Front Velocity Selection

The fact that an infinity of front velocities occurs for pulled fronts gives rise to the problem of velocity selection. In this section we present two methods to tackle this problem. The first method employs the Hamilton-Jacobi theory to analyze the dynamics of the front position. It is equivalent to the marginal stability analysis (MSA) [448] and applies only to pulled fronts propagating into unstable states. However, in contrast to the MSA method, the Hamilton-Jacobi approach can also deal with pulled fronts propagating in heterogeneous media, see Chap. 6. The second method is a variational principle that works both for pulled and pushed fronts propagating into unstable states as well as for those propagating into metastable states. This principle can deal with the problem of velocity selection, if it is possible to find the proper trial function. Otherwise, it provides only lower and upper bounds for the front velocity. 

1 Hyperbolic Scaling and Hamilton-Jacobi Equation for the Front Position 

This method consists in finding the Hamilton-Jacobi equation for an RD equation with a KPP reaction term and was originally introduced by Freidlin [140]. The first step consists in performing the hyperbolic scaling on the RD equation (2.3)  

The second step consists in analyzing the behavior of solutions of (2.3) for large times, of order s and determine whether or not a front exists in the limit f oo (e 0). We expect that after the hyperbolic scaling the new field /o (x, t) = p x/e, t/s) takes only two values, 0 and 1, as e 0, which means that the solution of (2.3) converges to the indicator function of the set whose boundary can be considered as the position of a moving front that separates the stable and unstable states. In fact any initial condition with compact support will ensure a front propagating with the minimal velocity. After the hyperbolic scaling, (2.3) reads 

The hyperbolic scaling (4.33) and the transformation (4.35) of the field allows us to obtain, in the asymptotic limit e 0, a Hamilton-Jacobi equation for any given reaction-transport equation. In this chapter we focus only on RD equations, but in Chap. 5 we deal with other models. Regardless of the specific form of the Hamilton-Jacobi equation, its solution can be written as 

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The operational problems related to undesired internal gas flows and the difference in heat front velocities limited the execution of a systematic experimental program. In Table 1 typical gas composition arc given for some selected experiments the number... 

These experiments are slightly more difficult in that both the 29s and 27d states must be excited. In this case the atoms are allowed to collide for a few microseconds after pulsed laser excitation, and the 29p atoms are detected as the tuning field is slowly swept over many shots of the laser. The important difference from the Na experiment is that the K atoms can be velocity selected. As shown in Fig. 5, there is a slotted disc in front of the oven, and if the disc is turning, a burst of the atoms passes through the slit, and only those within a narrow velocity range are excited by the 5 ns dye laser pulse 200 /xs after the burst of atoms passes through the slit. 

To implement the Doppler-selected TOF measurement, the initial relative velocity is arranged to be parallel to the propagation vector of the probe laser. This critical configuration can readily be achieved in this rotating sources machine.36 Under this configuration, each Doppler-sliced 2D distribution exhibits a cylindrical symmetry The slit in front of the TOF spectrometer allows only those products with a rather small vx to be detected. Hence, only the -distribution, obtained by the TOF measurement, is needed to completely characterize the Doppler-sliced 2D (vx — vy) distribution. 

Figure 2. Effect of flow velocity on conversion and selectivity for CH4 oxidation over 10 layers of 40 mesh Pt-10% Rh gauze. The feed contained 16% CH4 in air and the front layer of gauze was maintained at 1227 5°C.

The peak dynamic pressure in the shock front is called DP of an explosive. It has been established that a linear regression plot of experimental detonation pressures (DP) or Chapman-Jouguet Pressure (PCj) versus detonation velocities D for selected explosives fits the relationship in Equation 1.13 ... 

Consider the propagation of a one-dimensional normal shock wave in a gas medium heavily laden with particles. Select Cartesian coordinates attached to the shock front so that the shock front becomes stationary. The changes of velocities, temperatures, and pressures of gas and particle phases across the normal shock wave are schematically illustrated in Fig. 6.12, where the subscripts 1, 2, and oo represent the conditions in front of, immediately behind, and far away behind the shock wave front, respectively. As shown in Fig. 6.12, a nonequilibrium condition between particles and the gas exists immediately behind the shock front. Apparently, because of the finite rate of momentum transfer and heat transfer between the gas and the particles, a relaxation distance is required for the particles to gain a new equilibrium with the gas. 

These equations relate the undisturbed explosive lying at rest with pressure Pq = 0 and specific volume Vq = to the state behind the detonation front, which is characterized by a pressure P, a specific volume V, and a particle flow velocity u. Both u and the detonation velocity, D, are measured in the reference frame of the undisturbed material. Because Pq and Vq are known, the Rankine-Hugoniot relations are a set of three equations for the four unknowns, u, D, P, and V. The first relation determines u in terms of D, P, and Vi, which leaves two equations with three unknowns. The first of the remaining equations, Eq. (4b) defines the Rayleigh line while Eq. (4c) defines the Hugoniot curve. The problem is formally determined by selecting the solution of Eqs. (4b) and (4c) that corresponds to the minimum value of D for an unsupported detonation. This additional condition is the Chapman—Jouguet hypothesis, which was put on a firmer foundation by Zel dovich. ... 

The condition for complete separation given in Eq. 17.16 is clearly illustrated in Figure 17.3. The switching time, t, (or, alternately, the solid phase velocity) must be selected in such a way that the first component front passes the raffinate node but that the second component front does not reach it. This latter condition is equivalent to j6 > 1. When = 1, the extract reaches just the raffinate node at time t. This is the limit situation. We can see from Eq. 17.19 that, in this case, Lq is equal to 0. 

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