Local Velocity Approach

The local velocity approach assumes that the front position changes adiabatically in time as the front moves into a region where the characteristic parameters D and r change. For (6.50) with a source term f p) = p (1 — p) the velocity of the front is locally given by w =. Jr eXf)/2, where Xy is the position of the front. To be more specific, let us also assume (x) = x and a = 1. The temporal dependence of the front velocity is obtained by integrating the differential equation 

In Fig. 6.8 we compare the numerical results for the front velocity of (6.50) for r sx) = l + 8r] sx), t] x) = x, and p x, 0) a Heaviside function, with the analytical solutions (6.73) and (6.76) for different values of e. We observe that the velocity calculated from the SPA is in better agreement with numerical solutions than that calculated from the LVA. 

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FLOW IN BOUNDARY LAYERS. A boundary layer is defined as that part of a moving fluid in which the fluid motion is influenced by the presence of a solid boundary. As a specific example of boundary-layer formation, consider the flow of fluid parallel with a thin plate, as shown in Fig. 3.6. The velocity of the fluid upstream from the leading edge of the plate is uniform across the entire fluid stream. The velocity of the fluid at the interface between the solid and fluid is zero. The velocity increases with distance from the plate, as shown in Fig. 3.6. Each of these curves corresponds to a definite value of x, the distance from the leading edge of the plate. The curves change slope rapidly near the plate they also show that the local velocity approaches asymptotically the velocity of the bulk of the fluid stream. 

The Hamilton-Jacobi formalism, on the other hand, only holds for KPP kinetics, but in contrast to singular perturbation analysis there is no need to assume either weak or smooth heterogeneities. The local velocity approach is based on the assumption that for weak and smooth heterogeneities the velocity of the front is given by the local value of the reaction rate r and the diffusion coefficient D at each spatial point, i.e., the front velocity coincides with the instantaneous Fisher velocity V 2y/r x)D x). In general, this simple-minded approach is not consistent with results from the other analytical methods or with numerical solutions. 

Fig. 6.8 Comparison of the temporal evolution of the velocity of fronts (in dimensionless units) between the singular perturbation analysis result given by (6.73) (solid lines), the local velocity approach (6.76) (dashed lines), and numerical results (symbols) for different values of s. Here = X and / = p ( — p). Reprinted with permission from [290]. Copyright 2003 by the American Physical Society...

The local velocity approach for KPP kinetics yields v = 2 r eXf). The differential equation for the front position is... 

This section deals principally with the application of fluid dynamics to those devices whose primary use is the measurement of either velocity or rate of discharge. The measurement of local velocities at different points in the cross section of a stream represents a two- or three-dimensional approach to the problem of determining the rate of discharge. The distribution of velocity over the section is found experimentally and inte-... 

Lagrangian Numerical Scheme. The Lagrangian approach defines cells of whose corners, and hence boundaries, move with the local velocity. Cell corner movement is used to update densities, strains and stresses in the fluid. The velocity is then updated to complete one step in the time evolution of fluid acoustic response. 

In the simplest case, all the droplets are of the same size and the droplet canopy affects the wind flow like an easily penetrable roughness mathematically expressed by the conjugation problem (3.33)—(3.35). The boundary layer approach is thus accepted. The distributed mass force / should depend, however, not on the local velocity P of the carried medium alone, but on the relative velocity between the two media V - T. To get /, the individual force (1.14) should be multiplied by the concentration of droplets n. 

In Fig. 3.6 the dashed line OL is so drawn that the velocity changes are confined between this line and the trace of the wall. Because the velocity lines are asymptotic with respect to distance from the plate, it is assumed, in order to locate the dashed line definitely, that the line passes through all points where the velocity is 99 percent of the bulk fluid velocity Line OL represents an imaginary surface that separates the fluid stream into two parts one in which the fluid velocity is constant and the other in which the velocity varies from zero at the wall to a velocity substantially equal to that of the undisturbed fluid. This imaginary surface separates the fluid that is directly affected by the plate from that in which the local velocity is constant and equal to the initial velocity of the approach fluid. The zone, or layer, between the dashed line and the plate constitutes the boundary layer. 

The hydrodynamic component of the net force hr is obtained, as shown in Fig. 8.4.IB, by considering a second flow in which the particle is held stationary. In this local flow the fluid velocity vanishes on the collector surface and on the particle, while far from the particle the velocity approaches the stagnation flow behavior characterized by Eq. (8.4.1). From dimensional considerations... 

The differences between Pitts (P) and Fuoss-Onsager (F-O) are first, the above mentioned omission by F-O of the effect of asymmetric potential on the local velocities of the solvent near the ions second, the use of the more usual boundary conditions 5.2.28b by F-O compared to the P assumption that perturbations cease to be important at r = a. Pitts, Tabor and Daly, who have analysed in detail both treatments, concluded that the discrepancy due to the different boundary conditions is small but has the effect of reducing ionic interactions in the P treatment with respect to the F-O. This is confirmed by the analysis of data with both theories. Usually P requires a smaller value of the a parameter than F-O. The third discrepancy between the theoretical treatments is in the expression of Vj, in eqn. 5.2.5, for which F-O add a term which involves the effect of the asymmetry of the ionic atmosphere upon the central ion surrounded by such atmosphere. The last difference lies in the hydrodynamic approaches and the corresponding boundary conditions. P imposes the condition that the velocity of the smoothed... 

There is also a Coriolis force that vanishes as the body s velocity in the rotating local frame approaches zero. The centrifugal and Coriohs forces are apparent or fictitious forces, in the sense that they are caused by the acceleration of the rotating frame rather than by interactions between particles. When we treat these forces as if they are real forces, we can use Newton s second law of motion to relate the net force on a body and the body s acceleration in the rotating frame (see Sec. G.6). 

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