Dirichlet condition

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

An arbitrary function /(0) which satisfies the Dirichlet conditions can be expanded as... 

For any function /(x) which satisfies the Dirichlet conditions over the range —00 X oo and for which the integral... 

Dirichlet conditions, electronic states, adiabatic-to-diabatic transformation, two-state system, 304-309... 

The equations written by Dr. Prigogine, by taking Dirichlet conditions, are by themselves equations of a membrane system. 

Here C is defined by the boundary value in the case of the Dirichlet conditions (3.1.3b), (3.1.3d) at one of the end points or by the space averages of the initial concentrations in the case of the Neumann conditions (3.1.3a), (3.1.3c) at both ends. In the spirit of a standard linear stability analysis consider a small perturbation of the equilibrium of the form... 

In the world of numerical analysis, one distinguishes formally between three kinds of boundary conditions [283,528] the Dirichlet, Neumann (derivative) and Robin (mixed) conditions they are also sometimes called [283,350] the first, second and third kind, respectively. In electrochemistry, we normally have to do with derivative boundary conditions, except in the case of the Cottrell experiment, that is, a jump to a potential where the concentration is forced to zero at the electrode (or, formally, to a constant value different from the initial bulk value). This is pure Dirichlet only for a single species simulation because if other species are involved, the flux condition must be applied, and it involves derivatives. Therefore, in what follows below, we briefly treat the single species case, which includes the Cottrell (Dirichlet) condition as well as derivative conditions, and then the two-species case, which always, at least in part, has derivative conditions. In a later section in this chapter, a mathematical formalism is described that includes all possible boundary conditions for a single species and can be useful in some more fundamental investigations. 

It remains to describe how to handle the boundary value Cq. Clearly, for the RL variant, there is no problem because the last concentration value calculated is C(, and Cq can then be computed from all the other C values, now known, according to the boundary condition. This leaves the LR problem. If the boundary concentration is determined as such (the Dirichlet condition, for example the Cottrell experiment), then this is simply applied. It is with derivative (Neumann) boundary conditions that there is a (small) problem. Here, we know an expression for the gradient G at the electrode. For simplicity, assume a two-point gradient approximation at time t + ST... 

The Neumann boundaries (which involve derivatives) are converted into finite difference form and substituted into the finite difference equations for the nodes in the specified region (e.g. above the electrode surface). The Dirichlet conditions (which fix the concentration value) may be substituted directly. 

At boundary nodes where the variable values are given by Dirichlet conditions, no model equations are solved. When the boundary condition involve derivatives as defined by Neumann conditions, the boundary condition must be discretized to provide the required equation. The governing equation is thus solved on internal points only, not on the boundaries. Mixed or Robin conditions can also be used. These conditions consist of linear combinations of the variable value and its gradient at the boundary. A common problem does arise when higher order approximations of the derivatives are used at... 

Let us assume a 3-D domain, Q, in a linear, homogeneous, and isotropic medium. To absorb outgoing waves, Q is surrounded by a set of PMLs that dissipate the waves propagating through their interior, as shown in Figure 4.1. If adequate field attenuation is conducted by these absorbers, zero field values may be presumed at their outer border, thus permitting for simple Dirichlet conditions to be imposed at the ends of the domain without creating spurious reflections. However, other local ABCs may also be utilized [27]. 

There are three options for the boundary condition at x = 0. Dirichlet Condition... 

Dirichlet Condition. The instantaneous surface heat flux is given by... 

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