Continuity equation order parameter

Analytical solutions for the closure problem in particular unit cells made of two concentric circles have been developed by Chang [68,69] and extended by Hadden et al. [145], In order to use the solution of the potential equation in the determination of the effective transport parameters for the species continuity equation, the deviations of the potential in the unit cell, defined by... 

Let us consider a dynamically symmetric binary mixture described by the scalar order parameter field < )(r) that gives the local volume fraction of component A at point r. The order parameter < )(r) should satisfy the local conservation law, which can be written as a continuity equation [143] ... 

The evolution of a system described by an equation related to eqn 2.26 was studied using cell dynamics simulations by Oono and co-workers (Bahiana and Oono 1990 Puri and Oono 1988). In the CDS method the continuous order parameter is discretized on a lattice and at time t is denoted where n labels... 

As we have illustrated above, properties associated with continuous or critical phase transitions behave similarly as one approaches the critical point. We have concentrated on two kinds of properties, the density or order parameter and the heat capacity, and have shown that they vary with temperature as given by equations (13.1), (13.2), and (13.4). That is,... 

An expression for the rate of change of the order parameter is required. Since the concentration is conserved, the following equation of continuity applies,... 

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(Z) are made at discrete time 0 < Zi < t2, , < tn < T. In this case the sequence z(iz) is a discrete sample of the stochastic process z(i). 

The radical reaction schemes for thermal cracking mentioned in Chapter 1 have not been used so far in design. They lead to a set of continuity equations for the reacting components that are mathematically stiff in nature, because of the orders of magnitude of difference between the concentrations of molecular and radical species. Only recently have satisfactory numerical integration routines for sets of stiff differential equations been worked out (see Gear [17]). In addition, the rate parameters of radical reactions are frequently not known with sufficient precision, so far. The radical scheme has therefore been approximated by a set of reactions containing only molecular species. 

In the equation, the first term of the Taylor expansion is zero because the free energy is a minimum at the average value of the order parameter in the parent phase. According to the Landau theory, near a continuous transition, the coefficient J goes to zero as... 

At small c values (with the cubic term of the G expansion), the temperature range of hysteresis is narrow, and the order parameter value Qn is small, in accordance with Equations 29 aind 24. Such trauisitions aire referred to as first-order continuous-like transitions (Figure 1.27). 

Once the model parameters are determined and correlated, the flow model can be used to predict the behavior of the reactor. With a first-order reaction occurring in a steady flow reactor with axial dispersion, the continuity equation for species A is ... 

Cell dynamics simulations are based on the time dependence of an order parameter, (i) (Eq. 1.23), which varies continuously with coordinate r. For example, this can be the concentration of one species in a binary blend. An equation is written for the time evolution of the order parameter, dir/dt, in terms of the gradient of a free energy that controls, for example, the tendency for local diffusional motions. The corresponding differential equation is solved on a lattice, i.e. the order parameter V (r) is discretized on a lattice, taking a value at lattice point i. This method is useful for modelling long time-scale dynamics such as those associated with phase separation processes. 

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