Constitutive equations simple fluid

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. 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

Even if satisfactory equations of state and constitutive equations can be developed for complex fluids, large-scale computation will still be required to predict flow fields and stress distributions in complex fluids in vessels with complicated geometries. A major obstacle is that even simple equations of state that have been proposed for fluids do not always converge to a solution. It is not known whether this difficulty stems from the oversimplified nature of the equatiorrs, from problems with ntrmerical mathematics, or from the absence of a lamirrar steady-state solution to the eqrratiorrs. 

This section summarizes results of the phenomenological theory of viscoelasticity as they apply to homogeneous polymer liquids. The theory of incompressible simple fluids (76, 77) is based on a very general set of ideas about the nature of mechanical response. According to this theory the flow-induced stress at any point in a substance at time t depends only on the deformations experienced by material in an arbitrarily small neighborhood of that point in all times prior to t. The relationship between stress at the current time and deformation history is the constitutive equation for the substance. 

For a Newtonian fluid in a simple elongational flow, the constitutive equation becomes... 

One physical restriction, translated into a mathematical requirement, must be satisfied that is that the simple fluid relation must be objective, which means that its predictions should not depend on whether the fluid rotates as a rigid body or deforms. This can be achieved by casting the constitutive equation (expressing its terms) in special frames. One is the co-rotational frame, which follows (translates with) each particle and rotates with it. The other is the co-deformational frame, which translates, rotates, and deforms with the flowing particles. In either frame, the observer is oblivious to rigid-body rotation. Thus, a constitutive equation cast in either frame is objective or, as it is commonly expressed, obeys the principle of material objectivity . Both can be transformed into fixed (laboratory) frame in which the balance equations appear and where experimental results are obtained. The transformations are similar to, but more complex than, those from the substantial frame to the fixed (see Chapter 2). Finally, a co-rotational constitutive equation can be transformed to a co-deformational one. 

Goddard (27) expressed the notion of the simple fluid constitutive equation in a co-rotational integral series. The integral series expansion had been used in the co-deformational frame by Green and Rivlin (28) and Coleman and Noll (29). The co-rotational expansion takes the form ... 

H. Giesekus, A Simple Constitutive Equation for Polymer Fluids Based on the Concept of Deformation-dependent Tensorial Mobility, J. Non-Newt. Fluid Mech., 11, 60-109 (1982). 

Leonov AI (1992) Analysis of simple constitutive equations for viscoelastic liquids. J Non-Newton Fluid Mech 42(3) 323-350... 

Pokrovskii VN, Pyshnograi GV (1990) Non-linear effects in the dynamics of concentrated polymer solutions and melts. Fluid Dyn 25 568-576 Pokrovskii VN, Pyshnograi GV (1991) The simple forms of constitutive equation of polymer concentrated solution and melts as consequence of molecular theory of viscoelasticity. Fluid Dyn 26 58-64... 

The two-way arrow between polymer rheology and fluid mechanics has not always been appreciated. Traditionally we look at polymer rheology as input to fluid mechanics, as a way to supply constitutive equations. Gary Leal pointed out the use of fluid mechanics to provide feedback to tell us whether the constitutive equation is satisfactory. In the past, we tested constitutive models by examining polymeric liquids with very simple kinematics, homogeneous flows as a rule, either simple shear or simple shear-free types of flows. We don t actually use polymers in such simple flows, and it s essential to understand whether or not these constitutive equations actually interpolate properly between those simple types of kinematics. So there s a two-way arrow that we have to pay more attention to in the future. 

Figure 13.36. These numerical methods also give results that are similar to experiments. Other approaches use the continuity equation and equation of motion developed for fluid flow (see Chapter 12) with a constitutive equation for the powder mass. The constitutive equation for powder flow is a problem that has no solution at this time. Several simple formulas in terms of tensor invariants and deviation tensors [83]...

From a numerical viev point, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the munerical problem, in relation to the nonlinearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry. 

A mathematical expression relating forces and deformation motions in a material is known as a constitutive equation. However, the establishment of constitutive equations can be a rather difficult task in most cases. For example, the dependence of both the viscosity and the memory effects of polymer melts and concentrated solutions on the shear rate renders it difficult to establish constitute equations, even in the cases of simple geometries. A rigorous treatment of the flow of these materials requires the use of fluid mechanics theories related to the nonlinear behavior of complex materials. However, in this chapter we aim only to emphasize important qualitative aspects of the flow of polymer melts and solutions that, conventionally interpreted, may explain the nonlinear behavior of polymers for some types of flows. Numerous books are available in which the reader will find rigorous approaches, and the corresponding references, to the subject matter discussed here (1-16). 

Many constitutive equations have been proposed in addition to those indicated above, which are special cases of fluids with memory. Most of these expressions arise from the generalization of linear viscoelasticity equations to nonlinear processes whenever they obey the material indifference principle. However, these generalizations are not unique, because there are many equations that reduce to the same linear equation. It should be noted that a determined choice among the possible generalizations may be suitable for certain types of fluids or special kinds of deformations. In any case, the use of relatively simple expressions is justified by the fact that they can predict, at least qualitatively, the behavior of complex fluids. 

In [26] the velocity fields and thereby the power for stirrers with simple geometry (anchor stirrer and gate stirrer) have been calculated for the laminar case (highly viscous liquid with Newtonian or pseudoplastic flow behavior) by the help of the numerical solution of the continuity and momentum balance in connection with the rheological constitutive equation. In the case of Newtonian fluids the power characteristic in the laminar flow range could be calculated for all three stirrers with the help of the expression ... 

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. 

This equation has the correct limiting behavior it reduces to an equation for a simple Newtonian fluid when dx/dt approaches to 0 for steady shear flow. When the stress changes rapidly with time, and X is negligible compared with dx/dt, it reduces to the constitutive equation of a Hookian solid. 

As discussed earlier, LADDs are complex, multicomponent mixtures consisting of both organic and inorganic compounds dispersed in a liquid matrix. Such compositions can exhibit a broad range of rheological characteristics from simple Newtonian to complex pseudoplastic flow. Shown in Figure 9.6 and Figure 9.7 are flow and viscosity profiles of Newtonian and non-Newtonian fluids as a function of applied shear rate. A number of mathematical models have been proposed [76] to describe the flow characteristics of various systems. These equations are called constitutive equations and are used to predict flow behavior in complex systems. 

The constitutive equation, (2-60), for the stress, on the other hand, will be modified for all fluids in the presence of a mean motion in which the velocity gradient Vu is nonzero. To see that this must be true, we can again consider the simplest possible model system of a hard-sphere or billiard-ball gas, which we may assume to be undergoing a simple shear flow,... 

The physical significance of pressure, as it first appeared in the constitutive equation for stress in a stationary fluid, (2 60), is clear. This is the familiar pressure of thermodynamics. When a fluid is undergoing a motion, however, the simple notion of a normally directed surface force acting equally in all directions is lost. Indeed, it is evident on examining the... 

The definition (2 85) is a purely mechanical definition of pressure for a moving fluid, and nothing is implied directly of the connection for a moving fluid between p and the ordinary static or thermodynamic pressure p. Although the connection between p and p can always be stated once the constitutive equation for T is given, one would not necessarily expect the relationship to be simple for all fluids because thermodynamics refers to equilibrium conditions, whereas the elements of a fluid in motion are clearly not in thermodynamic equilibrium. Applying the definition (2-85) to the general Newtonian constitutive model, (2-80), we find... 

One application of the solutions (4-55)-(4-61) is to evaluate the effect of viscous dissipation in the use of a shear rheometer to measure the viscosity of a Newtonian fluid. In this experiment, we subject the fluid in a thin gap between two plane walls to a shear flow by moving one of the walls in its own plane at a known velocity and then measuring the shear stress produced at either wall (by measuring the total tangential force and dividing by the area). In the absence of viscous dissipation, the velocity profile is linear and the shear rate is simply given by the tangential velocity U divided by the gap width d. Now, the constitutive equation, (2-87), for an incompressible Newtonian fluid applied to this simple flow situation takes the form... 

The simple fluid theory has not yet yielded general relations for ij" and Xk- A constitutive equation of Bernstein, Kearsley, and Zapas does give parallel relations for rj and x as discussed by Booij (/2a, p. 63). The rigid dumbbell results above are not consistent with the Bemstein-Kearsley-Zapas formulae. 

---