A Simple Mixture Theory for Isothermal and Non-isothermal Flows of n Newtonian and Generalized Newtonian Constituents including Interaction Effects

Abstract

Development of mathematical models based on conservation and balance laws including constitutive theories are presented for a saturated mixture of n homogeneous, isotropic, and incompressible constituents for isothermal and non-isothermal flows. The constituents and the mixture are assumed to be Newtonian or generalized Newtonian fluids. Power law and Carreau-Yasuda models are considered for generalized Newtonian shear thinning fluids. The mathematical model is derived for a n constituent mixture with volume fractions using principles of continuum mechanics: conservation 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, deviatoric Cauchy stress tensors, and heat vector in terms of the dependent variables related to the constituents and their material coefficients. In the derivation of the mathematical model effects of the interaction forces are accounted in the momentum and energy equations. In the development of the constitutive theories two approaches are considered. In the first approach we assume that the mixture stress is the sum of the constituent stresses. This approach requires derivation of the bulk properties of the constituents based on the constituent volume fractions and their properties which are then utilized in the constitutive theories for the constituents forming the mixture. In the second approach the mixture stress is assumed not to be the sum of the constituent stress. For a homogenous isotropic mixture we begin with its own constitutive theory for the deviatoric mixture stress defined using mixture material coefficients and the symmetric part of the velocity gradient tensor for the mixture. Mixture material coefficients are derived using volume and mole fractions of the constituents and a mixing rule. The mutual parameter in the mixing rule is described using arithmetic mean, geometric mean, and harmonic mean. The validity of the proposed models are demonstrated for degenerated cases of same constituents i.e., two of the constituents same etc. Dimensionless forms 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, that is, least squares processes in which local approximations are considered in H(k,p) 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.