Upper-convected Maxwell

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... 

In this section the discretization of upper-convected Maxwell and Oldroyd-B models by a modified version of the Luo and Tanner scheme is outlined. This scheme uses the subdivision of elements suggested by Marchal and Crochet (1987) to generate smooth stress fields (Swarbrick and Nassehi, 1992a). 

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... 

Using the upper-convective Maxwell fluid eqn. (9.166) it reduces to... 

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 ... 

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

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

In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. 

Remark 4.4 No result such as Theorem 4.2 seems to be known for Maxwell models. We however have to mention the result [44], where the upper-convected Maxwell model in the whole space R is considered. 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

It is worth noticing that Tlapa and Bernstein [71] have proven that the Squire theorem holds true for the PoiseuiUe flow of m upper-convected Maxwell fluid. It means that any instability, which may be present for three dimensional disturbances, is also present for two dimensional ones at a lower value of the Reynolds number. This property is not true, in general, for non-Newtonian fluids [72]. 

In [80], as in previous works (e.g. [81]) on viscoelastic flows, Chen assumes disturbances of the form for a steady Couette flow of two upper-convected Maxwell fluids, and... 

M. Renardy, The stress of an upper-convected Maxwell fluid in Newtonian field near a re-entrant corner, J. Non-Newtonian Fluid Mech., 50 (1993) 127-134. 

Y. Renardy, Stability of the interface in two-layer Couette flow of upper-convected Maxwell liquids, J. Non-Newtonian Fluid Mech., 28 (1988) 99-115. 

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 ... 

An upper-convected Maxwell model has been used with the full relaxation spectrum for the calculation of the stress, but for calculating the birefringence this spectrum has been restricted to long relaxation times as shown in Fig. 12. The model predictions for the data of the Fig. 9 are shown in Fig. 13. The deviations from the linear stress-optical nole are well accounted for by this very simple model. However, the model does not describe the stress data in simple elongation, and in particular the initial stress values at temperatures close to the Tg. 

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]. 

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 [Pg.127]

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 .)... 

Bagley (1992a) measured the apparent biaxial elongational viscosity of wheat flour dough. The upper convected Maxwell model was considered to be adequate in explaining both the effect of crosshead speed and sample... 

Bagley, E. G., Christianson, D. D., and Martindale, J. A. (1988). Uniaxial compression of a hard wheat flour dough Data analysis using the upper convected Maxwell model. J. Text. Stud. 19, 289-305. 

Figure 2.4 depicts the growth rates predicted from (2.26) for two jets of the upper-convected Maxwell liquids, which are shown by curves 1 and 3. They correspond to different values of the relative gas velocity (U for curve 1 is higher... 

Fig. 2.5 The amplitude Y of the bending perturbations of a jet of the upper-convected Maxwell liquid is shown by curve 1 [1], The values of the dimensionless groups are IT] = 10 IT2 = 0.156 x 10, II3 = 0.64, 114 = 115 = 0. Curve 2 depicts the amplitude of the corresponding jet of Newtonian liquid (Hz = 0) (Courtesy of Pearson Education)...

Generally, it is found that while the upper convected Maxwell fluid, Eq. (26), and the Lodge rubberlike liquid, Eq. (31), predict the correct qualitative features of polymeric fluid behavior, the representation is not quantitative. In particular, in a stress-relaxation experiment, the relaxation takes place over too broad a range of time to be described by a single exponential. One therefore uses a spectrum of relaxation times, and modifies Eq. (30) to... 

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. 

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