K-Version Least Squares Finite Element Processes for 2-D Generalized Newtonian Fluid Flows

Abstract

In this paper, a mathematical and computational framework based on h, p, k and variationally consistent (VC) integral form is utilized to present a finite element computational process for two dimensional steady, isothermal as well as non-isothermal fluid flows for power law and Carreau models of viscosity. Strong form of the governing differential equations is presented and utilized in dependent variables P, u, v and T. Since the differential operator is non-linear, Galerkin method, Petrov Galerkin method, method of weighted residuals and Galerkin method with weak forms yield integral form that are variationally inconsistent (VIC). Least Squares Processes on the other hand, yield integral forms that are VC. Computational processes based on VC integral forms are unconditionally stable and non-degenerate and hence do not require the use of problem dependent upwinding methods. The parameter k controlling global differentiability of the approximation permits higher order global approximations, which are necessitated by the physics in governing differential equations (GDEs). The behavior of the power law and Carreau models of shear thinning fluid behavior is investigated using theoretical solutions (when possible) as well as numerical simulations using the proposed methodology. Fully developed flow between parallel plates, steady pressure driven Couette flow and lid driven cavity are used as model problems.