A Nonlinear Constitutive Theory for Deviatoric Cauchy Stress Tensor for Incrompressible Viscous Fluids

Abstract

Newton’s law of visocosity is a commonly used constitutive theory for deviatoric Cauchy stress tensor. In this constitutive theory originally constructed based on experimental observation, the deviatoric Cauchy stress is proportional to the symmetric part of the velocity gradient tensor. The constant of proportionality is the viscosity of the fluid. For all continuous media if the deforming matter is in thermodynamic equilibrium then all constitutive theories including those considered here must satisfy conservation and balance laws. It is well known that only the second law of thermodynamics provides possible conditions or mechanisms for deriving constitutive theories. The constitutive theory for deviatoric stress tensor used here can be shown to be a simplified form of the constitutive theory derived using conditions resulting from the entropy inequality in conjunction with the theory of generators and invariants that contains up to fifth degree terms in the components of the symmetric part of the velocity gradient tensor. In general the constitutive theory for deviatoric stress tensor is basis (covariant, contravariant, or Jaumann) dependent as it uses convected time derivatives of the Green and Almansi strain tensors of orders higher than one. However, the first convected time derivative of the Green and Almansi strain tensors are in fact symmetric part of the velocity gradient tensor which is basis independent. Thus, if the constitutive theory for deviatoric Cauchy stress tensor is only dependent on the symmetric part of the velocity gradient tensor, then it is basis independent. This is the case for the theory presented in this paper. In this paper we limit the constitutive theory for deviatoric Cauchy stress tensor to contain only up to quadratic terms in the components of the symmetric part of the velocity gradient tensor. The objective is to study the resulting flow physics due to the constitutive theory for deviatoric Cauchy stress tensor that contains up to quadratic terms in the velocity gradient tensor. Model problems consisting of fully developed flow between parallel plates, square lid-driven cavity, and asymmetric sudden expansion are used to present numerical solutions. Numerical solutions of the model problems are calculated using least squares finite element formulation based on residual functional in which the local approximations are considered in higher order scalar product spaces that permit higher order global differentiability solutions. Nonlinear system of algebraic equations are solved using Newton’s linear method with line search.