Rheological stress tensor

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

There are basically three types of approaches to define the solid stress tensor, or more specifically the solid viscosity. In the early hydrodynamic models— developed by Jackson and his co-workers (Anderson and Jackson, 1967 Anderson et al., 1995), Kuipers et al., (1992), and Tsuo and Gidaspow (1990)—the viscosity is defined as an empirical constant, and also the dependence of the solid phase pressure on the solid volume fraction is determined from experiments. The advantage of this model is its simplicity, the drawback is that it does not take into account the underlying characteristics of the solid phase rheology. 

The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

The symbols used follow the recent recommendations of the Society of Rheology SI units are used. We follow the stress tensor convention used by Bird et al., namely, n = P6 + x, where n is the total stress tensor, P is the pressure, and x is that part of the stress tensor that vanishes when no flow occurs both P and x, are positive under compression. 

Cooke BJ, Matheson AJ (1976) Dynamic viscosity of dilute polymer solutions at high frequencies of alternating shear stress. J Chem Soc Faraday Trans II 72(3) 679-685 Curtiss CF, Bird RB (1981a) A kinetic theory for polymer melts. I The equation for the single-link orientational distribution function. J Chem Phys 74 2016—2025 Curtiss CF, Bird RB (1981b) A kinetic theory for polymer melts. II The stress tensor and the rheological equation of state. J Chem Phys 74(3) 2026—2033 Daoud M, de Gennes PG (1979) Some remarks on the dynamics of polymer melts. J Polym Sci Polym Phys Ed 17 1971-1981... 

Here are the components of the stress tensor as defined in rheology Tn—T22 is the first normal stress difference and T21 the shear stress, equal to Nt and rxsh, respectively. Hence, from dynamic mechanical measurements it is possible to determine the zero shear first normal stress coefficient Fq0 and zero shear viscosity y0. 

Later, Landau and Lifshitz (1959) obtained the same result by averaging the stress tensor over the entire space, thereby initiating one of the first dynamical (i.e., nonenergetic) approaches to calculating the rheological properties of suspensions. Attempts to extend Eq. (4.1) to higher concentrations are legion. Most propose a power series expansion of the form... 

In addition to the microstructural geometrical features described above, macroscopic, dynamical, rheological properties of the suspensions are derived by Brady and Bossis (1985). Dual calculations are again performed, respectively with and without DLVO-type forces. When such forces are present, an additional contribution (the so-called elastic stress) to the bulk stress tensor exists. In such circumstances, the term (Batchelor, 1977 Brady and Bossis, 1985)... 

This was an introduction to rheology. If you want to go further, you will find that it is a difficult subject. For three-dimensional problems you need sophisticated mathematics to get anywhere. You will need tensor equations the three stresses Xxy, Xyx and Xyy that you have seen are three of the nine components of the stress tensor (Figure C4-17). And that is just... 

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. 

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 ... 

Rheological properties of food materials over a wide range of phase behavior can be expressed in terms of viscous (viscometric), elastic and viscoelastic functions which relate some components of flie stress tensor to specific components of the strain or shear rate response. In terms of fluid and solid phases, viscometric... 

In formal rheology, relations between these three tensors are formulated and analyzed. Only for the two extremes of viscoelastic behaviour are such relations simple. For purely elastic materials there is a relation between the stress tensor and the strain tensor it contains the elasticity modulus and the Poisson ratio, accounting for the extent to which extension in one direction is accompamied by concomitant compression in the other two. For purely viscous fluids there is a relation between the stress tensor and the strain rate tensor. As extension in one direction is concomitant with (viscous) compression in the other two, in this case only one viscosity is required. For incompressible Newton fluids eventually an expression with only one viscosity results, see (1.6.1.131. 

The possible development of gradients in the components of the interfacial stress tensor due to flow of an adjacent fluid implies that the momentum flux caused by the the flow of liquid at one side of the interface does not have to be completely transported across the interface to the second fluid but may (partly or completely) be compensated in the interface. The extent to which this is possible depends on the rheological properties of the interface. For small shear stresses the interface may behave elastically or viscoelastically. For an elastic interfacial layer the structure remains coherent the layer will only deform, while for a viscoelastic one it may or may not start to flow. The latter case has been observed for elastic networks (e.g. for proteins) that remciln intact, but inside the meshes of which liquid can flow leading to energy dissipation. At large stresses the structure may yield or fracture (collapse), leading to an increased flow. 

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton s law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written ... 

The stress tensor t in Eq. (6) is related to the rate-of-strain tensor by a rheological equation of state such as ... 

The dependence of the viscous stress on the velocity gradient in the fluid is a constitutive law, which is usually called the bulk rheological equation. The general linear relation between the viscous stress tensor, T, and the rate of strain tensor,... 

In fact, Equation 5.281 describes an interface as a two-dimensional Newtonian fluid. On the other hand, a number of non-Newtonian interfacial rheological models have been described in the literature. Tambe and Sharma modeled the hydrodynamics of thin liquid films bounded by viscoelastic interfaces, which obey a generalized Maxwell model for the interfacial stress tensor. These authors also presented a constitutive equation to describe the rheological properties of fluid interfaces containing colloidal particles. A new constitutive equation for the total stress was proposed by Horozov et al. ° and Danov et al. who applied a local approach to the interfacial dilatation of adsorption layers. 

Before leaving the stress-tensor expressions we should note that for the special case of Hookean dumbbells, it is possible to eliminate the quantity between Eqs. (4.19) and (4.20) and obtain an equation for r in terms of kinematic quantities — that is, a properly formulated constitutive equation (or rheological equation of state) is obtained10 ... 

An important class of non-Newtonian fluids is formed by isotropic rheological stable media whose stress tensor [ry] is a continuous function of the shear rate tensor [e,j] and is independent of the other kinematic and dynamic variables. One can rigorously prove that the most general rheological model satisfying these conditions is the following nonlinear model of a viscous non-Newtonian Stokes medium [19] ... 

We will now examine the simplest rheological model, the Hooke law for elasticity and Newton law for viscosity. In Hooke s law the tensor of strain Un is a linear fimction of the stress tensor of S, i.e. the deformation is proportional the acting forces. If the inertial stress, the elastic or Hooke s stress and the viscous stress are additive, we can write... 

First of all, surface rheology is completely described by four rheological parameters elasticity and viscosity of compression/dilatation and of shear. In every case surface flow is coupled with the hydrodynamics of the adherent liquid bulk phase. From interfacial thermodynamics we know that the integration over the deviation of the tangential stress tensor from the bulk pressure represents the interfacial tension y (after Bakker 1928). 

For the description of surface rheological properties as we have said we need to consider the surface stress caused by dilatation/compression as well as by shear. As in the case of area changes two coefficients of surface shear flow exist. Using the same symbols for the shear and stress tensors, as given in Fig. 3.5., we obtain... 

Surface Rheology. Surface rheology deals with the functional relationships that link the dynamic behavior of a surface to the stress that is placed on the surface. The complex nature of these relationships is often expressed in the form of a surface stress tensor, P8. Both elastic and viscous resistances oppose the expansion and deformation of surface films. The isotropic (diagonal) components of this stress tensor describe the di-latational behavior of the surface element. The components that are off-diagonal relate the resistance to changing the shape of the surface element to the applied shear stresses. Equation 7 demonstrates the general form of the surface stress tensor. 

More recenfly, a complete set of governing relationships was derived from the requirements of the compatibility of dynamics and thermodynamics [Grmela and Ait-ICadi, 1994, 1998 Grmela et al, 1998, 2001]. The authors developed a set of equations governing the time evolution of the functions Q and q. (see Eqs 7.95), as well as the extra stress tensor expressed in their terms. The rheological and morphological behavior was expressed as controlled by two potentials thermodynamic and dissipative. Under specific conditions for these potentials, Lee and Park formalism can be recovered. 

A number of fluids mentioned throughout the text that are of importance in physicochemical hydrodynamics do not behave in the Newtonian fashion outlined in Section 2.2. That is, the stress tensor is not a linear function of the rate of strain tensor. Such nonlinear fluids are termed non-Newtonian and the study of their behavior falls under the science of rheology, which deals with the study of the deformation and flow of matter. The materials encompassed by this broad subject cover a spectrum from Newtonian fluids at one end to elastic materials at the other with such fluids as tars, liquid crystals, and silly putty in between. Among the fluids we have discussed in the text that do not exhibit a Newtonian behavior are some polymeric liquids, some protein solutions, and suspensions. 

Another important quantity of general use in fluid mechanics and polymer rheology is the total stress tensor. This tensor is defined in terms of the stress tensor and the hydrostatic pressure, as indicated in the following equation ... 

Unlike simple differential constitutive equations as the one previously addressed, constitutive equations may present special types of derivatives such as the substantial derivative, or other types of derivatives in which a hypothetical frame of observation of the flow is allowed to translate, rotate, and/or deformate [33], The Criminale-Ericsen-Filbey (CEF) equation, written here as Equation 22.21, is an example of this type of equations. The CEF equation is relatively simple, and it is explicit in the stress tensor. The latter is a feature not shared by all rheological relationships belonging in the category of equations with special types of derivatives [35]. 

As a part of continuum mechanics, rheology has been developed assuming continuity, homogeneity, and isotropy. In multiphase systems such as nanocomposites, these assumptions are rarely vaUd. Thus, the rheology of multiphase systems (MPSs) determines volume-average properties or bulk quantities [Hashin, 1965], The volume-averaged rate of strain tensor, (yy>, and the corresponding stress tensor are expressed, respectively, as... 

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