M. Behr, Rice MEMS - Flows of Viscoelastic Fluids

Modeling of Flows of Viscoelastic Fluids

[CFD Picture]

Figure 7: Stick-slip problem: Streamwise (x-axis) velocity profile with (top) and without (bottom) LS stabilization.

Viscoelastic constitutive equations, which govern the flow of common liquids such as oil, pose unique challenges for numerical analysis. The components of the deviatoric stress tensor have to be typically modeled as separate variables, increasing the computational burden. The time-dependent constitutive equations for many non-Newtonian fluids of interest have highly nonlinear and advection-dominated nature, leading to numerical stability problems. The method developed here takes advantage of stabilization techniques developed for incompressible flows of Newtonian fluids, and extends them to flows involving Maxwell-B and Oldroyd-B non-Newtonian fluids. The stabilization terms based on the LS form of the constitutive equation provide the necessary stability at large Deborah (or Weissenberg) numbers, and also allow for equal order interpolation of the deviatoric stress, velocity and pressure variables. This leads to significant reduction in the computational effort over the typical formulations, in which the compatibility between unknowns is achieved by using high-order, such as quadratic, velocity interpolation, and even higher-order, discontinuous quadratic or higher, stress interpolation. The formulation has been tested on simple benchmark problems, such as flow of a Maxwell-B fluid in a stick-slip configuration, where no-slip boundary condition in a portion of a channel is suddenly relaxed to allow full slip. This is one of the cases which demonstrate the need for either satisfying the compatibility conditions between the stress and velocity fields, or circumventing it through the use of LS stabilization, as is done with the present method. The equal-order interpolation results with and without the LS terms are shown in Figure 7. Flow of an Oldroyd-B fluid in a 4:1 contraction was also modeled at Deborah numbers up to 3.2.