Constitutive relations fluids

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

Pseudoplastic fluids are the most commonly encountered non-Newtonian fluids. Examples are polymeric solutions, some polymer melts, and suspensions of paper pulps. In simple shear flow, the constitutive relation for such fluids is... 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

Establishing the necessary constitutive and closure equations (the former relate fluid stresses with velocity gradients the latter relate unknown Navier-Stokes-equation coiTclations witli known quantities). 

In the broadest sense, I found the analogy with fluid mechanics to be very helpful. Just as kinematics provides the geometrical framework of fluid mechanics by exploring the motions that are possible, so also stoicheiometry defines the possible reactions and the restrictions on them without saying whether or at what rate they may take place. When dynamic laws are imposed on kinematic principles, we arrive at equations of motion so, also, when chemical kinetics is added to stoicheiometry, we can speak about reaction rates. In fluid mechanics different materials are distinguished by their constitutive relations and allow equations for the density and velocity to be formulated thence, various flow situations are examined by adding appropriate boundary conditions. Similarly, the chemical kinetics of the reaction system allow the rates of reaction to be expressed in terms of concentrations, and the reactor is brought into the picture as these rates are incorporated into appropriate equations and their boundary conditions. 

The constitutive relations along with the conservation equations give the basic equations of fluid mechanics, which are a set of five nonlinear partial differential equations involving the seven variables, p, g,e, P, and T. Because five equations [Eqs. (1), (2), (3), (5), and (6)] cannot determine seven quantities, the equations are closed by expressing any two variables of the set (p,e,P,T) in terms of the other two remaining variables. This is done by using the assumption of local equilibrium and thermodynamic equations of state. 

The derivation of the fiber spinning equations for a non-Newtonian shear thinning viscosity using a power law model are also derived. For a total stress, axx, in a power law fluid, we write the constitutive relation... 

As with any theory of material behavior, we have to make constitutive assumptions in order to define the peculiar mixture of a poroelastic material and a compressible bi-component fluid. Among other quantities, we must state constitutive relations for the mass supply a and the momentum supply m, which give rise to adsorption/desorption and to diffusion, respectively. 

The fundamental theory of fluid mechanics is expressed in the mathematical language of continuum tensor field calculus. An exhaustive treatment of this subject is found in the treatise by Truesdell and Toupin (1960). Two fundamental classes of equations are required (1) the generic equations of balance and (2) the constitutive relations. 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

The formulation of proper constitutive relations is a complex problem and is the basis of the science of rheology, which cannot be covered here. This section presents only four relatively simple constitutive relations that have proved to be practically useful to chemical engineers. Elastic fluid behavior is expressly excluded from consideration. The following equations are a listing of these constitutive relations many others are possible ... 

In laminar flow the velocity distribution, and hence the frictional energy loss, is governed entirely by the rheological constitutive relation of the fluid. In some cases it is possible to derive theoretical expressions for the friction factor. Where this is possible, a three-step procedure must be followed. 

Equation (70) is clearly independent of any constitutive relation and applies universally to all fluids in a pipe of this geometry. 

In some special cases it is possible to solve the equations of motion [Eq. (11)] entirely independently of any knowledge of the constitutive relation and to obtain a universal shear stress distribution that applies to all fluids. In other cases it is not possible to do this because the evaluation of certain integration constants requires knowledge of the specific constitutive relation. Because of space limitations, we illustrate only one case of each type here. 

The ambiguity of definition of Re encountered in the concentric annulus case is compounded here because of the fact that no viscosity is definable for non-Newtonian fluids. Thus, in the literature one encounters a bewildering array of definitions of Re-like parameters. We now present friction factor results for the non-Newtonian constitutive relations used above that are common and consistent. Many others are possible. 

Johnson, PC. and Jackson, R. (1987), Frictional collisional constitutive relations for granular materials with application to plain shearing, J. Fluid Meek, 176, 67. 

The bulk flow curve of the system studied by Isa et al. fitted a Herschel-Bulkley (HB) form for a yield-stress fluid [8] at small to moderate flow rates. The velocity profile predicted from a HB constitutive relation consists of a central, unsheared plug and shear zones adjacent to the channel walls, which fits qualitatively with what was observed see Fig. 12a. At the same time, however, the size of the central... 

If the driving forces for mass transfer are taken to be the difference in compositions between the interface (x j) and bulk fluid (x ), then the constitutive relations for (/) may be written as... 

In fluid d mamics there is no specific use of the transport equation for entropy other than being a physical condition indicating whether a constitutive relation proposed has a sound physical basis or not (nevertheless, this may be a constraint of great importance in many situations). In this connexion we usually think of the second law of thermodynamics as providing an inequality, expressing the observation that irreversible phenomena lead to entropy production. 

Johnson PC, Jackson R (1987) Erictional-coUisional constitutive relations for granular materials, with application to plane shearing. J Fluid Mech 176 67-93 Johnson PC, Nott P, Jackson R (1990) Frictional-colhsional equations of motion for particulate flows and their application to chutes. J Fluid Mech 210 501-535 Jung J, Gidaspow D, Gamwo IK (2006) Bubble Computation, Granular Temperatures, and Reynolds Stresses. Chem Eng Comm 193 946-975... 

Reeks MW (1993) On the constitutive relations for dispersed particles in nonuniform flows. I. Dispersion in simple shear flow. Phys Fluids A 5 750-761 Reyes Jr JN (1989) Statistically Derived Conservation Equations for Fluid Particle Flows. Nuclear Thermal Hydraulics 5th Winter meeting, Proc ANS Winter Meeting. 

The MTE was solved using a finite difference technique on a digital computer. It was necessary to supply a constitutive relation for the gas-fluid pair that was being studied. For 02-blood, the dissociation curve of Severinghaus (37) was used in a tabulated form. Because of its high solubility, C02 was assumed to be physically dissolved in blood without considering its dissociation curve. 

The two expansion coefficients ijs and rjv are called the shear and bulk viscosities, respectively. The shear viscosity is all that is required to describe our gedanken experiment. The bulk viscosity describes the viscous or dissipative part of the response to a compression. This linear constitutive relation is called the Newtonian stress tensor. A fluid correctly described by this form is called a Newtonian fluid.16... 

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