Finite Element Analysis of Non-Newtonian Flow

1 Introduction.- 1.1 Preamble.- 1.2 Objectives and Layout.- 2 Governing Differential Equations.- 2.1 Introduction.- 2.2 Governing Equations.- 2.3 Constitutive Equation.- 2.3.1 Viscous shear flow.- 2.3.2 Inelastic generalised Newtonian flow.- 2.3.3 Power law fluid.- 2.4 Elastic Liquids.- 2.4.1 Maxwell model.- 2.4.2 Oldroyd's liquid B model.- 2.4.3 White-Metzner model.- 2.5 Equation in Steady State Flow.- 2.6 Initial and Boundary Conditions.- 2.7 Stream Function.- 3 Finite Element Method.- 3.1 Introduction.- 3.2 Variational Principle and Rayleigh-Ritz Method.- 3.3 Galerkin Weighted Residual Method.- 3.4 Virtual Work Principle.- 3.5 Finite Element Method in Two Dimensions.- 3.5.1 Introduction.- 3.5.2 K Matrix and load vector.- 3.5.3 Triangular elements.- 3.5.4 Natural coordinate system.- 3.5.5 Numerical integration.- 4 Steady Non-Newtonian Flow.- 4.1 Introduction.- 4.2 Creeping Motion.- 4.2.1 Extra stress and strain rate relations.- 4.3 Finite Element Formulation for Creeping Flow...57.- 4.3.1 Finite element spatial discretisation.- 4.3.2 Slippage and friction.- 4.3.3 Numerical examples.- 5 Temporal Discretisation.- 5.1 Introduction.- 5.1.1 Transient Newtonian flow equations in two dimensions.- 5.1.2 Boundary conditions.- 5.2 Finite Element Formulation.- 5.3 Temporal Discretisation.- 5.3.1 Generalized trapezoidal and mid-point family of methods.- 5.3.2 High order scheme - Taylor series in time.- 5.3.3 Predictor-corrector scheme.- 5.3.4 Automatic time step selection.- 5.4 Solution of Nonlinear System of Equations.- 6 Finite Element Analysis for Transient non-Newtonian Flow.- 6.1 Introduction.- 6.2 The Taylor-Galerkin Method.- 6.3 The Pressure Correction Method.- 6.3.1 Pressure correction method with midpoint rule.- 6.3.2 Pressure correction method with higher order scheme.- 6.3.3 Stability considerations.- 6.4 Transient Non-Newtonian Flow.- 6.4.1 Transient Generalised Newtonian Flow.- 6.4.2 Oldroyd's liquid B model.- 6.5 Non-isothermal Flow.- 6.6 Adaptive Analysis.- 6.7 Error Estimation for Fluid Flow.- 6.8 Numerical Examples.- 6.8.1 Newtonian flow.- 6.8.2 Non-Newtonian flow.- 6.8.3 Adaptive method.- A Software Description for NSTEAD.- A.1 Introduction...129.- A.2 Glossary of Variable Names.- A.2.1 Main variables.- A.2.2 Main arrays.- A.2.3 Main subroutines.- A.3 Program Overview.- A.4 Input Instructions.- A.5 Element Stiffness Calculations.- A.6 Documented Examples.- A.6.1 Example 1: Flow in a plane channel...147.- A.6.2 Example 2: Flow over a step.- B Software Description for NFLOWG.- B.1 Introduction.- B.2 Glossary of Variable Names.- B.2.1 Main logical variables for the code.- B.2.2 Main variables for the geometry and mesh data...181.- B.2.3 Main arrays used in the Code.- B.2.4 Main subroutines.- B.3 Main Routine of NFLOWG.- B.4 Input Instructions.- B.4.1 Input instruction for file PARA.DAT.- B.4.2 Input instruction for file GEOM.DAT.- B.5 Documented Example.