<i>p</i>‐version least squares finite element formulation for two‐dimensional, incompressible, non‐Newtonian isothermal and non‐isothermal fluid flow

Abstract

Abstract This paper presents a p ‐ version least squares finite element formulation (LSFEF) for two‐dimensional, incompressible, non‐Newtonian fluid flow under isothermal and non‐isothermal conditions. The dimensionless forms of the diffential equations describing the fluid motion and heat transfer are cast into a set of first‐order differential equations using non‐Newtonian stresses and heat fluxes as auxiliary variables. The velocities, pressure and temperature as well as the stresses and heat fluxes are interpolated using equal‐order, C 0 ‐continuous, p ‐version hierarchical approximation functions. The application of least squares minimization to the set of coupled first‐order non‐linear partial differential equations results in finding a solution vector {δ} which makes the partial derivatives of the error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. The paper presents the implementation of a power‐law model for the non‐Newtonian Viscosity. For the non‐isothermal case the fluid properties are considered to be a function of temperature. Three numerical examples (fully developed flow between parallel plates, symmetric sudden expansion and lid‐driven cavity) are presented for isothermal power‐law fluid flow. The Couette shear flow problem and the 4:1 symmetric sudden expansion are used to present numerical results for non‐isothermal power‐law fluid flow. The numerical examples demonstrate the convergence characteristics and accuracy of the formulation.