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A simple mixture theory for ν Newtonian and generalized Newtonian constituents

Content Provider	Springer Nature Link
Author	Surana, K. S. Powell, M. Reddy, J. N.
Copyright Year	2012
Abstract	This work presents the 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 generalized Newtonian fluids. Power law and Carreau–Yasuda models are considered for generalized Newtonian shear thinning fluids. The mathematical model is derived for a ν constituent mixture with volume fractions $${\phi_\alpha}$$ 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 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 constituents 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 Newtonian fluids between parallel plates as a model problem, it is shown that partial pressures p α of the constituents must be expressed in terms of the mixture pressure p. In this work, we propose $${p_\alpha=\phi_\alpha p}$$ and $${\sum_\alpha^\nu p_\alpha = p}$$ which implies $${\sum_\alpha^\nu \phi_\alpha = 1}$$ 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. Modifications 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 is derived and used to present numerical studies for boundary value problems using finite element processes based on a residual functional, that is, least squares finite element processes in which local approximations are considered in $${H^{k,p}\left(\bar{\Omega}^e\right)}$$ scalar product spaces. Fully developed flow between parallel plates and 1:2 asymmetric backward facing step is used as model problems for a mixture of two constituents.
Starting Page	33
Ending Page	65
Page Count	33
File Format	PDF
ISSN	09351175
Journal	Continuum Mechanics and Thermodynamics
Volume Number	26
Issue Number	1
e-ISSN	14320959
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2012-12-01
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mixture theory Saturated mixture Newtonian and generalized Newtonian constituents Classical Continuum Physics Engineering Thermodynamics, Heat and Mass Transfer Theoretical and Applied Mechanics Structural Materials
Content Type	Text
Resource Type	Article
Subject	Physics and Astronomy Mechanics of Materials Materials Science

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