Constitutive equations comparison

So far, we are able to construct the constitutive equations for qc, Pc, and y. For moderate solids concentrations, we can neglect the kinetic contributions in comparison to the collisional ones. Thus, we can assume P Pc and qk qc. Substituting the constitutive relations into Eqs. (5.274), (5.275), and (5.281), after neglecting the kinetic contributions, yields five equations for the five unknowns ap, Up, and Tc (or ( 2)). Hence, the closure problem is resolved. 

The simulation of non-Newtonian fluid flow is significantly more complex in comparison with the simulation of Newtonian fluid flow due to the possible occurrence of sharp stress gradients which necessitates the use of (local) mesh refinement techniques. Also the coupling between momentum and constitutive equations makes the problem extremely stiff and often time-dependent calculations have to be performed due to memory effects and also due to the possible occurrence of bifurcations. These requirements explain the existence of specialized (often FEM-based) CFD packages for non-Newtonian flow such as POLYFLOW. 

The aim of this section is to perform comparisons between the predictions of some constitutive equations and experimental results in simple shear and uniaxial elongation on three polyethylenes. In addition, this is expected to provide well-defined sets of material parameters to be used in the model equations for the computation of complex flows. 

R.G.Larson, A critical comparison of constitutive equations for polymer melts, J. of Non-Newt. Fluid Mech. 23 (1987), 249-269. 

S.A.Khan, R.G.Larson, Comparison of simple constitutive equations for polymer melts in shear and biaxial and uniaxial extensions, J. Rheol. 21 (1987), 207-234. 

Birefringence measurements are often performed to compare theoretical and experimental stress distributions in an abrupt contraction (see Section III-l). Such comparisons have been already published for the White-Metzner [30], K-BKZ [27, 31] and Wagner [32, 33] constitutive equations. Generally speaking. 

The comparison of the predictions given by these constitutive equations in steady-state shear and extensional simple flows are summarized in Figs. 1 and 2. 

For comparison we briefly display the standard macroscopic transfer functions resulting from the averaging process, before we introduce the particular constitutive equations normally applied in chemical reaction engineering. This might help to elucidate the connection between the conventional fluid mechanics modeling framework and the customary chemical reaction engineering interfacial coupling terms. 

Here the situation is very similar to that encountered in connection with the need for continuum (constitutive) models for the molecular transport processes in that a derivation of appropriate boundary conditions from the more fundamental, molecular description has not been accomphshed to date. In both cases, the knowledge that we have of constitutive models and boundary conditions that are appropriate for the continuum-level description is largely empirical in nature. In effect, we make an educated guess for both constitutive equations and boundary conditions and then normally judge the success of our choices by the resulting comparison between predicted and experimentally measured continuum velocity or temperature fields. Models derived from molecular theories, with the exception of kinetic theory for gases, are generally not available for comparison with the empirically proposed models. We discuss some of these matters in more detail later in this chapter, where specific choices will be proposed for both the constitutive equations and boundary conditions. 

The obvious question is this What conditions should be imposed Without a molecular or microscopic theory for guidance, there is no deductive route to answer this question. The application of boundary conditions then occupies a position in continuum mechanics that is analogous to the derivation of constitutive equations in the sense that only a limited number of these conditions can be obtained from fundamental principles. The rest represent an educated guess based to a large extent on indirect comparisons with experimental data. In recent years, insights developed from molecular dynamics simulations of relatively simple... 

Following Doi and Ohta s work, a more general theory was derived for immiscible polymer blends by Lee and Park [1994]. A constitutive equation for immiscible blends was proposed. The model and the implied blending laws were verified by comparison with dynamic shear data of PS/LLDPE blends in oscillatory shear flow. 

Figure 2 shows the comparison of the fractal-layer (solid line a) and two-timescale (solid line b) models with the simulations in terms of effective diffusivity, eq. (13). Both the models furnish a satisfactory level of agreement with simulation data. We may therefore conclude that approximate models based on a Riemann-Liouville constitutive equation are able to furnish an accurate description of adsorption kinetics on fractal interfaces. These models can also be extended to nonlinear problems (e.g. in the presence of nonlinear isotherms, such as Langmuir, Freundlich, etc.). In order to extend the analysis to nonlinear cases, efficient numerical sJgorithms should be developed to solve partied differential schemes in the presence of Riemann-Liouville convolutional terms. 

In the case of trickle flow, it has been shown that under certain conditions the slit-flow approximation yields a very satisfactory set of constitutive equations for the gas-liquid and the liquid-solid drag forces [20, 21]. As a matter of fact, the slit flow becomes well representative of the trickle-flow regime when the liquid texture is contributed by solid-supported liquid Aims and rivulets. This generally occurs at low liquid flow rates that allow the transport of film-like liquids [20]. We will assume, without proof though, that such hypotheses also hold in the case of artificial-gravity operation. The validity of these assumptions and of the several others outlined above will be evaluated later in terms of model versus experiment comparisons. Choosing the drag force closures of the simplified Holub slit model [20], the equations system becomes ... 

Fig. 11.42. Comparison of the dilatancy boundary measured using volumetric measurements and AE measurements. The continuous line shows the calculated dilatancy boundary using the obtained constitutive equations.

These three parameters (or other equivalent dimensionless groups) must appear in whatever formulation of this type of problem (the esterification reaction in PVRs), possibly together with other parameters which take into account other aspects such as additional phenomena (for example concentration polarization of the membrane), the presence of products in the initial mixture, the concentration of the catalyst and more complex constitutive equations. The dimensionless parameters have a more general validity than the individual dimensional parameters that appear grouped into them and characterize more univocally the behaviour of the system. The adoption of the parameter 5, the ratio of the characteristic rate of permeation to the characteristic rate of reaction, can be extended to any PVR and in general also to any membrane reactor. With this approach the comparison between different studies on PVRs is more direct and meaningful. On the other hand, the less acceptable, though often employed, dimensional parameter A/V, is comprised in the definition of 5. 

A comparison of the constitutive (5.10) and (5.12) with the stress-strain (5.9) shows a favourable analogy. This is often used to model smart materials as a part of an adaptronic structure e. g. by substituting cut by analogous parts of the constitutive equations of the smart material in the finite element model of the adaptronic structure (see also below). 

White, J.L. and Tanaka, H. (1981) Comparison of a plastic-viscoelastic constitutive equation with rheological measurements on a polystyrene melt reinforced with small particles, /. Non-Newtonian Fluid Mech., 8,1-10. 

Note that the normal stresses can become quite large relative to the shear stresses. (The simplest comparison is at 1 s, where the viscosity is numerically equal to the shear stress in consistent SI units.) The lines in the figure refer to a constitutive equation that is discussed subsequently. 

It is evident in the foregoing examples that deviations from linear viscoelastic behavior are evoked by both large strains and large strain rates. Phenomenological constitutive equations have been developed in which one or the other has a dominant role, as described for example by memory functions which depend either on strain invariants or on strain rate invariants. - " In critical comparisons of pre-... 

The numerical modeling methods for polymer blends have been reviewed in this chapter, with different categories such as volume-of-fluid, molecular dynamics and diffusion-controlled methods being introduced. Use of the Cahn-Hilliard method was emphasized for binary and ternary polymer systems with no obvious mechanical flux, while specific factors such as elastic energy and functionalized substrate were considered for purposes of comparison. The diffusion-controlled model described, using the Cahn-Hilliard equation as the constitutive equation, can be used to depict the gradient of the interface as well as the composition profile of partially miscible blends hence, it is feasible to implement this equation in a polymer blend system. It should be noted that although these examples do not consider mechanical flux, additional constitutive equations (e.g., Navier-Stokes) can easily be added to this diffusion-controlled model. 

In order to avoid the necessity of measuring the stress-strain curves of the adhesive as a function of rate and temperature, we would like to have a constitutive equation for the adhesive, the parameters of which can be determined from relatively simple rheological measurements. Authors have taken multiple approaches to this problem. In some of the work, the adhesive stress-strain curves in uniaxial extension were measured which allows a direct comparison between the constitutive model and the data. In other cases, the model was used as a tool to... 

The constitutive equations for the particle phase and boundary conditions are adopted from Lun et al. [2] and Johnson et al. [9] respectively. The numerical method used is that described in Reese et al. [13] and Zhang Reese [14]. The profiles of particle velocity, volume fraction and the granular temperature of dry granular and granular-air mixture flows down a smooth inelastic chute have been calculated, as well as the energy dissipation profiles. The lack of suitable experimental data for comparison means that our results can only, at present, be compared to other simulations. However, this does allow some conclusions to be drawn regarding the quantitative and qualitative differences our model introduces when compared... 

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