Upper-convected Maxwell equation

Comparing eq. 4.3.10 with eq. 4.3.1, we see that to second order in the velocity gradient the upper-convected Maxwell equation for small strain rates reduces to a special case of the equation of the second-order fluid with V i.o = 2kr]o and V 2,o = 0. All properly formulated constitutive equations for which the stress is a smooth functional of the strain history reduce at second order in the velocity gradient to the equation of the second-order fluid. Example 4.3.3, however, illustrates that the equation of the second-order fluid cannot be trusted except for slow nearly steady flows. 

Calculate the predictions of the upper-convected Maxwell equation in (a) start Up of steady shear and (b) steady state uniaxial extension for arbitrary shear rate y and extension rate e, and compare these predictions with those for the Newtonian and second-order fluids. 

Using the definition of eq. 4.3.2 for die upper-convected derivative, the upper-convected Maxwell equation (eq. 4.3.7) can be written in expanded form as follows  

The term Vr is zero because we are considering a homogeneous flow. The symmetry of the shearing flow leads us to expect that the stress tenstMT will contain only the components T12.r21.r11.r22 and T33. Assuming this form for the stress tensor and using eq. 2.2.10 for the velocity gradient, we obtain 

To obtain the steady state results, we set the time derivative to zero. We find immediately that T33 = T22 = 0. With this result for T22. we find that T12 = mY f m which we can obtain Th = 2riokY. This result implies that the shear viscosity is a constant tjo. the fitst normal stress coefficient is also a constant equal to 2X o end the second normal stress coefficient is zero. This is the same result that we obtained in the second-order fluid limit of the UCM equation  

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Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

The Lodge equation can also be obtained in a differential form known as the Upper Convected Maxwell equation (UCM) ... 

Equation (3-32) is the upper-convected Maxwell equation. Note that in a state of rest (d = 0), with no flow (Vv = 0), the stress tensor is an isotropic tensor, G8. 

It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

Each upper-convected Maxwell equation... 

Equation (3-77) differs from the upper-convected Maxwell equation, Eq. (3-32), in that it includes the term (2/3G)D aa, which imparts strain softening and shear thinning to the behavior of the model. 

Note first that if the fluid is at a state of equilibrium with no flow, then the time derivative d is equal to zero, and the velocity gradient Vv is also zero. This implies from the above equations that = G8. Hence cr, i = <7 2 = ct t, = G at equilibrium, and aj = 0, for i j. Thus, although the diagonal stress components are not zero at equilibrium, they are all equal to each other, and the nondiagonal components are all equal to zero. Hence, the stress tensor is isotropic, but nonzero at equilibrium. (If one redefines the stress tensor as H = a — G8, then S " = 0 at equilibrium. The upper-convected Maxwell equation can then be rewritten in terms of Z .)... 

Recall from Chapter 3 that the linear Maxwell equation can be written in an integral form. The nonlinear upper-convected Maxwell equation can also be written in an integral form, namely... 

To within an added isotropic constant, the stress r,-, given by eq. 4.3.31 for each mode, satisfies an upper-convected Maxwell equation ... 

Example 14.7 Obtain an expression for the net tensile stress (negleeting the solvent contribution) in a uniaxial extension experiment aceording to the upper-convected Maxwell equation. 

The Maxwell class of viscoelastic constitutive equations are described by a simpler form of Equation (1.22) in which A = 0. For example, the upper-convected Maxwell model (UCM) is expressed as... 

The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than l/(2A,i). The steady-state elongational wscosity depends then on strain rate ... 

For the upper-convected Maxwell model, the full equations for reads... 

This difficulty can be overcome by the use of a viscoelastic model limiting the effect of the singularity in the transport equations. In the Modified Upper Convected Maxwell (MUCM) proposed by Apelian et al. (see [1]), the relaxation time X is a function of the trace of the deviatoric part of the extra stress tensor ... 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

Apelian, M. R., e. a. (1988). hnpad of the constitutive equation and singularity on the calculation of stick-slip flow The modified upper-convected maxwell model, /. Non-Newtonian Fluid Mech. 27 299-321. 

The K-BKZ and other integral constitutive equations discussed above can be regarded as generalizations of the Lodge integral, eq4.3.18. The upper-convected Maxwell (UCM) equation, which is the differential equivalent of the Lodge equation, can also be generalized to make possible more realistic predictions of nonlinear phenomena. 

This is the constitutive equation or rheological equation of state for the elastic dumbbell suspensions. It is identical to the upper-convected Maxwell model, eq. 4.3.7. The molecular dynamics have led to a proper (frame-indifferent) time derivative and to a definition... 

Let us consider the upper convected Maxwell model given by Eq. (3.4). Since we are only interested in small-amplitude oscillations with Uj = Vi(t,x2), all nonlinear terms appearing in the convected derivative of stress tensor a (see Eq. (2.107)) can be neglected and thus Eq. (3.4) reduces to the classical Maxwell equation, Eq. (3.3). Applying Eq. (3.79) to (3.3) we obtain ... 

Figure 3.11 gives plots of n /irjo versus A.jC that are predicted from two constitutive equations (1) the upper convected Maxwell model, and (2) the Oldroyd three-constant model. It is seen in Figure 3.11 that both models predict values of increasing very rapidly without bound as e increases, in contrast to the experimental results given in Figure 3.10. As a matter of fact, all the expressions summarized in Table 3.3 predict similar elongational behavior, which is considered to be physically unrealistic.

Giesekus (1982) summarized nicely a series of his papers on the formulation of a new class of constitutive equations. The origin of Eq. (3.23) comes from a modification of the upper convected Maxwell model as applied to a dilute polymer solution, namely... 

This is also the integral form of the differential constitutive equation called the upper convected Maxwell model , which is given in the next section. 

Just as there are various possible finite strain tensors, there are various time derivatives that can be used in place of the ordinary derivative of stress in Eq. 10.21 to satisfy the continuum mechanics requirements for a model to be able to describe large, rapid deformations in arbitrary coordinate systems. The derivative that yields a differential model equivalent to Lodge s Eq. 10.6 is the upper convected time derivative (defined in Eq. 11.19), and the resulting model is called the upper-convected Maxwell model. Other possibilities include the lower-convected derivative and the corotational derivative. Furthermore, a weighted-sum of two of these derivatives can be used to formulate a differential constitutive equation for polymeric liquids. In particular, the Gordon-Schowalter convected derivative [7] is defined in this manner. 

A purely viscous non-Newtonian approach was followed by Han and Park (1975b). They used the power-law model and the energy equation, assuming that the effects of crystallization were insignificant. The agreement of this model with experimental data in terms of the bubble radius and thickness as a function of the axial distance for LDPE and HDPE was reported to be reasonable. In terms of viscoelastic models, Luo and Tanner (1985) considered the Leonov model, and Cain and Denn (1988) considered the upper convected Maxwell and Marrucci models in nonisothermal cases of film blowing. In some of the cases analyzed, multiple steady-state solutions were present (see also Problem 9C.2). 

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