Closure, constitutive equations

Nozad et al. [8] discuss a set of closure constitutive equations (or transformations) given by... 

Closure constitutive equations are developed similar to those used when the existence of the local thermal equilibrium was assumed. This requires relating the disturbances in the temperature fields to the gradients of the volume-averaged temperatures and to the difference between the phase volume-averaged temperatures. 

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. 

It is probably an understatement to say that the quest in integral equation studies is the search for accurate closures. Perhaps closures constitute the most... 

The unsteady governing equations that apply in laminar flow also apply in turbulent flow but essentially cannot be solved in their full form at present In turbulent flow, therefore, the variables are conventionally split into a time averaged value plus a fluctuating component and an attempt is then made to express the governing equations in terms of the time averaged values alone. However, as a result of the nonlinear terms in the governing equations, the resultant equations contain extra variables which depend on the nature of the turbulence, i.e., there are, effectively, more variables than equations. There is, thus, a closure problem and to bring about closure, extra equations which constitute a turbulence model" must be introduced. 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

MTE methods require closure assumptions for the last three terms in Eq. (23), and there has been heated debate on this point. There seems to be universal agreement that the dissipation term should be modeled by the constitutive equation... 

To save space the governing time averaged equations for each of the primary variables are not listed here as their mathematical form coincides with the volume averaged model formulation given in sect 3.4.1. Nevertheless, it is important to note that the physical interpretations of the mean quantities and the temporal covariance terms differ from their spatial counterparts. Furthermore, the conventional constitutive equations for the unknown terms are discussed in chap 5, and the same modeling closures are normally adopted for any model formulation even though their physical interpretation differ. 

The macroscopic multi-phase models resulting from the local averaging procedures must be supplemented with state equations, constitutive equations, boundary and initial conditions. The constitutive equations specify how the phases interact with themselves and with each other. The closure laws or constitutive laws can thus be divided into three types [16] Topological, constitutive and transfer laws, where the first type describes the spatial distribution of phase-specific quantities, the second type describes physical properties of the phases and the third type describes different interactions between the phases. 

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. 

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 ... 

As shown above, insights from thermodynamics principles have lead to constitutive equations (100) and (101). These equations thus allow us to develop a set of governing equations which is applicable to the cavity closure problem. These equations are then solved with respect to the initial and boundary conditions for a spherical cavity to obtain a set of analytic solutions, of which a detailed discussion is given in Wong et al. (2008). 

The constitutive equations (115), (116), (118) are then used to developed governing equations for the closure of a long cylindrical tunnel in poroviscoelastic massif. Laplace transform solutions have been develofjed and discussed in detail in Dufour et al. (2009) to which interested readers may refer. 

Incremental model budding Our intelligent model editor (Lakner et al. 1999) obtains the minimal model by asking for the model elements (balance volumes, extensive quantities and transport mechanisms) and automatically constructs the balance equations in their extensive forms. The intensive-extensive relationships and the reaction rate equations must be defined as additional constitutive equations. Besides these relations, the closure constraint among total mass and component masses, namely is also produced automatically. This equation describes a relationship among the extensive (and naturally among the intensive) quantities and serves to make the resultant model minimal. 

To understand the processing and dispensing behavior of adhesives we often need to construct flow models of the process. While all models are approximations, they still have to satisfy continuity of mass and the balance of momentum and thermal energy. The constitutive equation, plus the appropriate initial and/or boundary conditions of the problem at hand, provide closure to these balance laws. While any realistic solution to a particular processing problem will generally involve numerical computations, several generic problems of interest may be amenable to analytical development. For instance, when a pressure-sensitive adhesive is pressed down unto a surface, we have an example of squeeze flow. Similarly, if a paste is spread onto a surface via a knife, we have an example of wedge flow. These two flows will be discussed for the PLF. 

The liquid is an example of a viscous fluid while the rubber band is an example of an elastic solid. There are materials with mechanical responses that span the entire range between these extremes. Mechanical constitutive equations are relations between the dynamics (stresses and their time rates of change) and kinematics (deformations and their time rates of change) of materials. As such they provide closure to the balance equations (see Section 3), thereby allowing the solution to a specific mechanical problem involving a specific material. 

It is important to note that Equation (14.16), Equation (14.17), and Equation (14.18) do not constitute a closed set of equations the evolution of the size distribution, Equation (14.10), still has to be calculated. For efficient simulations of flow-enhanced nucleation in polymer processing, closure of Equation (14.16), Equation (14.17), and Equation (14.18) is necessary. This is possible by introducing some assumptions. The mathematical structure of many existing flow-enhanced nucleation models, which do not contain all the details considered here, can be reproduced in this way, as shown in Sections 14.4.1.2-14.4.1.4. The influence of flow, for example on the rate of creation of precursors, is subsequently specifled for a number of these models. A common assumption is that all FIPs are active, so that their size distribution need not be considered. Quiescent precursors are then treated as a separate species because it is known from experiments that they do have a distribution of sizes or, equivalently, of activation temperatures. This is discussed next. 

To illustrate these ideas let us summarize the general system of equations that constitute the SCGLE theory. In principle, these are the exact results for A (f), F k, f), and t) in Equations 1.20,1.23, and 1.24, complemented with the simplified Vineyard approximation in Equation 1.37 and the simplified interpolating closure in Equation 1.38. This set of equations define the SCGLE theory of colloid dynamics. Its full solution also yields the value of the long-time self-diffusion coefficient which is the order parameter appropriate to detect the glass transition from the fluid side. This is, however, not the only method to detect dynamic arrest transition, as we now explain. 

Establishing the necessary constitutive and closure equations (the former relate fluid stresses with velocity gradients the latter relate unknown Navier-Stokes-equation coiTclations witli known quantities). 

Carbanions derived from side chain tertiary amides have also been cyclized to provide isoquinolones and isoindoles (equation 36).125 126 While benzyne intermediacy in the formation of the former is likely, the latter seems to arise through a SrnI reaction pathway. Synthesis of indole from the meta bromo compound (87), on the other hand, clearly involves an aryne cyclization. 27 A more versatile route to indoles is based on intramolecular addition of aminyl anions to arynes (equation 38).128 A somewhat similar dihydroindole preparation constitutes the first step in a synthesis of lycoranes (equation 39).129 The synthesis of (88) also falls in the same category of reactions, but it is noteworthy because only a few examples of ring closure of heteroarynes are mentioned in literature.27 28... 

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. 

Assuming that the effects of turbulence can be reasonably modeled such that the correlations of turbulent fluctuations can be linked to the averaged quantities, the dependent variables for a -phase flow are k, Uk, pk, and Tk, where k varies from 1 to n. Thus, the total number of dependent variables is 6n. Although Eqs. (5.168) through (5.171) constitute 5n + 1 equations, additional n - 1 equations (pk = Pk( k, 7k)) are needed to reach a closure. The development of these n — 1 equations requires the mechanistic understanding of the pseudofluid nature of each phase. 

The four equations governing two-dimensional turbulent flow are, therefore, Eqs. (2.95), (2.106), and (2.116). These contain, beside the four mean flow variables u, v, p, and T, additional terms which depend on the turbulence held. In order to solve this set of equations, therefore, information concerning these turbulence terms must be available and the difficulties associated with the solution of turbulent flow problems arise basically from the difficulty of analytically predicting the values of these terms. The set of equations effectively contains more variables than the number of equations. This is termed the turbulence closure problem . In order to bring about closure of this set of equations, extra equations must be generated, these extra equations constituting a turbulence model . 

---