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A continuum theory for multiphase particulate suspensions in liquid media was formulated. The generalized fundamental balance laws for the particulate phases and the continuous fluid phase were presented. The thermodynamics of the multiphase system was studied, and the method of Lagrangian multipliers was employed to include the incompressibility of materials constraint and the criterion for fully saturated media. Constitutive equations were developed, which included micro-rotational and micro-dilatational effects. The basic equations of motion of different phases were derived and discussed. Special considerations were given to the case of incompressible constituents. Applications of this theory to nanofluids, dilute suspension flows, and multiphase sedimentations were also described.

Abstract: In this paper, we discuss the importance of constitutive modeling of the stress tensors in certain engineering issues, related to the pressure drop and skin friction, encountered in solids transport. To study this problem, we first give a brief account of the formulation of a two-component mixture based on the theory proposed by Massoudi, Rajagopal and co-workers. The mixture consists of a linearly viscous fluid infused with solid particles. The solids particles are modeled as a granular media and it is assumed that the mixture is dense enough so that we can use the theory of interacting continua. The subsequent boundary value problem, flow between two flat plates, is then solved numerically and results for various dimensionless numbers are presented for velocities and volume fraction profiles. The engineering quantities of interests discussed are related to the pressure drop and skin friction at the walls. [ABSTRACT FROM AUTHOR]

In this paper we study the flow and heat transfer in a chemically reacting non‐linear fluid between two long horizontal parallel flat plates that are at different temperatures. The top plate is sheared, whereas the bottom plate is fixed. The fluid is modeled as a generalized power‐law fluid whose viscosity is also assumed to be a function of the concentration. The effects of radiation are neglected. The equations are made dimensionless and the boundary value problem is solved numerically; the velocity and temperature profiles are obtained for various dimensionless numbers. Published in 2009 by John Wiley & Sons, Ltd.

In this paper, we study the flow of a compressible (density‐gradient‐dependent) non‐linear fluid down an inclined plane, subject to radiation boundary condition. The convective heat transfer is also considered where a source term, similar to the Arrhenius type reaction, is included. The non‐dimensional forms of the equations are solved numerically and the competing effects of conduction, dissipation, heat generation and radiation are discussed. Copyright © 2008 John Wiley & Sons, Ltd.

Abstract: In this paper we provide a brief review of the basic equations for the flow of two linearly viscous fluids using the mixture theory equations given in Atkin and Craine [R.J. Atkin, R.E. Craine, Continuum theories of mixtures: applications, J. Inst. Math. Appl. 17 (1976) 153; R.J. Atkin, R.E. Craine, Continuum theories of mixtures: basic theory and historical development, Quart. J. Mech. Appl. Math. 29 (1976) 290]. We then look at certain principles (or more accurately assumptions) due to Truesdell [C. Truesdell, Sulle basi della thermomeccanica, Rand Lincei, Series 8 22 (1957) 33–38, and 158–166] and Adkins [J.E. Adkins, Non-linear diffusion, 1. Diffusion and flow of mixtures of fluids, Philos. Trans. Roy. Soc. London A 255 (1963) 607–633; J.E. Adkins, Non-linear diffusion, 2. Constitutive equations for mixtures of isotropic fluids, Philos. Trans. Roy. Soc. London A 255 (1963) 635–648] and show that if the ‘assumption of the limiting cases’ of Adkins is to hold, then a very specific structure on the material properties of the two fluids has to be imposed. This new hypothesis provides one such condition for this requirement. An attempt is made to derive a relationship for the mixture viscosity using these ideas. [Copyright &y& Elsevier]

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as non‐linear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematical parameters such as temperature, shear rate, porosity or concentration, etc. In Part I, we will give a brief review of the basic equations of thermodynamics and heat transfer to indicate the importance of the modelling of the heat flux vector. We will also discuss the concept of effective thermal conductivity (ETC) in granular and porous media. In Part II, we propose and subsequently derive a properly frame‐invariant constitutive relationship for the heat flux vector for a (single phase) flowing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat flux vector in addition to being proportional to the temperature gradient (the Fourier's law), could also depend on the gradient of density (or volume fraction), and D(the symmetric part of the velocity gradient) in an appropriate manner. The emphasis in this paper is on the idea that for complex non‐linear materials it is the heat flux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or sufficient. Copyright © 2006 John Wiley & Sons, Ltd.

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as non‐linear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematical parameters such as temperature, shear rate, porosity or concentration, etc. In Part I, we will give a brief review of the basic equations of thermodynamics and heat transfer to indicate the importance of the modelling of the heat flux vector. We will also discuss the concept of effective thermal conductivity (ETC) in granular and porous media. In Part II, we propose and subsequently derive a properly frame‐invariant constitutive relationship for the heat flux vector for a (single phase) flowing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat flux vector in addition to being proportional to the temperature gradient (the Fourier's law), could also depend on the gradient of density (or volume fraction), and D(the symmetric part of the velocity gradient) in an appropriate manner. The emphasis in this paper is on the idea that for complex non‐linear materials it is the heat flux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or sufficient. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract: We study the unsteady shear flow of fluids with pressure-dependent viscosity, situated between two parallel horizontal plates with the upper plate moving while the flow is subjected to an oscillating pressure gradient. The dimensionless form of the momentum equation is solved numerically using a central difference approximation for the spatial derivative terms and a forward difference approximation for the time derivative term. In addition to providing the velocity profiles at the midsection between the two plates, the values of shear stress at the lower (stationary) plate for various values of the dimensionless numbers are also plotted. [Copyright &y& Elsevier]

We will derive a constitutive relationship for the stress tensor of an anisotropic rod-like assembly of granular particles where not only the transverse isotropy (denoted by a unit vector n , also called the fiber direction) is included, but also the dependence of the stress tensor T on the density gradient, a measure of particle distribution, is studied. The granular media is assumed to behave as a continuum, and the effects of the interstitial fluid are ignored. No thermodynamical considerations are included, and using representation theorems, it is shown that in certain limiting cases, constitutive relations similar to those of the Leslie-Ericksen liquid crystal type can be obtained. It is also shown that in this granular model, one can observe the normal stress effects as well as the yield condition, if proper structures are imposed on the material coefficients.

In this paper, we discuss the importance of the Principle of Material Frame-Indifference, sometimes referred to as Objectivity, in multiphase flows. We first give a brief account of the formulation of a two-phase mixture and then indicate where and how this principle should be utilized. We will then give a brief review of the meaning of this principle in continuum mechanics and how it has been used in various fields of physics. We also give examples of frame-indifferent and frame-dependent constitutive relations which have been used in multiphase flow studies. The importance of lift forces is also discussed. Finally, we derive a general constitutive relation which is frame-invariant and general enough to be used in many applications.

The flow of a multiphase mixture consisting of a viscous fluid and solid particles between two vertical plates is studied. The theory of interacting continua or mixture theory is used. Constitutive relations for the stress tensor of the granular materials and the interaction force are presented and discussed. The flow of interest is an ideal one where we assume the flow to be steady and fully developed; the mixture is flowing between two long vertical plates. The non-linear boundary value problem is solved numerically, and the results are presented for the dimensionless velocity profiles and the volume fraction as functions of various dimensionless numbers.

•A thermodynamic model is developed to predict CO2properties when it is compressed into a high-pressure, constant-volume storage cylinder.•The model incorporates the effects of the compressibility of the fluid.•Heat loss to the surrounding environment and heating of the cylinder wall are used as the parameters.•CO2pressure is not sensitive to the heat loss, but it does depend on the compression power.•The compression can be considered as a constant temperature process.•The compression can be modeled as a polytropic process, where PVn=constantwith the polytropic exponential power n = 0.995.

We study the linearlized stability of the state of uniform fiuidization, within the context of the theory of mixtures. The mixture is assumed to be made up of a classical linearly viscous fluid infused with particles. In marked departure from most of the previous studies, we model the solid as a granular material and assume a constitutive relation that stems from classical motions in continuum mechanics. The linearized stability analysis of the state of uniform fiuidization, in general, for a three-dimensional disturbance leads to an eighth order equation for the characteristic whose root implies instability, when positive.

The near-wall enrichment of platelets strongly influences thrombus formation in vivo and in vitro . This paper develops a multi-constituent continuum approach to study this phenomenon. A mixture theory model is used to describe the motion of plasma and red blood cells (RBCs) and the interactions between the two components. A transport model is developed to study the influence of the RBCs field on the platelets. The model is used to study blood flow in a rectangular micro-channel, a sudden expansion micro-channel, and a channel containing micro crevices (representing a practical problem encountered in most blood-wetted devices). The simulations show that in the rectangular channel the concentration of the platelets near the walls is about five times higher than the concentration near the channel centerline. It is also noticed that in the channel with crevices, a large number of platelets accumulate in the deep part of the crevices and this may serve as nidus for excessive thrombus formation occurring in medical devices. [ABSTRACT FROM AUTHOR]

We discuss three boundary value problems in the flow and heat transfer analysis in flowing granular materials: (i) the flow down an inclined plane with radiation effects at the free surface; (ii) the natural convection flow between two heated vertical walls; (iii) the shearing motion between two horizontal flat plates with heat conduction. It is assumed that the material behaves like a continuum, similar to a compressible nonlinear fluid where the effects of density gradients are incorporated in the stress tensor. For a fully developed flow the equations are simplified to a system of three nonlinear ordinary differential equations. The equations are made dimensionless and a parametric study is performed where the effects of various dimensionless numbers representing the effects of heat conduction, viscous dissipation, radiation, and so forth are presented.

The flow of a multiphase mixture consisting of a viscous fluid and solid particles between two vertical plates is studied. The theory of interacting continua or mixture theory is used. Constitutive relations for the stress tensor of the granular materials and the interaction force are presented and discussed. The flow of interest is an ideal one where we assume the flow to be steady and fully developed; the mixture is flowing between two long vertical plates. The non-linear boundary value problem is solved numerically, and the results are presented for the dimensionless velocity profiles and the volume fraction as functions of various dimensionless numbers.

Abstract: In this paper, we consider the two dimensional flow of blood in a rectangular microfluidic channel. We use Mixture Theory to treat this problem as a two-component system: One component is the red blood cells (RBCs) modeled as a generalized Reiner–Rivlin type fluid, which considers the effects of volume fraction (hematocrit) and influence of shear rate upon viscosity. The other component, plasma, is assumed to behave as a linear viscous fluid. A CFD solver based on OpenFOAM® was developed and employed to simulate a specific problem, namely blood flow in a two dimensional micro-channel, is studied. Finally to better understand this two-component flow system and the effects of the different parameters, the equations are made dimensionless and a parametric study is performed. [Copyright &y& Elsevier]

Abstract: Computational prediction of blood damage has become a crucial tool for evaluating blood-wetted medical devices and pathological hemodynamics. A difficulty arises in predicting blood damage under turbulent flow conditions because the total stress is indeterminate. Common practice uses the Reynolds stress as an estimation of the total stress causing damage to the blood cells. This study investigates the error introduced by making this substitution, and further shows that energy dissipation is a more appropriate metric of blood trauma. [ABSTRACT FROM AUTHOR]

It is known that in a vessel whose characteristic dimension (e.g., its diameter) is in the range of 20–500 μm, blood behaves as a non-Newtonian fluid, exhibiting complex phenomena, such as shear-thinning, stress relaxation, and also multi-component behaviors, such as the Fahraeus effect, plasma-skimming, etc. For describing these non-Newtonian and multi-component characteristics of blood, using the framework of mixture theory, a two-fluid model is applied, where the plasma is treated as a Newtonian fluid and the red blood cells (RBCs) are treated as shear-thinning fluid. A computational fluid dynamic (CFD) simulation incorporating the constitutive model was implemented using OpenFOAM® in which benchmark problems including a sudden expansion and various driven slots and crevices were studied numerically. The numerical results exhibited good agreement with the experimental observations with respect to both the velocity field and the volume fraction distribution of RBCs. [ABSTRACT FROM AUTHOR]

This study was undertaken to develop a numerical/computational simulation of von Willebrand Factor (vWF) - mediated platelet shear activation and deposition in an idealized stenosis. Blood is treated as a multi-constituent mixture comprised of a linear fluid component and a porous solid component (thrombus). Chemical and biological species involved in coagulation are modeled using a system of coupled convection-reaction-diffusion (CRD) equations. This study considers the cumulative effect of shear stress (history) on platelet activation. The vWF activity is modeled as an enhancement function for the shear stress accumulation and is related to the experimentally-observed unfolding rate of vWF. A series of simulations were performed in an idealized stenosis in which the predicted platelets deposition agreed well with previous experimental observations spatially and temporally, including the reduction of platelet deposition with decreasing expansion angle. Further simulation indicated a direct relationship between vWF-mediated platelet deposition and degree of stenosis. Based on the success with these benchmark simulations, it is hoped that the model presented here may provide additional insight into vWF-mediated thrombosis and prove useful for the development of more hemo-compatible blood-wetted devices in the future. [ABSTRACT FROM AUTHOR]