Upper convective model

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

For the generalized linear Maxwell model (GLM) and the upper convected model (UCM), the only material parameters needed are contained in the relaxation time spectrum of the material which can be obtained from simple linear viscoelastic measurements. For the Giesekus model, one needs in addition the mobility factors which Christensen and McKinley obtained by fitting the stress-strain curves of the adhesive. The advantage of the Giesekus model was that it provided them with a better description of the stress-strain curves. This, of course, is to be expected since those curves were used to deduce the parameters of the model. 

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... 

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 first kind of modification to the UCM model that may be conceivable is that of the convected derivative. This leads one to consider that the motion of the network junctions is no more that of the continuum and thus, the afiine assumption of the Lodge model is removed. Among the various possibilities, Phan Thien and Tanner suggested the use of the (Jordon-Schowalter derivative [47], which is a linear combination of the upper- and lower-convected derivatives, instead of the upper-convected derivative ... 

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

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. 

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]. 

Figure 6 A range of mantle models for the distribution and fluxes of noble gases in the Earth. Layered mantle models with the atmosphere derived from the upper mantle involve either progressive unidirectional depletion of the upper mantle (A) or an upper mantle subject to inputs from subduction and the deeper mantle, and has steady state concentrations (B). Whole mantle convection models involve degassing of the entire mantle, with helium with high He/ He ratios found in OIB stored in either a deep variable-thickness layer (C), a layer of subducted material at the core-mantle boundary (D), or the core (E). The models are discussed in the text and Chapter 2.06 (source Porcelli and...

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. 

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. 

Figure 3 Predicted plate spacings without (dashed curve) and including the convection model with two upper limits in the Nusselt number (solid and dotted lines). Crosses refer to observations compiled from sea ice field studies and circles to laboratory data at comparable NaCl concentrations from- 25 to 31 %o. The low velocity points refer to thick arctic sea ice (square) and the bottom of an Antarctic ice shelf (triangles).

Figure 4. Comparison of the oxygen yield Mo as a function of the initial stellar mass Mzams for models with and without stellar wind mass loss, and assuming a remnant mass of the order of 2Mq. Upper panel Models with slow semiconvective mixing. Weaver Woosley s (1993 WW93) models are computed without mass loss, Langer Henkel (1995 LH95) included mass loss. Lower panel Models with fast semiconvective mixing resp. Schwarzschild criterion for convection. Woosley Weaver (1995 WW95) neglected mass loss, Maeder (1992 M92) explored models with standard and excessive mass loss.

One of the most fundamental, contemporary debates about the nature of the Earth s mantle centers on the subject of mantle convection. There are two conflicting views which are commonly described as "layered convection" and "whole-mantle convection" (see Fig. 3.17). The layered convection model is championed by geochemists who prefer to see the mantle as two separate convecting layers. In this model the upper and lower mantle are geo-chemically isolated from each other and convect separately. Whole mantle convection is advocated by geophysicists, who that believe there is evidence for a significant exchange of mass between the upper and lower mantle. 

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. 

Letwimolnun et al. [2007] used two models to explain the transient and steady-state shear behavior of PP nanocomposites. The first model was a simplified version of the stmcture network model proposed by Yziquel et al. [1999] describing the nonlinear behavior of concentrated suspensions composed of interactive particles. The flow properties were assumed to be controlled by the simultaneous breakdown and buildup of suspension microstructure. In this approach, the stress was described by a modified upper-convected Jeffery s model with a modulus and viscosity that are functions of the suspension structure. The Yziquel et al. model might be written ... 

Ata = -l,b = c = 0, equations [7.2.24] correspond to the Maxwell liquid with a discrete spectrum of relaxation times and the upper convective time derivative. For solution of polymer in a pure viscous liquid, it is convenient to represent this model in such a form that the solvent contribution into total stress tensor will be explicit ... 

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