann Simulations of Drop Deformation and Breakup in a Simple Shear Flow
A new lattice Boltzmann method for multicomponent immiscible fluids with the same density is proposed. In the method, the Swift et al. model (1995) is used as an index function for the calculation of interface profiles, and another distribution function i
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Abstract. A new lattice Boltzmann method for multicomponent immiscible fluids with the same density is proposed. In the method, the Swift et al. model (1995) is used as an index function for the calculation of interface profiles, and another distribution function is introduced for the calculation of pressure and velocity of the fluids. The method is applied to simulations of drop deformation and breakup in a simple shear flow. Two types of drop breakup behavior can be simulated, and the results agree well with experimental results.
1
Introduction
The problems of drop deformation and breakup in viscous flows are very important in many science and engineering fields, such as mixing in multiphase viscous systems, blending of molten polymers, emulsion formation and rheology, deformation of biological cells, and so on [1]. The numerical approach to the problems has difficulties due to deformable boundaries of immiscible fluids, rupture of domain boundaries, and modeling of interfacial tension. Recently, the lattice Boltzmann method (LBM) has developed into an alternative and promising numerical scheme for simulating multicomponent fluid flows [2-5]. In this paper, a new lattice Boltzmann method for multicomponent immiscible fluids with the same density is proposed. In the method, by using the idea by He et a1. [4], two distribution functions are introduced for the calculation of interface profiles and for the calculation of pressure and velocity of the fluids. The Swift et a1. model [5] is used for the calculation of interface profiles. The method is applied to simulations of drop deformation and breakup in a simple shear flow.
2
Numerical Method
Non-dimensional variables, which are defined by using a characteristic length L, a characteristic particle speed c, a characteristic time scale to = L/U where U is a characteristic flow speed, and a reference density Po, are used as in Ref. [6]. In the LBM, a modeled fluid, composed of identical particles whose velocities are restricted to a finite set of N vectors ci(i = 1,2,··· , N), is considered. The fifteen-velocity model (N = 15) [7] is used in the present paper. The velocity vectors of this model are given by N. Satofuka (ed.), Computational Fluid Dynamics 2000 © Springer-Verlag Berlin Heidelberg 2001
500
T. Inamuro et aI. [Cl, C2, Ca, C4, C5, C6, C7, Cs, Cg, C10, Cll, C12, Cla, C14, C15]
[
=
0 1 0 0 -1 0 0 1 -1 1 1 -1 1 -1 -1] 1 0 0 -1 0 1 1 -1 1 -1 -1 1 -1 . 0 1 0 0 -1 1 1 1 -1 -1 -1 -1 1
o0 o0
(1)
The physical space is divided into a cubic lattice, and the evolution of particle population at each lattice site is computed. Two particle distribution functions, Ii and gi, are used. Ii is used as an index function for the calculation of interface profiles, and gi is used for the calculation of pressure and velocity of the fluid. The evolution of the particle distribution functions Ii(~,t) and gi(~,t) with velocity Ci at the point ~ and at time t is computed by the following equations:
rr
+ CiLlx, t + Llt) -
1 = -[Ii(~, t) Tf
f(!(~, t)l,
(2)
gi(~ + CiL
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