Variable field

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

Equation (2.45) represents the weighted residual statement of the original differential equation. Theoretically, this equation provides a system of m simultaneous linear equations, with coefficients Q , i = 1,... m, as unknowns, that can be solved to obtain the unknown coefficients in Equation (2.41). Therefore, the required approximation (i.e. the discrete solution) of the field variable becomes detemfined. 

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). 

Descriptions given in Section 4 of this chapter about the imposition of boundary conditions are mainly in the context of finite element models that use elements. In models that use Hermite elements derivatives of field variable should also be included in the set of required boundai conditions. In these problems it is necessary to ensure tluit appropriate normality and tangen-tiality conditions along the boundaries of the domain are satisfied (Petera and Pittman, 1994). 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

The field variables are recalculated replacing (5.82b) with = 0 along CD. 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

Note that s/° is, therefore, not an electromagnetic field variable it is determined by the matter field operators. 

The requirement of Postulate 3 that the equations of motion be form-invariant (i.e., that fix ) and A u(x ) satisfy the same equation of motion with respect to a as did fx) and Au x) with respect to x demands that the field variables transform under such transformations according to a finite dimensional representation of the Lorentz group. In other words it demands that transform like a spinor... 

The constant of Integration /i Is a field variable similar to the chemical potential (/i = -3.6227 In our calculations). 

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

The last, and less extensively studied field variable driving percolation effects is chemical potential. Salinity was examined in the seminal NMR self-diffusion paper of Clarkson et al. [12] as a component in brine, toluene, and SDS (sodium dodecylsulfate) microemulsions. Decreasing levels of salinity were found to be sufficient to drive the microemulsion microstructure from water-in-oil to irregular bicontinuous to oil-in-water. This paper was... 

A somewhat different water, decane, and AOT microemulsion system has been studied by Feldman and coworkers [25] where temperature was used as the field variable in driving microstructural transitions. This system had a composition (volume percent) of 21.30% water, 61.15% decane, and 17.55% AOT. Counterions (sodium ions) were assigned as the dominant charge transport carriers below and above the percolation threshold in electrical... 

Several unifying conclusions may be based upon the order parameter results illustrated here for microstructural transitions driven by three different field variables, (1) disperse phase volume fraction, (2) temperature, and (3) chemical potential. It appears that the onset of percolating cluster formation may be experimentally and quantitatively distinguished from the onset of irregular bicontinuous structure formation. It also appears that... 

Cloughley, J. B., Factors influencing the caffeine content of black tea Part 1 — The effect of field variables. Food Chem., 9 69, 1982. 

An NMR system using a field-variable, cryogen-free superconducting magnet... 

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