and

Dr. Comer Duncan

Physics Department

Bowling Green State University

gcd@chandra.bgsu.edu

haowen@bgnet.bgsu.edu

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

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.