Numerical Modeling of Viscoelastic Drops

John Gemmer , Millersville University Tobin Isaac, Rice University

Mark Sussman , Florida State University

In an experiment at Pennsylvania State University [Belmonte & Sostarecz], viscoelastic drops were observed falling through a viscous medium. They observed that the trailing surface of thedrops would dimple inwards similarly to Rayleigh-Taylor instability. In the case of larger drops, a toroidal shape formed.These effects could not be attributed to inertial forces, so they were conjectured to be the result of the viscoelastic properties of the fluid. To better understand these effects, we used numerical methods to model the experiment. Numerical modeling allowed us to track not only the shape of a falling drop but the interior velocity of the fluid and the configuration tensor of the elasticity.

Abstract

Constitutive Equations

Numerical Methods

Numerical Methods (cont.)

Geometry 2.0

The program that ran these numerical simulations was Geometry 2.0. In each simulation, the drop started as a sphere with no initial velocity. Because both the initial state and the forces are symmetrical about the z-axis through the middle of the drop, only half of the boundary needs to be calculated. The effects in these simulations were slow to form, so the drop often fell over large distances. Therefore, instead of having a bubble falling, we treat the problem as a stationary drop in an upward-flowing medium, where the terminal velocity equals the current.

Results of the Simulations

In all our simulations the surface tension was held at a constant 25 dyne/cm, the density of the bubble was held at 1.27 g/cm3 and the density of the medium was held at .98g/cm3. The following parameters were allowed to vary explicitly: elastic viscosity, viscosity of the medium, viscosity of the bubble, and the radius of the bubble. The speed of the falling bubble also varied as a function of the other parameters.

Results of the SimulationsComparison of a Viscoelastic bubble to a

Newtonian bubbleSide by side comparison of a viscoelastic bubble with an elastic viscosity of 0.45P to a Newtonian bubble. Both of the bubbles have a viscosity of 4.9P and are falling through a medium with a viscosity of 9.8P. The radius of the bubbles in both cases is 1.2cm and both bubbles have a terminal velocity of 10cm/s.

Results of the Simulations (cont.)

We see in the above cross sections that both the Newtonian and the viscoelastic drops form a torus as they descend. Our experiments showed that for dimpling or toroidal effects to occur in a Newtonian drop, the medium must be much more viscous than the drop itself. In the case of viscoelastic drops, these effects could be observed even when the drop was more viscous than the medium.

Results of the Simulations (cont.)Comparison of Simulation to Experimental

Results

Clockwise from upper left: 0.76 ml 0.16% xanthan gum drop; 0.76 ml glycerol/water drop without xanthan gum; simulation of 0.76 ml Newtonian drop with 11.88 P viscosity; simulation of 0.76 ml viscoelastic drop with 11.88 P viscosity and 0.064% viscoelasticity.

Results of the Simulations (cont.)The Role of the Configuration Tensor

The image above shows the normal components of the configuration tensor. A tremendous amount of viscoelastic stress builds as the dimpling begins to occur.

Results of the Simulations (cont.)Solution Convergence

Results of the Simulations (cont.)Dimple Formation in 3 Dimensions

A 1.2cm radius viscoelastic bubble with a viscosity of 4.0P falling at a rate of 9.0cm/s through a medium with viscosity of 16.0P. The elastic viscosity of the bubble was .35P. The time step between each image is .75 seconds. Each image was created in Tecplot 10.0 using iso-surface reconstruction of the zero contour.

Results of the Simulations (cont.)Circulation Inside a Viscoelastic Drop

Viscoelastic bubble after falling for 3.75s at a rate of 9.00cm/s. The viscosity of the inside of the bubble is 4.00P while the surrounding medium has a viscosity of 16.00P. The vector field is a plot of the surrounding medium’s velocity field. As the bubble flattens out circulation begins to form inside the bubble. This causes a change in the hydrodynamic pressure of the inside and outside of the bubble which forces the bubble to dimple on top.

Reference

Depiction of the boundary marker method and the level-set method taken from:• Sethian, J.A. “Tracking Interfaces with Level Sets.” American Scientist. May-June 1997. pg.254-263

Images of experimental drops taken from:• Belmonte, A. & Sostarecz, M.C. “Motion and shape of viscoelastic drop falling through a viscous fluid.” J. Fluid. Mech. (2003), vol. 497, pp. 235-252