Constitutive equation second

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

For the special cases in which the normality conditions may be solved for k, it is possible to find alternate forms of the constitutive equations. Using (5.59) in (5.58) and (5.62) in (5.61) and dividing the first resulting equation by the second... 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

The system is non-Newtonian and viscosity is a function of temperature and shear rate. A constitutive equation including a second order term for the logarithm of shear rate was used. 

As seen, the Halpin-Tsai equation has a term a, raised to the power of one, to accommodate the filler aspect ratio. Since IAF intends to supplant the same, the new equation is expected to have a reduced dependence on the aspect ratio. Thus, the presence of aspect ratio in the equation needs to be diluted. Two constitutive equations are suggested the first one contains a correction term in the form of a shape reduction factor (a0 5) (24), while the second (25), is devoid of any extrashape related corrections Modified Halpin-Tsai I ... 

Second Level. At the second level, the constitutive equations must involve two (or more) additional variables. For instance ... 

For almost steady flows one can expand yl1 or y about t t and obtain second-order fluid constitution equations in the co-deforming frame. When steady shear flows are considered, the CEF equation is obtained, which, in turn, reduces to the GNF equation for T i = 2 = 0 and to a Newtonian equation if, additionally, the viscosity is constant. 

The same statement can be made about inelastic non-Newtonian fluids, such as the Power Law fluid, from a mathematical solution point of view. In reality, most non-Newtonian fluids are viscoelastic and exhibit normal stresses. For fluids such as those (i.e., fluids described by constitutive equations that predict normal stresses for viscometric flows), theoretical analyses have shown that secondary flows are created inside channels of nonuniform cross section (78,79). Specifically it can be shown that a zero second normal stress difference is a necessary (but not sufficient) condition to ensure the absence of secondary flow (79). Of course, the analyses of flows in noncircular channels in terms of constitutive equations—which, strictly speaking, hold only for viscometric flows—are expected to yield qualitative results only. Experimentally low Reynolds number flows in noncircular channels have not been investigated extensively. In particular, only a few studies have been conducted with fluids exhibiting normal stresses (80,81). Secondary flows, such as vortices in rectangular channels, have been observed using dyes in dilute aqueous solutions of polyacrylamide. Interestingly, these secondary flow vortices (if they exist) seem to have very little effect on the flow rate. 

With regard to constitutive equations, White (13) notes that, in view of the short residence time of the polymer in the nip region (of the order of magnitude of seconds), it would be far more realistic to use a constitutive equation that includes viscoelastic transient effects such as stress overshoot, a situation comparable to that of squeezing flows discussed in Section 6.6. 

The three constant Oldroyd model is a nonlinear constitutive equation of the differential corrotational type, such as the Zaremba-Fromm-Dewitt (ZFD) fluid (Eq. 3.3-11). [For details, see R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Second Edition, Vol. 1, Wiley, New York, 1987, Table 7.3-2.]... 

To simplify the situation, one can keep only one internal variables with the smallest number from each set, that is x k and u k. It allows one to specify equations (8.28) for this case and to write a set of constitutive equations for two internal variables - the symmetric tensors of second rank. The particular case of general equations are equations (9.24)-(9.27) - constitutive equations for strongly entangled system of linear polymer. For a weakly entangled system, one can keep a single internal variable to obtain an approximate... 

The set of constitutive equations for the dilute polymer solution consists of the definition of the stress tensor (6.16), which is expressed in terms of the second-order moments of co-ordinates, and the set of relaxation equations (2.39) for the moments. The usage of a special notation for the ratio, namely... 

There are two main reasons for the departure of the present model from the DLVO theory. First, the constitutive equations, which relate the polarization to the electric potential, are different. Second, the boundary conditions are different, since the average polarization in the DLVO theory is directly related to the surface charge, while in the present treatment it depends also on the surface dipole density. 

This defines the magnetic field intensity H /x is the 3x3 linear magnetic permeability tensor. Equation (2.7.13), linking B to H, is the "second constitutive equation." The magnetic field H is expressed explicitly by... 

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

The constitutive equations are the Oldroyd-B model and a modified Oldroyd-B model in which the viscosity depends on the rate of strain. In [79], Laure et al. study the spectral stability of the plane Poiseuille flow of two viscoelastic fluids obeying an Oldroyd-B law in two configurations the first one is the two layer Poiseuille flow in the second case the same fluid occupies the symmetric upper and lower layers, surrounding the central fluid. (See Figure 9.)... 

The constitutive equation of second-order fluids indicates that the stress tensor is determined by — u), and consequently Gy t) is independent of t and depends only on and Af Thus the function dy reduces to a function y, and the constitutive equation for second-order fluids [Eq. (13.9)] can be expressed as... 

Substitution of Eqs. (9) into Eqs. (8) and subsequent differentiation with respect to lead to the equilibrium equations in terms of microstresses and microstrains (i.e. strains averaged across the layer thickness). To exclude the latter, constitutive equations for the damaged layer and the outer sublaminate, equations of the global equilibrium of the laminate as well as generalised plane strain conditions are employed. Finally, a system of coupled second order non-homogeneous ordinary differential equations is obtained... 

We begin by inserting the constitutional equation of state in the caloric equation of state, Eq. (1.13.16) this leads to the important finding that (dE/dV)T = 0, regarded as a second criterion to be imposed on ideal gases. Thus, the energy of an ideal gas depends solely on temperature. As a result we now write out the differential energy in the abbreviated form dE = dE/dT)y dT, whence... 

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