### The Dumbbell Model

To begin understanding the behavior of polymers in flow, Leal and his colleagues have initially adopted a simple mathematical model: the elastic dumbbell model, two spheres with a spring between them. "The dumbbell is oriented and stretched by the flow," says Leal, "and the resulting tension in the spring makes it want to return to its non-stretched state." Though the dumbbell is a simple shape, it provides a reasonable model of the polymer chain dynamics, and the resulting mathematical model for flow is more complicated than previous models that lack a description of molecular structure.

For the last three years, Leal and colleagues have been running and modifying their computer code at Pittsburgh, complementing this theoretical research with laboratory work. "What we're finding," he says, "is that the relatively simple computational models we're studying do a surprisingly good job of capturing many important qualitative features of the flows we're studying in the laboratory."

Leal sees these simple models as a first step. "They help to explain the connection between what you see macroscopically in the fluid mechanics and what's actually happening at the microscopic level. In most of the problems we're looking at computationally, there's a one-to-one counterpart with a flow geometry we're looking at in the laboratory."

An example is how polymers flow between two rotating side-by-side cylinders in an experimental device called a two-roll mill. Leal's simulations of flow characteristics such as velocity and the polymer's shape and how much it is stretched show good agreement with observed data. "We find encouraging qualitative and even quantitative comparisons," says Leal, "between the model calculations and what we see experimentally."

Simulating a Stretched Polymer

These graphics represent simulations of polymer flow in a planar cross-slot device. Dilute polymer solution enters from both sides of the input channel (horizontal axis). The two inflow streams splash into one another, creating a stagnation flow pattern at the center, where they merge and flow out both directions of the cross channel (vertical axis). The black lines indicate flow streamlines. The vertical colored stripe (right) is a closeup of the stagnation region.

Colors emanating from the stagnation point represent how much the polymers stretch with reference to equilibrium state. The polymers enter unstretched, and near the stagnation point stretch out. Since the relaxation process takes time, the polymers remain stretched a distance downstream. With this model (a "dumbbell model," adapted from Hinch and DeGennes), maximum stretch is 300 (dimensionless length squared), compared to two at equilibrium. Thus at the stagnation point, the polymer is stretched nearly to maximum.

"The velocity gradient is approximately constant at the stagnation point," says Leal. "Maximum extension occurs along the outflow axis beginning from the stagnation point, which indicates that polymer conformation is dominated by residence time in this constant velocity gradient region. In other words, conformation depends on the total strain experienced by the polymer molecule, rather than the strain rate."