Constitutive hydrodynamic equations

The constitutive hydrodynamic equations for uniaxial nematic calamitic and nematic discotic liquid crystals are identical. In comparison to nematic phases the hydro-dynamic theory of smectic phases and its experimental verification is by far less elaborated. Martin et al. [17] have developed a hydrodynamic theory (MPP theory) covering all smectic phases but only for small deformations of the director and the smectic layers, respectively. The theories of Schiller [18] and Leslie et al. [19, 20] for SmC-phases are direct continuations of the theory of Leslie and Ericksen for nematic phases. The Leslie theory is still valid in the case of deformations of the smectic layers and the director alignment whereas the theory of Schiller assumes undeformed layers. The discussion of smectic phases will be restricted to some flow phenomena observed in SmA, SmC, and SmC phases. 

Although the constitutive hydrodynamic equations for nematic and polymeric liquid... 

On the other hand, in a pure liquid crystal system, liquid crystalline order, such as orientation order in nematic or layer order in smectic, is created under phase transition point, and the symmetry of the system is reduced. At the same time, new hydrodynamic fluctuation motions appear to be associated with new degrees of freedom. The modes of hydrodynamic fluctuations are characterized by a dispersion relation that can be obtained by solving the constitutive hydrodynamic equations of the system, giving the angular frequency wave number q of the fluctuations. It can be said that in a uniform alignment of the pure liquid crystal, the system universally satisfies the dispersion relation from the micrometer scale up to the length of the sample chamber, which means that the material keeps spatial homogeneity for the dynamics in pure system. 

Accdg to Dunkle (Ref 28), Brode (Ref 14), in order to solve detonation problems without recourse to empirical values derived from explosion measurements, integrated the hydrodynamical equations of motion (which constitute a set of nonlinear partial... 

The model yields a set of hydrodynamic equations for the solid phase. For equation closure, additional constitutive relations, which can be obtained by using the kinematic argument of the collision and by assuming the Maxwellian velocity distribution of the solids, are needed. Two examples are given to illustrate the applications of this model in this chapter. 

Equations (1-5) are a set of 18 coupled, nonlinear, partial differential equations that, along with material equations of state and constitutive relations for mass, energy, and momentum exchange, form the hydrodynamic equation set in IFCI. 

Following the scheme of MNET, a Fokker-Planck equation was obtained from which a coarse-grained description in terms of the hydrodynamic equations was derived in turn. Molecular deformation and diffusion effects become coupled and a class of non-linear constitutive relations for the kinetic and elastic parts of the stress tensor are obtained. The expression for the stress tensor can be written in terms of dimensionless quantities like... 

Transport coefficients occur in all forms of continuum, hydrodynamic equations concerned with mass, momentum and energy conservation once constitutive equations for the fluids of interest are introduced. Such equations are frequently encountered in trying to model mathematically technological processes with a view to their refinement. Attempts to model such processes mathematically (usually numerically) are frequently limited by a lack of knowledge of the physical properties of the materials involved including the transport coefficients of the fluids. 

In the improvement of all of these processes a complete understanding of heat transfer by both conduction and convection is essential. Since the governing hydrodynamic equations are well known, the accuracy of models of such processes depends sensitively on, and is currently limited by, our knowledge of the constitutive equations of the molten materials and, in particular, upon the transport coefficients which ento- than. Significant advances in the quality and uniformity of a number of matoials might be attainable were accurate data for the thermal conductivity and viscosity of molten materials at high temperature available. 

All these approaches are essentially alternative ways of solving the Navier-Stokes equation and its generalizations. This is because the hydrodynamic equations are expressions for the local conservation laws of mass, momentum, and energy, complemented by constitutive relations which reflect some aspects of the microscopic details. Frisch et al. [10] demonstrated that discrete algorithms can be constructed which recover the Navier-Stokes equation in the continuum limit as long as these conservation laws are obeyed and space is discretized in a sufficiently symmetric manner. 

The void fraction should be the total void fraction including the pore volume. We now distinguish Stotai from the superficial void fraction used in the Ergun equation and in the packed-bed correlations of Chapter 9. The pore volume is accessible to gas molecules and can constitute a substantial fraction of the gas-phase volume. It is included in reaction rate calculations through the use of the total void fraction. The superficial void fraction ignores the pore volume. It is the appropriate parameter for the hydrodynamic calculations because fluid velocities go to zero at the external surface of the catalyst particles. The pore volume is accessible by diffusion, not bulk flow. 

Equation (14.55) is known as the Clausius-Duhem or the fundamental inequality for a single-component system. The selection of the constitutive independent variables depends on the type of system considered. For example, the density, velocity, and temperature fields in hydrodynamics are customarily chosen. A process is then described by solving the balance equations with a consideration of constitutive relations and the Clausius-Duhem inequality. 

When the solute is spherical, or close to be so, its radius is easily obtained otherwise, estimations can be made on the basis of the geometry and arrangement of the constituting atoms or ions. For solutes having a complex stucture (e.g., micelles), a distinction should be made between the hydrodynamic radius (which appears in the Stokes-Einstein equation of the diffusion coefficient) and the reaction radius [98]. For Ps, RPs should represent the bubble radius. However, as shown in Table 4.4, the experimental data are systematically very well recovered by using the free Ps radius, RPs = 0.053 nm using the bubble radius results in a calculated value of kD (noted kDb) that is too small by a factor of 2 or 3. Table 4.4 does not include such cases where k kD, as these do not correspond to purely diffusion-controlled reactions. 

Closure of such differential equations requires the definitions of both constitutive relations for hydrodynamical functions and also kinetic relations for the chemistry. These functions are specified by recourse both to theoretical considerations and to rheological measurements of fluidization. We introduce the ideal gas approximation to specify the gas phase pressure and a caloric equation-of-state to relate the gas phase internal energy to both the temperature and the gas phase composition. It is assumed that the gas and solid phases are in local thermodynamic equilibrium so that they have the same local temperature. 

For each continuous phase k present in a multiphase system consisting of N phases, in principle the set of conservation equations formulated in the previous section can be applied. If one or more of the N phases consists of solid particles, the Newtonian conservation laws for linear and angular momentum should be used instead. The resulting formulation of a multiphase system will be termed the local instant formulation. Through the specification of the proper initial and boundary conditions and appropriate constitutive laws for the viscous stress tensor, the hydrodynamics of a multiphase system can in principle be obtained from the solution of the governing equations. 

In addition to these impediments to rheological measurements, some complex fluids exhibit wall slip, yield, or a material instability, so that the actual fluid deformation fails to comply with the intended one. A material instability is distinguished from a hydrodynamic instability in that the former can in principle be predicted from the constitutive relationship for the material alone, while prediction of a flow instability requires a mathematical analysis that involves not only the constitutive equation, but also the equations of motion (i.e., momentum and mass conservation). 

The tendency of LCs to resist and recover from distortion to their orientation field bears clear analogy to the tendency of elastic solids to resist and recover from distortion of their shape (strain). Based on this idea, Oseen, Zocher, and Frank established a linear theory for the distortional elasticity of LCs. Ericksen incorporated this into hydrostatic and hydrodynamic theories for nematics, which were further augmented by Leslie with constitutive equations. The Leslie-Ericksen theory has been the most widely used LC flow theory to date. 

Ishii M, Mishima K (1984) Two-Fluid Model and hydrodynamic Constitutive Equations. Nuclear Engineering and Design 82 107-126... 

These examples, and others like them, allow us to discern three distinct levels of model building, though admittedly the boundary between them is blurred. In particular, the level of such modeling might be divided into (i) fundamental laws, (ii) effective theories and (iii) constitutive models. Our use of the term fundamental laws is meant to include foundational notions such as Maxwell s equations and the laws of thermodynamics, laws thought to have validity independent of which system they are applied to. As will be seen in coming paragraphs, the notion of an effective theory is more subtle, but is exemplified by ideas like elasticity theory and hydrodynamics. We have reserved constitutive model as a term to refer to material-dependent models which capture some important features of observed material response. 

During our discussion of linear momentum balance in chap. 2, we noted that the fundamental governing equations of continuum mechanics as embodied in eqn (2.32) are indifferent to the particular material system in question. This claim is perhaps most evident in that eqn (2.32) applies just as well to fluids as it does to solids. From the standpoint of the hydrodynamics of ordinary fluids (i.e. Newtonian fluids) we note that it is at the constitutive level that the distinction... 

In fact, Equation 5.281 describes an interface as a two-dimensional Newtonian fluid. On the other hand, a number of non-Newtonian interfacial rheological models have been described in the literature. Tambe and Sharma modeled the hydrodynamics of thin liquid films bounded by viscoelastic interfaces, which obey a generalized Maxwell model for the interfacial stress tensor. These authors also presented a constitutive equation to describe the rheological properties of fluid interfaces containing colloidal particles. A new constitutive equation for the total stress was proposed by Horozov et al. ° and Danov et al. who applied a local approach to the interfacial dilatation of adsorption layers. 

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