Heat Transfer in Complex Fluids

Amongst the most important constitutive relations in Mechanics, when characterizing the behavior of complex materials, one can identify the stress tensor T, the heat flux vector q (related to heat conduction) and the radiant heating (related to the radiation term in the energy equation). Of course, the expression 'complex materials' is not new. In fact, at least since the publication of the paper by Rivlin & Ericksen (1955), who discussed fluids of complexity (Truesdell & Noll, 1992), to the recently published books (Deshpande et al., 2010), the term complex fluids refers in general to fluid-like materials whose response, namely the stress tensor, is 'non-linear' in some fashion. This non-linearity can manifest itself in variety of forms such as memory effects, yield stress, creep or relaxation, normal-stress differences, etc. The emphasis in this chapter, while focusing on the constitutive modeling of complex fluids, is on granular materials (such as coal) and non-linear fluids (such as coal-slurries). One of the main areas of interest in energy related processes, such as power plants, atomization, alternative fuels, etc., is the use of slurries, specifically coal-water or coal-oil slurries, as the primary fuel. Some studies indicate that the viscosity of coal-water mixtures depends not only on the volume fraction of solids, and the mean size and the size distribution of the coal, but also on the shear rate, since the slurry behaves as shear-rate dependent fluid. There are also studies which indicate that preheating the fuel results in better performance, and as a result of such heating, the viscosity changes. Constitutive modeling of these non-linear fluids, commonly referred to as non-Newtonian fluids, has received much attention. Most of the naturally occurring and synthetic fluids are non-linear fluids, for example, polymer melts, suspensions, blood, coal-water slurries, drilling fluids, mud, etc. It should be noted that sometimes these fluids show Newtonian (linear) behavior for a given range of parameters or geometries; there are many empirical or semi-empirical constitutive equations suggested for these fluids. There have also been many non-linear constitutive relations which have been derived based on the techniques of continuum mechanics. The non-linearities oftentimes appear due to higher gradient terms or time derivatives. When thermal and or chemical effects are also important, the (coupled) momentum and energy equations can give rise to a variety of interesting problems, such as instability, for example the phenomenon of double-diffusive convection in a fluid layer. In Conclusion, we have studied the flow of a compressible (density gradient type) non-linear fluid down an inclined plane, subject to radiation boundary condition. The 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. It is observed that the velocity increases rapidly in the region near the inclined surface and is slower in the region near the free surface. Since R{sub 7} is a measure of the heat generation due to chemical reaction, when the reaction is frozen (R{sub 7}=0.0) the temperature distributions would depend only on R{sub 1}, and R{sub 2}, representing the effects of the pressure force developed in the material due to the distribution, R{sub 3} and R{sub 4} viscous dissipation, R{sub 5} the normal stress coefficient, R{sub 6} the measure of the emissivity of the particles to the thermal conductivity, etc. When the flow is not frozen (RP{sub 7} > 0) the temperature inside the flow domain is much higher than those at the inclined and free surfaces. As a result, heat is transferred away from the flow toward both the inclined surface and the free surface with a rate that increases as R{sub 7} increases. For a given temperature, an increase in {zeta} implies that the activation energy is smaller and thus, the reaction rate is increased leading to an increase in the heat of the reaction. As a result the flow is chemically heated and its temperature increase. The results shown here indicate that for all values of {zeta} used the chemical effects are significant and the temperature is always higher than both the surface temperature and the free surface temperature. The heat transfer is always from the flow toward both the inclined surface and the free stream. It is also noticed that for all values of m chosen in this study, the temperature is higher than the surface and the free stream temperature. The heat transfer at the inclined surface and at the free stream increase slowly for negative values of m to about m=0.5, but it begins to significantly increase for m greater than 0.5

Numerical simulation of a two-component mixture (fluid-particles) between two plates

In this study, we study a two-component mixture, composed of rigid solid particles and a fluid. Mixture theory is used to model the interaction between the two different components. For the granular materials, we assume the Cauchy stress tensor depends on the shear rate and the density gradient of the granular materials, and the fluid is assumed to behave as a Newtonian fluid. We study the fully developed flow of this mixture between two flat plates. A parametric study is performed to study the effects of the material parameters, especially those related to the normal stress differences. We then consider the effects the slip boundary condition

Flow of granular materials with slip boundary condition

2014 , Purdue University, West Lafayette, IN, October 1‑3, 2014. We study the steady fully developed flow of granular materials between two horizontal flat plates, subject to slip at the walls. The constitutive equation used in our study is a model proposed by Rajagopal et al. [1], and the material properties such as viscosity and the normal stress coefficients are derived using the kinetic theory approximation proposed by Boyle and Massoudi [2] which includes the effect of the gradient of volume fraction. The slip boundary condition is based on the particle dynamics simulation results of Rosato and Kim [3]. The governing equations are nondimensionalized, and the resulting system of nonlinear differential equations is solved numerically. The results for the velocity profiles and the volume fraction profiles are presented. REFERENCES [1] Rajagopal, K.R., Massoudi, M., Wineman, A.S. Flow of granular materials between rotating disks. Mechanics Research Communications. 1994, 21, 629–634. [2] Boyle, E.J., Massoudi, M. A theory for granular materials exhibiting normal stress effects based on Enskog’s dense gas theory. Int. J. Engng. Sci. 1990,28(12), 1261–1275. [3] Rosato, A.D., Kim, H. Particle dynamics calculations of wall stresses and slip velocities for Couette flow of smooth inelastic spheres. Continuum Mech. Thermodyn. 1994, 6, 1‑20

An anisotropic constitutive relation for the stress tensor of a rod-like (fibrous-type) granular material

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

Modeling granular materials as compressible nonlinear fluids: Heat transfer boundary value problems

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

Vertical flow of a multiphase mixture in a channel

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

Unsteady flows of inhomogeneous incompressible fluids

In this paper, we study the unsteady motion of in homogeneous in compressible viscous fluids. We present the results corresponding to Stokes second problem and for the flow between two parallel plates where one is oscillating