Boundary conditions in mixture theory and in CFD applications of higher order models

Abstract

We discuss the importance of and the need for (additional) boundary conditions in Mixture Theory (also known as the Theory of Interacting Continua). Specifically, we will give an overview of the model due to Rajagopal and Massoudi which is appropriate for the flow of a linearly viscous fluid infused with solid particles. The solid particles are modeled as granular materials. In this formulation the need for additional boundary condition arises due to higher gradients of density (or volume fraction). The challenging issue of how to 'split' the total stress or the total velocity at the boundary is also discussed.

Modeling and numerical simulation of blood flow using the theory of interacting continua

“…In this approach, we consider a mixture of an incompressible fluid infused with solid particles, wherein the principles of mechanics of granular materials are used to describe the behavior of particles. This model has been used and discussed in Massoudi (Massoudi 1986; Massoudi 1988; Massoudi 2001; Massoudi 2002; Massoudi 2003; Massoudi 2003; Massoudi 2007; Massoudi 2008), Johnson et al, (Johnson, Massoudi et al 1991; Johnson, Massoudi et al 1991), Rajagopal, et al (Rajagopal, Troy et al 1992; Rajagopal, Massoudi et al 1994), Massoudi et al (Massoudi, Rajagopal et al 1999), Massoudi and Johnson (Massoudi and Johnson 2000), and Massoudi and Rao (Massoudi and Rao 2001), Ravindran et al (Ravindran, Anand et al 2004), Massoudi (Massoudi 2008; Massoudi 2010). We need to mention that alternatively blood can be viewed as a suspension where it can be modeled using the techniques of non-Newtonian fluid mechanics.…”

“…Nevertheless, in general, due to the higher order gradients of the volume fraction, it is necessary to provide additional boundary conditions [see Massoudi (2007) for a discussion of boundary conditions]. For most practical applications, these can be satisfied by certain symmetry conditions; in certain cases the values of the unknowns or their derivatives have to be specified as surface conditions at the solid walls or at the free surface.…”

A numerical study of blood flow using mixture theory

“…It should be remarked that in general, sometimes the additional boundary conditions can be provided from experimental data, and sometimes they can be based on other theories, such as kinetic theories, or physical insights. In certain cases, due to the higher order gradients of volume fraction, it is necessary to provide additional boundary conditions for solving practical and simple boundary value problems [see Massoudi (2007) for a discussion of boundary conditions]. For some practical applications, symmetry conditions can be used; in certain cases the values of the unknowns or their derivatives have to be specified as surface conditions at the walls or at the free surface.…”

Stability of wall bounded, shear flows of dense granular materials: the role of the Couette gap, the wall velocity and the initial concentration

“…For the volume fraction, we first note that, in weakly non-local theories that involve gradients as internal variables such as the one that (2.1)-(2.3) have been based upon, the prescription of boundary conditions cannot be circumvented, unless the flow under consideration enjoys multiple symmetries, (Massoudi 2007). For the problem at hand, the presence of the second-order differential operator in the right-hand side of the compaction equation in conjunction with the quasi-parallel flow postulate, imply that two boundary conditions are needed for the volume fraction as well.…”