Modelling and Simulations of non-Newtonian fluid flows

Open Access
Author:
Lee, Young-Ju
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
June 17, 2004
Committee Members:
  • Jinchao Xu, Committee Chair
  • Chun Liu, Committee Member
  • Andrew Leonard Belmonte, Committee Member
  • James Spiro Vrentas, Committee Member
  • Nigel David Higson, Committee Member
Keywords:
  • conformation tensor
  • non-Newtonian models
  • constitutive equation
  • positivity preserving scheme
  • Riccati differential equation
  • Lie derivative
  • multigrid
  • singular problems
  • falling sphere simulations
  • the Johnson-Segalman model
Abstract:
We observe that various rate-type non-Newtonian constitutive equations can be recast into the well-known symmetric Riccati equations. From the careful study on the Riccati form of various models, some robust and stable discretizations of the rate-type models are constructed. Discrete analogue of energy estimates have been derived and confirm the stability of our new schemes. As a consequence of discrete energy estimates, a global existence and uniqueness of an approximate solution is established regardless of the size of the ``Weissenberg number". We discuss how to solve the resulting discrete problem based upon the preconditioned MINRES (Minimum Residual) method and fast and efficient solver has been constructed. Moreover, based on the fact that the efficient preconditioner should be constructed for the Laplace equation with pure Neumann boundary condition, a certain framework on the convergence analysis of the method of successive subspace corrections for singular problems has been provided in a Hilbert space setting. Extensive numerical studies on a falling sphere through the Johnson-Segalman fluids are performed. The main motivation behind our numerical studies is from some belief that a range of parameters exhibiting a non-monotonic relation between the shear rate and the strain rate displayed for the steady shear flow of the Johnson-Segalman model might produces a continual and sustained oscillation of a falling sphere in a worm-like micellar fluid. In contrast to such a belief, our numerical experiments did not show a sustaining oscillation of a falling sphere, rather a transient oscillation. We then conclude that a property of model like a non-monotonic shear stress-strain rate can not alone be used for explaining a continual oscillation of a falling sphere in a worm-like micellar fluid. Finally, we report some intriguing numerical experiments that show how the slip parameter ``a" affects the pattern of oscillations of sphere and also a negative wake.