Constitutive equations continuum mechanical approach

It can be noted that other approaches, based on irreversible continuum mechanics, have also been used to study diffusion in polymers [61,224]. This work involves development of the species momentum and continuity equations for the polymer matrix as well as for the solvent and solute of interest. The major difficulty with this approach lies in the determination of the proper constitutive equations for the mixture. Electric-field-induced transport has not been considered within this context. 

The proposed mechanisms of models to explain the drag reduction phenomenon are based on either a molecular approach or fluid dynamical continuum considerations, but these models are mainly empirical or semi-empirical in nature. Models constructed from the equations of motion (or energy) and from the constitutive equations of the dilute polymer solutions are generally not suitable for use in engineering applications due to the difficulty of placing numerical values on all the parameters. In the absence of a more generally accurate model, semi-empirical ones remain the most useful for applications. 

Time constants are related to the relaxation times and can be found in equations based on mechanical models (phenomenological approaches), in constitutive equations (empirical or semiempirical) for viscoelastic fluids that are based on either molecular theories or continuum mechanics. Equations based on mechanical models are covered in later sections, particularly in the treatment of creep-compliance studies while the Bird-Leider relationship is an example of an empirical relationship for viscoelastic fluids. 

Only a few continuous source term closures are available, hence the discrete PBE model closures are used in practice. The macroscopic statistical mechanical PBE model thus coincides with the macroscopic PBE derived from continuum mechanical principles. In this way there are little or no differences in employing these two approaches. However, the formulation of the constitutive equations are strongly influenced by the concepts of kinetic theory of dilute gases. Nevertheless, the present closures are still at an early stage of development and future work should continue developing more reliable pa-rameterizations of the kernels. These must be validated for the application in question. 

Given the apparent arbitrariness of the assumptions in a purely continuum-mechanics-based theory and the desire to obtain results that apply to at least some real fluids, there has been a historical tendency to either relax the Newtonian fluid assumptions one at a time (for example, to seek a constitutive equation that allows quadratic as well as linear dependence on strain rate, but to retain the other assumptions) or to make assumptions of such generality that they must apply to some real materials (for example, we might suppose that stress is a functional over past times of the strain rate, but without specifying any particular form). The former approach tends to produce very specific and reasonable-appearing constitutive models that, unfortunately, do not appear to correspond to any real fluids. The best-known example is the so-called Stokesian fluid. If it is assumed that the stress is a nonlinear function of the strain rate E, but otherwise satisfies the Newtonian fluid assumptions of isotropy and dependence on E only at the same point and at the same moment in time, it can be shown (see, e.g., Leigh29) that the most general form allowed for the constitutive model is... 

The applicability of the CONNFFESSIT (Calculation Of Non-Newtonian Flows Finite Elements and Stochastic Simulation Technique) in its present form is limited to the solution of fluid mechanical problems of incompressible fluids under isothermal conditions. The method is based on a combination of traditional continuum-mechanical schemes for the integration of the mass- and momentum-conservation equations and a simulational approach to the constitutive equation. 

There are several approaches leading to the formulation of these non-linear relationships. Unfortunately it is not possible to summarize them here because the field is very rich in the variety of systems and behaviors. Consequently, such a summary hes beyond the scope of this study. An excellent and detailed review is given in Ref. (Beris Edwards, 1994). However, let us mention that many attempts have been made to bring together continuum mechanics theories and molecular models in order to formulate appropriate constitutive equations for... 

Continuum mechanics is the approach most frequently used to develop constitutive equations and predict material functions. This approach to constitutive modeling has established the framework for all constitutive modeling, including molecular modeling approaches and thermodynamic and stochastic methods. 

We have presented in this chapter an account on the development of diffusion theory. Various modes of flow are identified Knudsen, viscous, continuum diffusion and surface diffusion. Constitutive flux equations are presented for all these flow mechanisms, and they can be readily used in any mass balance equations for the solution of concentration distribution and fluxes. Treatment of systems containing more than two species will be considered in a more systematic approach of Stefan-Maxwell in the next chapter. 

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