#### Date of Award

8-2007

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Legacy Department

Mathematical Science

#### Advisor

Ervin, Vincent J

#### Committee Member

Jenkins , Eleanor W

#### Committee Member

Warner , Daniel D

#### Abstract

In this work computational approaches to the numerical simulation of steady-state viscoelastic fluid flow are investigated. In particular, two aspects of computing viscoelastic flows are of interest: 1) the stable computation of high Weissenberg number Johnson-Segalman fluids and 2) low-order approaches to simulating the flow of fluids obeying a power law constitutive model.

The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge. For the nonlinear Johnson-Segalman constitutive model, defect-correction and continuation methods are examined for computing steady-state viscoelastic flows at high Weissenberg numbers. A two-parameter defect-correction method for viscoelastic fluid flow is presented and analyzed. In the defect step the Weissenberg number is artificially reduced to solve a stable nonlinear problem. The approximation is then improved in the correction step using a linearized correction iteration. Numerical experiments support the theoretical results and demonstrate the effectiveness of the method. Continuation methods with natural and pseudo-arclength parametrizations are also examined. The implementation of these methods is discussed and computations with the methods are performed. Numerical results indicate that, for the discrete approximation of the flow equations, a limiting Weissenberg value exists and represents an elastic limit for stable simulation.

Some shear-thinning viscoelastic fluids (e.g. paint, blood) are modeled using a power law constitutive equation. Through the introduction of an auxiliary variable with relevant physical meaning, the variational formulation for the steady-state flow of these fluids can be written as a two-fold saddle point problem. This approach leads to a larger system than the usual finite element approximations, however, the regularity requirements for test and trial functions are relaxed, which leads to an approximation method that is ideally suited for adaptive computation. We analyze this problem in in the Sobolev spaces $L^r$ and $W^{1,r}$, where $r$ is determined by the nonlinearity of the problem. We show existence and uniqueness of the continuous and discrete variational formulations by proving appropriate inf-sup conditions in the Sobolev spaces. Numerical approximations are computed using lowest-order polynomial and Raviart-Thomas elements.

#### Recommended Citation

Howell, Jason, "Numerical Approximation of Shear-Thinning and Johnson-Segalman Viscoelastic Fluid Flows" (2007). *All Dissertations*. 102.

https://tigerprints.clemson.edu/all_dissertations/102