Rheological equation of state

Doi and Ohta [249], and then Doi and Onuki [259], considered a simplifled model of two immiscible liquids at 0j = 02 = /2 having the same viscosity and density, focusing primarily on the interface and the flow-induced evolution of the interfadal area and orientation. Instead of the interphase the authors assumed an interface without thickness - consequently, postulating that the interface does not perturb the flow field (the passively advected interface). Thus, a semi-phenomenological kinetic equation describes the time evolution in a given flow field of the interfadal area per unit volume, Q, and its anisotropy, tensor 

Hi is the unit vedor normal to interface, V is the system volume. 

Lee and Park [263] derived a more general constitutive equation for immiscible blends. which compared well with the dynamic shear data of PS/ LLD P E blends over a full range of frequency and composition. Their model included the dissipative time evolution of Qand qjj, written as functions of (i) the degree of total relaxation, (ii) the size relaxation strongly dependent on concentration, viz. a and (iii) the breakup and shape relaxation, assumed dependent on (1— )- The parameters contain adjustable constants and depend on concentration as well as on the deformation mode. The interfacial tension coefficient was assumed to be constant, independent of As before [249, 264], the constitutive equation was written in form of three functions dq /dt, and Oy. The model predicts well the dynamic moduli of 

the first term on the right-hand side of Eq. (2.44) expresses compatibility with mechanics, and the second term that with thermodynamics. The Poisson bivector transforms a potential gradient into a vector, that is, kinematics of the state variable x. The GENERIC formalism was applied to the flow of polymer blends [271]. 

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De Witt T., Mezner. W. A rheological equation of state which predicts non-Newtonian viscosity, normal stresses and dynamics module. J. Appl.Phys., 1985, v. 26, p. 889-892. 

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... 

The fundamental rheological characterization of a material requires the experimental determination of a constitutive equation (a rheological equation of state) that mathematically relates stress and strain, or stress and strain rate. The constants in the constitutive equation are the rheological properties of the material. 

The relationships between stress and strain, and the influence of time on them are generally described by constitutive equations or rheological equations of state (Ferry, 1980). When the strains are relatively small, that is, in the linear range, the constitutive... 

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 ... 

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 ... 

On the basis of these observations, Maxwell suggested to combine Hooke s law (for elastic bodies) and Newton s law (for viscous fluids) additively into a single rheological equation of state, which has the following form in the onedimensional case ... 

Let us consider power-law fluids in more detail. Experiments [141] show that the index n of non-Newtonian behavior of a substance may be treated as a constant if the temperature differences in the flow region do not exceed 30 to 50 K. The medium consistence k = k(T) is much more sensitive to temperature inhomogeneities and decreases with increasing T. Therefore, the rheological equation of state for a power-law fluid in the nonisothermal case can be written as follows ... 

In this section we present the equations of motion and heat transfer for incompressible non-Newtonian fluids governed by the rheological equation of state (7.1.1) when the apparent viscosity p = p(h, T) arbitrarily depends on the second invariant I2 of the shear rate tensor and on the temperature T. This section contains some material from the books [47, 320, 443], For the continuity equation in cylindrical and spherical coordinates, see Supplement 5.3. 

Leonov [1994] introduced kinetics of interactions into his rheological equation of state. The new relation can describe systems with a dynamic yield stress, without resorting to a priori introducing the yield stress as a model parameter (as it has been done in earlier models). 

A thorough analysis of forming processes relies on mathematical descriptions of the physical steps. These are independent of the material and are formulated with the equations of continuity, momentum and energy. These equations applied to the particular geometry must be combined with the rheological equation of state of the material. Furthermore, the relations expressing the temperature and pressure dependencies of such properties such as density, thermal conductivity, etc., are required. The combination of these relations constitutes the mathematical formulation of the flow process. These formulations are beyond the scope of the Chapter. 

Resistivity, thermal Retardation spectrum Reynolds number. Re Rheological equation of state Rheological measurements... 

The simplest rheological equation of state is for the Newtonian fluid where viscosity is the only material property needed to characterize the... 

V. The study of simple structural models may be quite helpful in the formulation of rheological equations of state ( constitutive equations ) for macromolecular solutions. 

In rheology, we study the mathematical expression for the relation between stress and deformation. Such an equation is usually referred to as the constitutive equation or the rheological equation of state. For expressing the constitutive equation in three dimensions, we need to find the proper way to express the deformation in the tensorial form as well. 

In order to characterize polymeric fluids and to test rheological equations of state it is customary to use simple, well defined flows. The two main flows are simple shear and simple elongational. These are shown schematically in Figure 1. In shear flow, material planes (see Figure 1) move relative to each other without being stretched, whereas in extensional flow the material elements are stretched. These two different flow histories generate different responses in not only flexible chain polymers but in liquid crystalline polymers. When these flows are carried... 

Marrucci G, lannirubertok G (2004) Interchain pressure effect in extensional flows of entangled polymer. Macromolecules 37 3934—3942 Morris FA (2001) Understanding rheology. Oxford University Press, Oxford Oldroyd JG (1950) On the formulation of rheological equations of state. Proc R Soc A200 523-541... 

Rheology of Immiscible Blends 7.5.1 Rheological Equation of State... 

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