Constitutive Equations for Fluids

For Newtonian fluids the relationship between stress tensor and the velocity vector is given in Table 12.1 for various coordinate systems [1]. These relationships can be generalized to be 

TABLE 12.1 Components of the Stress Tensor for Newtonian Fluids 

FIGURE 12.2 Constitutive equations for ceramic suspension rheology. 

The stress for pseudo-plastic and dilatent fluids is not a linear function of shear rate. For non-Newtonian fluids, the relation between t and A is not a simple proportionality because the viscosity is a function of A. For a Bingham plastic fluid, the following relationship holds  

FIGURE 12.3 Newtonian concept of viscosit3r F is the applied shear force, A is the area of the plates, t is the shear stress (t = F/A) the shear rate, y, is dVJdy, the viscosity, 7), is defined as rly. 

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There are two general types of constitutive equations for fluids Newtonian and non-Newtonian. For Newtonian fluids, the relation between the stress tensor, t, and the rate of deformation tensor or the shear stress is linear. For non-Newtonian fluids the relation between the stress tensor and the rate of deformation tensor is nonlinear. The various Newtonian and non-Newtonian rheologies of fluids are shown in Figure 12.2. There are four types of behavior (1) Newtonian, (2) pseudo-plastic, (3) Bingham plastic, and (4) dilatent. The reasons for these different rheological behaviors will also be discussed in subsequent sections of this chapter. But first it is necessary to relate the stress tensor to the rate of deformation tensor. 

Samoh l, I. Reduction of thermodynamic constitutive equations for fluid mixtures using form invariance. Collect. Czechoslov. Chem. Commun. 54, 277-283 (1989)... 

The Oldroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950)... 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

Single-integral constitutive equations for viscoelastic fluids... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

Leslie, F. M., Some constitutive equations for anisotropic fluids. Quart J Mech Appl Math, 1966,19(3), 357 370. 

So if we substitute the complex stress and strains into the constitutive equation for a Maxwell fluid the resulting relationship is given by Equation (4.21) ... 

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. 

Using a Maxwell model as a constitutive equation for a viscoelastic fluid, one can show that the instantaneous shear stress at the wall is smaller in the viscoelastic fluid than in the corresponding Newtonian fluid. 

We are now left to deal with the constitutive equation. For a generalized Newtonian fluid, we can write... 

In this case, p is an arbitrary constant, chosen as the zero shear rate viscosity. The expression for the non-Newtonian viscosity is a constitutive equation for a generalized Newtonian fluid, like the power law or Ostwald-de-Waele model [6]... 

The volumetric constitutive equations for a chemoporoelastic material can be formulated in terms of the stress S = a,p, it and the strain 8 = e, (, 9, i.e., in terms of the mean Cauchy stress a, pore pressure p, osmotic pressure it, volumetric strain e, variation of fluid content (, and relative increment of salt content 9. Note that the stress and strain are measured from a reference initial state where all the stress fields are equilibrated. The osmotic pressure it is related to the change in the solute molar fraction x according to 7r = N Ax where N = RT/v is a parameter with dimension of a stress, which is typically of 0( 102) MPa (with R = 8.31 J/K mol denoting the gas constant, T the absolute temperature, and v the molar volume of the fluid). The solute molar fraction x is defined as ms/m with m = ms + mw and ms (mw) denoting the moles of solute (solvent) per unit volume of the porous solid. The quantities ( and 9 are defined in terms of the increment Ams and Amw according to... 

A. I. Leonov, On a Class of Constitutive Equations for Viscoelastic Liquids, J. Non-Newt. Fluid Meek, 25, 1-59 (1987). 

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). 

The set of relations (8.27) determines the fluxes as quasi-linear functions of forces. The coefficients in (8.27) are unknown functions of the thermodynamic variables and internal variables. We should pay special attention to the fourth relation in (8.27) which is a relaxation equation for variable The viscoelastic behaviour of the system is determined essentially by the relaxation processes. If the relaxation processes are absent (all the =0), equations (8.27) turn into constitutive equations for a viscous fluid. 

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

Equation (3.6a) is now introduced into the following set of constitutive equations for simultaneous accelerative one-dimensional particle-fluid motion in the vertical direction (Kwauk, 1964) ... 

For simplicity, a set of constitutive equations for a Stokesian fluid without memory is... 

All the non-Newtonian constitutive equations just given are simplifications of the most general time-independent constitutive equation for isotropic, incompressible non-Newtonian fluids that do not exhibit elasticity [4,5],... 

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]...

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

P.KCurrie, Constitutive equations for polymer melts predicted by the Doi-Edwards and Curtiss-Bird kinetic theory models, J. of Non-Newt. Fluid Mech. 11 (1982), 53-68. 

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