The effect of shear flow on droplets of one fluid freely suspended in another immiscible fluid is a problem of longstanding interest. Long ago, Taylor considered a droplet of Newtonian fluid suspended in the shear flow of a second Newtonian fluid. He estimated the largest stable droplet radius r T by balancing the surface stresses due to interfacial tension and viscous stress due to shear flow. For two fluids with equal viscosity and neutrally buoyant drops (i.e. equal density), Taylor found that the , where is the viscosity, is the interfacial tension coefficient, and is the shear rate. This simple estimate shows that large shear rates or reduced surface tension results in smaller droplets. Droplets larger than the scale of r T break up in the shear flow, while droplets smaller than r T are stable.
From a numerical point of view, the droplet deformation and breakup problem is extremely challenging. The traditional modeling approach, which involves solving hydrodynamic partial differential equations, has had only limited success. The equations of motion must be solved for the flow both inside and outside the droplet, with the boundary condition applied on its surface. However, the shape of the droplet is not known a priori, and must be determined as part of the solution. Because of these complications, there have not been many successful numerical studies of droplet deformation and breakup.
We are currently investigating the droplet deformation and breakup using a recently discovered novel numerical technique, known as the Lattice-Boltzmann Method (LBM). We believe that this method will provide a powerful alternative to the standard Navier-Stokes equations, and will dramatically change the way people think about the simulation of multi-phase flow dynamics. In particular, the information about the interface boundary, the droplet size and shape, droplet breakup process, and the flow field can all automatically arise from the solutions. One of the greatest advantages of LBM is its simplicity for massive parallel computing. We have developed an very efficient code and our preliminary results are very encouraging. We have achieved the first numerical evidence of droplet breakup in simple shear flow.
At low shear rate Figure 1 =0.2 (dimensionless), shows the deformation of droplet which results from balancing the interfacial tension force with the viscous force. When the interfacial tension forces can no longer balance the viscous forces as one increases the shear rate, the deformations become unstable and the droplet breaks up.
Figure 2 depicts a shear rate of =0.3.
Figure 3 depicts a shear rate of =0.4.
Figure 4 depicts a shear rate of =0.5.
For a view of a droplet breakup in three dimensions, look at droplet_breakup.mpg in this directory.