FIRST BOUNDARY CONDITIONS

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-... 

Thus, a fourth-order differential equation such as Equation (D.11) has four boundary conditions which are the second and third of the conditions in Equation (D.8) at each end of the beam. The first boundary condition in Equation (D.8) applies to the axial force equilibrium equation, Equation (D.2), or its equivalent in terms of displacement (u). 

By comparing the first boundary condition (67) with (69) for i = 1 we find that... 

In such a setting it is required to find the values of the parameter A such that these homogeneous equations have nontrivial solutions y(x) 0. In contrast to the first boundary-value problem, here the parameter A enters not only the governing eqnation, but also the boundary conditions. The introduction of new sensible notations... 

We call the nodes, at which equation (1) is valid under conditions (2), inner nodes of the grid uj is the set of all inner nodes and ui = ui + y is the set of all grid nodes. The first boundary-value problem completely posed by conditions (l)-(3) plays a special role in the theory of equations (1). For instance, in the case of boundary conditions of the second or third kinds there are no boundary nodes for elliptic equations, that is, w = w. 

At time t = 0 an electric current of constant strength begins to flow in the system. At this time the uniform initial concentration distribution is still not disturbed, and everywhere in the solution, even close to the electrode surface, the concentration is the same as the bulk concentration Cyj. Hence, the first boundary condition (for any value of jc) is given by... 

Equation (11.2) remains valid as the first boundary condition in this case. The surface concentrations, c, of the reactants will remain constant, in accordance with the Nemst equation, when the electrode potential is held constant during current flow (and activation polarization is absent). Hence, the second boundary condition can be formulated as... 

Consider the case of transient diffusion at constant potential (constant surface concentration). The first boundary condition, (11.2), is preserved and the second boundary condition can be written (for any time t) as... 

The second boundary condition assures total finite existence probability at any time the first boundary condition implies that the recombination is fully diffusion-controlled, which has been found to be true in various liquid hydrocarbons (Allen and Holroyd, 1974). [The inner boundary condition can be suitably modified for partially diffusion-controlled reactions, which, however, does not seem to have been done.]... 

Here we are interested in escape out of the domain L specified by a single cycle of the potential that is out of a domain of length n that is the domain of the well. Because the bistable potential of Eq. (5.42) has a maximum at x = n/2 and minima at x = 0, x = 7t, it will be convenient to take our domain as the interval —7t/2 < x < n/2. Thus we will impose absorbing boundaries at x = —n/2, x = n/2. Next we shall impose a second condition that all particles are initially located at the bottom of the potential well so that x0 = 0. The first boundary condition (absorbing barriers at —n/2, n/2) implies that only odd terms in p in the Fourier series will contribute to Y (x). While the second ensures that only the cosine terms in the series will contribute because there is a null set of initial values for the sine terms. Hence... 

The restriction clearly implies the boundary conditions on the wave function, tp 0) = ip L) = 0. Since cosO = 1 0, the first boundary condition requires that 6 = 0 and hence that... 

For some biological systems, the species that eventually crosses the cell membrane has travelled through different media, each one with its own mass transfer characteristics. As an example, we deal with the case where the two media are the bulk solution and the cell wall (with the separation surface parallel to the cell membrane) with diffusion as the only relevant mass transfer phenomenon in each medium. Apart from having different parameters in the differential equations in each medium (due to the unequal diffusion coefficients), we need to impose two new boundary conditions at the separating plane which we denote as a. The first boundary condition follows from the continuity of the material flux ... 

Thus N dynamic equations are obtained for each component at each position, within each segment The equations for the first and last segment must be written according to the boundary conditions. The boundary conditions for this case correspond to the following the bulk tank concentration is S0 at the external surface of the biofilm where Z=0 a zero flux at the biofilm on the wall means that dS/dZ=0 at Z = L. 

The field amplitudes are written as scalars because reflection and transmission at normal incidence are independent of polarization. At the first boundary (z = 0), the amplitudes satisfy the usual boundary conditions ... 

The first boundary condition to be satisfied by solutions of the differential equations (7.207a) arises from the fact that the only source for the material D is the... 

Since xs must be bounded as r approaches zero according to the first boundary condition, we must choose Ct = 0. The second boundary condition requires that C2 = l/sinh30, leaving... 

Ci and C2 are constants, which are defined by the boundary conditions. The boundary conditions require that, at the surface, the potential is equal to the surface potential, ip x = 0) = rfo, and that, for large distances from the surface, the potential should disappear ip(x — oo) = 0. The second boundary condition guarantees that, for very large distances, the potential becomes zero and does not grow infinitely. It directly leads to C2 = 0. From the first boundary condition we get C = Hence, the potential is given by... 

A black box approach as the endpoint of the analysis raw materials and products, overall process-relevant data (e.g., yields, reactor conditions), and boundary conditions are given, leading to a first evaluation of the overall process economics are based on raw materials and products. 

The first boundary condition at the surface is provided by integrating the Poisson Eqs. (la) and (lb) over the volume of a flat box, which includes the surface, with the large sides parallel to the surface and a vanishingly thin width. After using the Gauss theorem, one obtains ... 

Integrating Eq. (4.27) once and using the first boundary condition gives ... 

Integrating Eq. (7.43) from the center line where y = R to some point distance y from the wall and using the first boundary condition listed in Eq. (7.44) then gives ... 

The boundary conditions are then applied to determine the form of the functions X and T. The basic assumption as given by Eq. (3-4) can be justified only if it is possible to find a solution of this form which satisfies the boundary conditions. First consider the boundary conditions with a sine-wave temperature dis-... 

Applying the first boundary condition, we find that... 

From the first boundary condition describing the cooling at the surface we obtain the constant /, and temperature becomes... 

The first boundary condition is the expression of an obvious point, namely, that very far from the boundary at which the diffusion source or sink is set up, the concentration of the diffusing species is unperturbed and remains the same as in the initial condition... 

The first boundary condition for the convective-diffusion equation shows that, at the interface, the property flux is written using the transfer property coefficient ... 

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