Scalar variables

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

Let jP be a vector whose components are functions of a scalar variable (e.g. time-dependent position vector of a point F in a three-dimensional domain)... 

Suppose that the vector field u(f) is a continuous function of the scalar variable t. As t varies, so does u and if u denotes the position vector of a point P, then P moves along a continuous curve in space as t varies. For most of this book we will identify the variable t as time and so we will be interested in the trajectory of particles along curves in space. 

For simplicity, we will use the fictitious chemical species as the subscripts whenever there is no risk of confusion. As shown above, this reaction can be rewritten in terms of two reacting scalars and thus two components for S. The closure problem, however, cannot be eliminated by any linear transformation of the scalar variables. 

Local mixing is best defined in terms of stochastic models. However, this condition is meant to mle out models based on jump processes where the scalar variables jump large distances in composition space for arbitrarily small df. It also rales out interactions between points in composition space and global statistics such as the mean. 

Let u(t) be an n-vector with components (u1,u2,..u ) depending on a single scalar variable t. The derivative of the vector u with respect to the scalar t is the n-vector defined as... 

A particular experiment provides observations of n independent variables Xj (j = 1,..., n) and of a single dependent scalar variable Y which we suspect to be related through a linear relationship such as... 

Starting at line 900 you find the user subroutine. In this routine the mole numbers occupy the array elements NW(1), NW(2),. .., NW(5) and the scalar variable NW stores the total mole number. At the current value X(l) and X(2) of the reaction extents we first calculate tine mole numbers. If any of them is negative or zero, the error flag ER is set to a nonzero value. If the mole numbers are feasible, the values computed according to (2.31) will occupy the array elements G(l) and G(2). The initial estimates are X(1) = 1 and X(2) = 0.1, the first corrections are D(l) = D(2) = 0.01. The following output shows some of the iterations. 

Replacing the Dirac c>-function of vectorial argument by the corresponding value of scalar variable, for H3 in (1.20) we find... 

Remark 1 Note that since y Y = 0 - 1, the master problem is a 0-1 programming problem with one scalar variable pB. If they variables participate linearly, then it is a 0-1 linear problem which can be solved with standard branch and bound algorithms. In such a case, we can introduce integer cuts of the form ... 

By introducing a scalar variable Hook, the master problem can be formulated as... 

The order parameter f(r) may be a scalar or vector field. An ordinary scalar variable may play the role of an order parameter in some systems. Landau free energy represents in this case and ordinary function of . 

It is possible to add and subtract matrices using + and —, but remember to ensure that the two (or more) mattices have the same dimensions as has the destination. It is also possible to mix matrix and scalar operations, so that the syntax =2 MMULT(X,Y) is acceptable. Furthermore, it is possible to add (or subtract) matrices consisting of a constant number, for example =Y +2 would add 2 to each element of Y. Other conventions for mixing matrix and scalar variables and operations can be determined by practice, although in most cases the result is what we would logically expect. 

Leave no space between simple variables being multiplied xy. Do not use a centered dot ( ) or the times sign (x) with single-letter scalar variables. 

The time-averaged values of scalar variables, 0. (species mole fractions and temperature) is calculated as ... 

It follows from Eq. (A 124) that the vector derivative operator changes the grade of the object it operates on by 1. For example, the vector derivative of the scalar X(xa) is a vector (because a X = 0 for any scalar X, so aX = a A X), and the vector derivative of the vector f(xa) is a scalar plus a bivector. The differentiation with respect to the vector variable xa greatly resembles the differentiation with respect to some scalar variable xa. For example, the vector differentiation is distributive,... 

The boundary conditions of (9) are the same as of the usual classical action in (2) (fixed end points and total time). However, separation of variables is now self-evident. The second term on the right hand side of the equation Et) is independent of the coordinates (the energy is a constant). Similarly the first term is independent of time. In fact, it is not necessary to write dQ = Qdr. Any parameterization of the trajectory with respect to a scalar variable s which is monotonic between the points Q (0) and Q t) will do dQ = dQ/ds) ds). It means that variation of the integral part of the right hand side of (9) with respect to the coordinates can be done with no reference to time. We therefore consider a variation with the abbreviated action [5]... 

Iso-surfaces can be constructed by connecting all points in the flow domain that have exactly the same value of a given variable. These surfaces can be colored by the contours of any other variable. Iso-surfaces are useful to visualize certain flow-field features, such as the central vortex in the cyclone separator shown in Fig. 4. Contour plots can be used to show the local values of scalar variables. A cross section of the flow domain is created and colored by the local value of the variable of interest, e.g., temperature. A color scale is used indicating how each color corresponds to a certain value of the variable. Such plots can be used to quickly see the variation of important variables throughout the domain. An example of a contour plot is found in Fig. 5, which is described in the example in the following section. 

An impermeable solid wall is specified assuming that the no-slip velocity condition at the boundary is valid. In general, the scalar variable boundaries are specified by Neumann, Dirichlet- or mixed conditions. 

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