Velocity distribution, expansion

The fragment recoil velocity resolution depends on the divergence of the molecular beam, molecular beam velocity distribution in the direction of the molecular beam axis, and the distance of fragments expanded in the velocity axis of the two-dimensional detector. If the divergence of the molecular beam is small and the fragment recoil velocity is much larger than the velocity difference of parent molecules, the recoil velocity resolution can be simply expressed as AV/V = s/L, where L is the length of expansion of... 

Another method is to measure the disappearance rate of the excited parent molecules, that is, the intensity changes of the disk-like images at various delay times (therefore, at various photolysis laser positions) along the molecular beam. This is very useful when the dissociation rate is slow and the method described above cannot be applied. This measurement requires a small molecular beam velocity distribution and a large variable distance between the crossing points of the pump and probe laser beams with the molecular beam. The small velocity distribution can be obtained through adiabatic expansion, and the available distances between the pump and probe laser beams depend on the design of the chamber. For variable distances from 0 to 10 cm in our system and AV/V = 10% molecular beam velocity distribution, dissociation rates as slow as 3 x 103 s 1 under collisionless condition can be measured. 

The primary reaction zone is a hollow cone-like zone, only lO- -lQ- m thick. The actual shape of the cone is determined largely by the velocity distribution of the gas mixture leaving the burner. While the velocity of the gases at the burner walls is virtually zero, it reaches a maximum in the centre. The rounding at the top is caused, in part, by thermal expansion of the gases, which also produces a backpressure which distorts the base... 

At sufficiently large Schmidt numbers [as in liquids, where the diffusion coefficient is very small (D 10 5 cm2/sec)], only the region very near the wall is affected by diffusion. Therefore, the velocity distribution can be approximated by the first term in the Taylor expansion... 

After the shock wave reaches the surface, the star starts to expand and soon the expansion becomes homologous. In Fig. 4, the velocity distribution for the homologous expansion is shown for 11E1 and 11E2. The velocity gradient with respect to the enclosed mass, Mr, is very steep near the surface, while it is almost flat in the helium layer and the heavy element core. This is because the core material is decelerated and forms a dense shell due to the reverse shock. The... 

For the heated vertical plate and horizontal cylinder, the flow results from natural convection. The stagnation configuration is a forced flow. In each case the flow is of the boimdai7 Kiyer type. Simple analytical solutions can be obtained when the thickness of the du.st-free space is much smaller than that of the boundary layer. In this case the gas velocity distribution can be approximated by the first term in an expansion in the distance norroal to the surface. Expressions for the thickness of the dust-free space for a heated vertical surface and a plane stagnation flow are derived below. 

Hydrodynamic models are derived from the mesoscale model (e.g. the Boltzmann equation) using a Chapman-Enskog expansion in powers of the Knudsen number (Bardos et al., 1991 Cercignani et al, 1994 Chapman Cowling, 1961 Ferziger Kaper, 1972 Jenkins Mancini, 1989). The basic idea is that the collision term will drive the velocity distribution n towards an equilibrium function eq (i-e. the solution to C( eq) = 0), and thus the deviation from equilibrium can be approximated by n -i- Knui. From the... 

In order to increase the number of degrees of freedom in a systematic manner, a functional expansion can be used to represent the NDF (Grad, 1949b). Using the velocity distribution as an example, the formal expansion is... 

Friction loss from sudden contraction of cross section. When the cross section of the conduit is suddenly reduced, the fluid stream cannot follow around the sharp corner and the stream breaks contact with the wall of the conduit. A jet is formed, which flows into the stagnant fluid in the smaller section. The jet first contracts and then expands to fill the smaller cross section, and downstream from the point of contraction the normal velocity distribution eventually is reestablished. The cross section of minimum area at which the jet changes from a contraction to an expansion is called the vena contracta. The flow pattern of a sndden contraction is shown in Fig. 5.14. Section CC is drawn at the vena contracta. Vortices appear as shown in the figure. 

As well as producing higher beam intensities, the supersonic nozzle sources have a further advantage. In the supersonic expansion through the Laval slit the gas is adiabatically cooled to very low temperatures of ca. 40°K [187] but increases its translational energy in the beam direction to the flow velocity of the gas, e.g. Mach number ca. 15 or peak velocity 70% higher than the most probable velocity of a 900°K oven [187]. Supersonic nozzles produce very narrow velocity distributions compared with the Maxwell—Boltzmann distribution obtainable from an effusive oven at the same temperature. 

To find an approximate solution of the kinetic equation, an orthogonal expansion of the velocity distribution with respect to the direction v/v of the velocity v is commonly used in the treatment of the kinetic equation. Depending on the... 

If the electric field and the inhomogeneity in the plasma are parallel to a fixed space direction—for example, the direction of the coordinate space—the velocity distribution becomes symmetrical around the field E z, t) = E z, t)e, gets the reduced dependence F U, vjv,z, t), and can be given the expansion (Shkarofsky et al, 1966 Golant et al, 1980)... 

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