A Simple Mixture Theory for ν Newtonian and Generalized Newtonian Constituents

Abstract

This work presents development of mathematical models based on conservation laws for a saturated mixture of ν homogeneous, isotropic, and incompressible constituents for isothermal flows. The constituents and the mixture are assumed to be Newtonian or gen- eralized Newtonian fluids. Power law and Carreau-Yasuda models are considered for gen- eralized Newtonian shear thinning fluids. The mathematical model is derived for a ν con- stituent mixture with volume fractions φα using principles of continuum mechanics: con- servation of mass, balance of momenta, first and second laws of thermodynamics, and principles of mixture theory yielding continuity equations, momentum equations, energy equation, and constitutive theories for mechanical pressures and deviatoric Cauchy stress tensors in terms of the dependent variables related to the constituents. It is shown that for Newtonian fluids with constant transport properties, the mathematical models for con- stituents are decoupled. In this case one could use individual constituent models to obtain constituent deformation fields, and then use mixture theory to obtain the deformation field for the mixture. In the case of generalized Newtonian fluids, the dependence of viscosities on deformation field does not permit decoupling. Numerical studies are also presented to demonstrate this aspect. Using fully developed flow of Newtonian and generalized Newto- nian fluids between parallel plates as a model problem, it is shown that partial pressures pα of the constituents must be expressed in terms of the mixture pressure p. In this work we propose pα = φα p and ν α pα = p which implies ν α φα = 1 which obviously holds. This rule for partial pressure is shown to be valid for a mixture of Newtonian and generalized Newtonian constituents yielding Newtonian and generalized Newtonian mixture. Modifi- cations of the currently used constitutive theories for deviatoric Cauchy stress tensor are proposed. These modifications are demonstrated to be essential in order for the mixture theory for ν constituents to yield a valid mathematical model when the constituents are the same. Dimensionless form of the mathematical models are derived and used to present numerical studies for boundary value problems using finite element processes based on a residual functional i.e. least squares finite element processes in which local approximations are considered in H k,p Ωe scalar product spaces. Fully developed flow between parallel plates and 1:2 asymmetric backward facing step are used as model problems for a mixture of two constituents.