THIRD BOUNDARY CONDITIONS

Aitemativeiy, the beam end couid have compiete rotational restraint and no transverse displacement, i.e., clamped. However, a third boundary condition exists in Rgure D-3 just as in Figure D-2. That is, an axial condition on displacement or force must exist in addition to the conditions usually thought of as comprising a clamped-end condition. Note that the block-like device at the end of the beam prevents rotation and transverse deflection. A similar device will be used later for plates. Whether all of the three boundary conditions can actually be enforced depends on the order of the differential equation set when (necessarily approximate) force-strain and moment-curvature relations are substituted in Equations (D.2), (D.4), and (D.7). 

The mass transfer process is again governed by equation 10.66, but the third boundary condition is applied at y = L, the film thickness, and not at y = oo. As before, the Laplace... 

Locally one-dimensional schemes find a wide range of applications in solving the third boundary-value problem. If, for example, G is a rectangle of sides /j and or a step-shaped domain, then equations (21) should be written not only at the inner nodes of the grid, but also on the appropriate boundaries. When the boundary condition du/dx = cr u- -v[ is imposed on the side = 0 of the rectangle 0 < < / , a = 1,2, the main idea... 

Thus, boundary conditions (1.89) and (1.90) define the potential within the volume V up to some constant. Correspondingly, the third boundary value problem can be formulated as ... 

The concentration of particles in contact with the collector is taken to be zero because these particles are no longer part of the disperse phase. At large distances from the collector the particle concentration must tend to that of the approaching fluid. The third boundary condition arises from the symmetry about the forward stagnation path 6 = 0). 

Since the second term of the right-hand side does not satisfy the second boundary condition (jt— oo, y- 0), then B (s) = 0. The third boundary condition in Laplace space is... 

The third boundary condition considers that planes ys = s-y, and ys = sy are impermeable to the transferred property ... 

By attention to Figures 4(b), 6(b) and 8(b), it is seen that the exit mass flux from the leakage location have an intense fluctuant behavior. The boundary conditions in upstream and downstream also affect the time average of the exit mass flux and the amplitude and frequency of the fluctuations. The average at the first type of boundary condition is about 118 kg/s, at the second state it is about 110 kg/s after 70 seconds and at the third boundary state this average reaches 350 kg/s. 

Flux defined in terms of a mass transfer coefficient, k, with an external, known concentration, cq, or a heat transfer coefficient, h, with an external, known temperature, To (called a Robin condition or boundary condition of the third kind) ... 

The condition (5-224) provides a third boundary condition through matching with the core solution for the radial velocity. Although (5-224)-(5-226) is a well-posed problem, we shall not solve it here. The solution for F0 was numerical, and thus the present problem must also be solved numerically. The most efficient approach is to solve the problems for F0 and F together as this avoids storing the solutions for F0 and/or interpolating to evaluate the coefficients in (5-225). 

Finally, if we substitute (12 288) into (12-294), and express the resulting equation in dimensionless form, we obtain the third boundary condition at z = 1 ... 

When reaction does take place and a > 0, the third boundary condition is nonlinear and to our knowledge no analytical solution to Eqs. (83 through (113 exists. To solve the equations, we used a finite difference Galerkin scheme (with the relaxation parameter set equal to 2/33 (29, 303. Numerical solutions were obtained for c as a function of x and y for different cases of Shyf, a and 0. The mixing cup concentration can be obtained from the solution of c and is given at any x by... 

Coal particle gas diffusion mathematical model was set up basing on the third boundary condition and taking into account gas mass transmission characteristics on borderline. The applying scope includes gas diffusion model under the first boundary condition, thus it is not only more scientific and reasonable, but also is provided with application more widely. 

The equation (5) can be solved given the first, the second or the third boundary conditions or any combination of them. 

If a particle moves to the right of jc = L we simply shift its position to x—L likewise any particle exiting to the left of jc = 0 has its position shifted to. c-l-L (see Fig. 1.15). This type of boundary condition is a common choice and considered more relevant in simulations of realistic systems than confining boundary conditions. Periodic boundary conditions allow us to preserve Newton s third law, the translational symmetry, and thus the conservation of momentum. 

At a planar electrode, the equation to be solved under conditions of linear semiinfinite diffusion is the same as for a potential step. The difference is the third boundary condition, which instead of defining the diffusion-limited current expresses the concentration gradient resulting from the applied current at the electrode surface. 

The third boundary condition, which is valid for short contact times, assumes that none of the diffusing solute has yet reached the wall so that the wall is effectively at infinity. 

This is a difference scheme for a nonuniform grid, implemented in the software. The calculation is conducted in two stages first flows are calculated, and then the enthalpy is calculated. It makes it convenient to formulate the second and third boundary conditions. Various conservative numerical methods differ from each other by different ways of the flow rate q calculating. Flows are calculated by approximation of Fourier s law ... 

When the heat is conducted through two adjacent materials with different thermal conductivities, the third boundary condition comes from a requirement that the temperature at the interface is the same for both materials (in cases when the contact resistance may be neglected) or there is a discontinuity in temperature distribution at the interface described by either contact resistance R,c or thermal contact coefficient h,c, defined by ... 

Since there is no conical intersection in the buffer zone, CTq, the second integral is zero and can be deleted so that we are left with the first and the third integrals. In general, the calculation of each integral is independent of the other however, the two calculations have to yield the same result, and therefore they have to be interdependent to some extent. Thus we do each calculation separately but for different (yet unknown) boundary conditions The first integral will be done for Gi2 as a boundary condition and the second for G23. Thus A will be calculated twice ... 

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). 

---