Field equations and boundary conditions

In the case of the quasi stationary field, which is the common case in geophysical exploration, we usually ignore the displacement currents (see Chapter 8), which results in a simplification of system (12.1)  

Solving the corresponding equation for the electric field, we can determine the magnetic field from the second Maxwell s equation  

Maxwell s equations (8.23), or the second order differential equations (12.1) and (12.3), are supplemented with a boundary-value condition, i.e. with the additional equations for the electric or magnetic fields on the boundary dV of the volume 

The accuracy of condition (12.8) is estimated as O (l/jr] ). One can find expressions for asymptotic boundary conditions of higher order of accuracy with respect to distance, r, in Berdichevsky and Zhdanov (1984). 

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As an example of this approach let us consider the constitutive equation arrived at (a) by adopting unchanged the field equations and boundary conditions of the linear theory, and (b) introducing cubic and higher order terms in the polynomial representation... ... 

The dependence of field equations and boundary conditions on retardation times, physical ageing, and stress is retained for the case of two-phase diffusion and remains analogous to (6.27)-(6.29). 

As before, the problem is governed by the creeping flow equations and boundary conditions given earlier [Eqs. (11)—(14)]. The far field boundary condition in this case is... 

Since MPC dynamics yields the hydrodynamic equations on long distance and time scales, it provides a mesoscopic simulation algorithm for investigation of fluid flow that complements other mesoscopic methods. Since it is a particle-based scheme it incorporates fluctuations, which are essential in many applications. For macroscopic fluid flow averaging is required to obtain the deterministic flow fields. In spite of the additional averaging that is required the method has the advantage that it is numerically stable, does not suffer from lattice artifacts in the structure of the Navier-Stokes equations, and boundary conditions are easily implemented. 

Apropos of the analogy noted by Batchelor between vortex velocity and a magnetic field, it should be noted that for the realization of truly steady turbulence a supply of mechanical energy is necessary. The supply of energy comes about either through nonpotential volume forces or through the motion of the surfaces bounding the fluid. With these factors taken into consideration, the set of equations and boundary conditions for a vortex does... 

Here the situation is very similar to that encountered in connection with the need for continuum (constitutive) models for the molecular transport processes in that a derivation of appropriate boundary conditions from the more fundamental, molecular description has not been accomphshed to date. In both cases, the knowledge that we have of constitutive models and boundary conditions that are appropriate for the continuum-level description is largely empirical in nature. In effect, we make an educated guess for both constitutive equations and boundary conditions and then normally judge the success of our choices by the resulting comparison between predicted and experimentally measured continuum velocity or temperature fields. Models derived from molecular theories, with the exception of kinetic theory for gases, are generally not available for comparison with the empirically proposed models. We discuss some of these matters in more detail later in this chapter, where specific choices will be proposed for both the constitutive equations and boundary conditions. 

The time-dependent fiinction Hit ) is determined by the rate of increase or decrease in the bubble volume. The governing equations and boundary conditions that remain to be satisfied are (1) the radial component of the Navier Stokes equation (2) the kinematic condition, in the form of Eq. (2 129), at the bubble surface and (3) the normal-stress balance, (2 135), at the bubble surface with = 0. Generally, for a gas bubble, the zero-shear-stress condition also must be satisfied at the bubble surface, but xrti = 0, for a purely radial velocity field of the form (4-193), and this condition thus provides no usefirl information for the present problem. 

The governing DE and boundary conditions for the temperature field are again (9 1) and (9-2), and the dimensionless equation and boundary conditions are (9-7) and (9-8), with the same definition for the dimensionless temperature (9-3) and the sphere radius as a characteristic length scale. Only the form of the velocity field, u, and the choice of characteristic velocity are different for this case of a linear shear flow far from the sphere. The appropriate choice for the characteristic velocity is... 

Precise solution of the multidimensional problem of heat conductivity by analytical methods is very complicated and laborious. Therefore, an approximate finite difference method was developed based on the differential heat conductivity equation and boundary conditions. In this method, the temperature of the vulcanized section of the covering fragment was subdivided into elementary volumes of unit thickness because it is necessary to define the temperature field of the vulcanized. 

When two different fields have similar governing equations and boundary conditions they are called analogous. A majority of momentum and thermal boundary layers, however, are not analogous. For example, a pressure gradient in the momentum boundary layer or an energy generation in the thermal boundary layer, or an incompatibility between momentum and thermal boundary conditions, eliminates this analogy. 

In Section 8.4 a superposition of f sti and is carried out, leading to Eq. (8.4.18), which defines the particle velocity with respect to the collector. Explain where these forces come from and describe the superposition of the velocity fields associated with each. Only the governing differential equations and boundary conditions along with simple sketches to indicate the solution domain are required. 

Equations and boundary conditions for the electric field potential in the area around the particles have been formulated in Section 12.1, and look like ... 

HENISCH One of the systems with asymmetric contacts discussed involved two carrier tra-n sport. In the ordinary way, this is governed by two current equations in terms of field and diffusion, two continuity relationships (incl. recombination), Poisson s equation, and boundary conditions for field, potential and carrier concentrations. These transport equations cannot be explicitly solved without simplifying assumptions. It would be interesting to see what assumptions can reasonably be made to render the problem algebraically tractable. The equations given appeared to envisage diffusion without space charge. How do (a) the local field. 

With these governing equations and boundary conditions in place, and if the input pressure distribution that drives the flow field is known, it is possible to develop a formal solution for the axial velocity. For an oscillatory flow, the input pressure would normally be expected to be of a sinusoidal form P(x, r, t) = const e . Following Reference 18, the method of characteristics shows the pressure distribution throughout the tube to be P(x, r, t) = A(x, where c is the wave speed in the... 

Step 4 - it is initially assumed that the flow field in the entire domain is incompressible and using the initial and boundary conditions the corresponding flow equations are solved to obtain the velocity and pressure distributions. Values of the material parameters at different regions of the domain are found via Equation (3.70) using the pseudo-density method described in Chapter 3, Section 5.1. 

The foregoing equations are coupled and are generally nonlinear no general solution exists. However, these equations serve as a starting point for most of the analysis that is relevant to electrophoretic transport in solutions and gels. Of course, the specific geometry and boundary conditions must be specified in order to solve a given problem. Boundary conditions for the electric field include specification of either (1) constant potential, (2) constant current, or (3) constant power. 

At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by finite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume V is defined by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss s theorem,... 

Equations (13.7)-(13.13) are used to evaluate the ignition processes of energetic materials with appropriate initial and boundary conditions. In general, the conditions in the thermal field for ignition are given by... 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

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