A major area of thrust has emerged in our group, in the development of asymptotic methods to describe high Weissenberg number viscoelastic flows (i.e. highly elastic and viscous liquids). Features associated with the high Weissenberg number limit, such as stress boundary layers and corner singularities, have been prominent and troublesome in numerical simulations of viscoelastic flows, and the lack of a mathematical understanding has been a major impediment to progress in the field. Michael Renardy's research over the past few years has, for the first time, developed asymptotic theories which have shed light on the nature of viscoelastic flows at high Weissenberg number. Our works form an important contribution to the Federal Government's strategic initiatives in the areas of materials and manufacturing, and have provided fundamental insight in key problems that relate to technology.
Within the last decade, numerical simulations of viscoelastic flows have made significant progress, and it has become feasible to study complex flows at moderate or even high Weissenberg numbers in a variety of geometries. Indeed, many of the remaining difficulties in numerical simulation can be linked to features which arise in the limit where the Weissenberg number (a dimensionless measure of the elasticity of the fluid) is high. Not surprisinly, these features are most pronounced for the upper convected Maxwell model, which, among the commonly used models, has the highest elastic stress response at high deformation rates.
Features associated with the high Weissenberg number limit include rapid variations of stresses near boundaries and separating streamlines, as well as stress singularities at reentrant corners. The inability to resolve these singular features continues to substantially impede numerical simulations of these flows.
Michael Renardy's research over the past few years has succeeded in elucidating many of the salient features of high Weissenberg number flows and has paved the way for a qualitative understanding of these flows. The first step is achieved by formally setting the Weissenberg number equal to infinity in the equations of motion. This is analogous to passing from the Navier-Stokes to the Euler equations in the limit of infinite Reynolds number. Rather curiously, for the case of the upper convected Maxwell model, the resulting equations for the infinite Weissenberg number limit can be transformed to the Euler equations, but with a rather unusual ``equation of state," which relates the divergence of the ``velocity field" to the ``density" instead of the usual pressure-density relationship. The incompressible Euler equations remain an important special case. Some known flows of the Euler equations, in particular potential flow, can be employed to describe viscoelastic flows at high Weissenberg number.
Near boundaries or stagnation points, the asymptotic description in terms of the Euler equations breaks down. This is because the constitutive law of the upper convected Maxwell fluid (and many similar models) involves the convected derivative of the stress tensor, and while this term enters the dominant balance in the formal limit of infinite Weissenberg number, it vanishes at points where the velocity is zero. Hence a different balance of terms applies near these points. The result is the formation of boundary layers, which become sharper as the Weissenberg number increases. The use of asymptotic methods to describe such boundary layers is pioneered in his paper for the upper convected Maxwell model. In the paper with Thomas Hagen, the approach is extended to other constitutive models. In agreement with the experience from numerical simulation, the boundary layers for the other models are predicted to sharpen much more slowly as the Weissenberg number is increased.
Reentrant corners (i.e. corners larger than 180 degrees) in the boundary of the flow domain lead to singularities where the velocity gradients and stresses are infinite. Hence, regardless of the value of the Weissenberg number for the global flow, we should view such flows as having an infinite local Weissenberg number at the corner, and the use of high Weissenberg number asymptotics is appropriate for the analysis of the local behavior. In Renardy's paper, the reentrant corner behavior of the upper convected Maxwell model is analyzed using the method of matched asymptotics. There is a core region away from the boundaries, where the flow can be described by a potential flow solution of the Euler equations. Next to each wall, there is a cusp-shaped boundary layer in which the solution is given by a similarity solution of the boundary layer equations. Other constitutive models again show boundary layers which are much less sharp.