Of momentum

If a beam of monoenergetic ions of mass A/, is elastically scattered at an angle 6 by surface atoms of mass Mg, conservation of momentum and energy requires that... 

Now each such particle adds its change in momentum, as given above, to the total change of momentum of the gas in time t. The total change in momentum of the gas is obtained by multiplying Af by the change in momentum per particle and integratmg over all allowed values of tlie velocity vector, namely, those for which V n< 0. That is... 

Figure A3.1.5. Steady state shear flow, illustrating the flow of momentum aeross a plane at a height z.

Following the method used above, we see that there will be an upward flux of momentum m the v-direetion,... 

The equation of momentum conservation, along with the linear transport law due to Newton, which relates the dissipative stress tensor to the rate of strain tensor = 1 (y. 4, and which introduces two... 

The first tenn is what one would expect to obtain classically for a particle of momentum hk, and it is much bigger than the second tenn provided k a 1. Since the de Broglie wavelength X is In/kQ, this condition is equivalent to the statement that the size of the wavepacket be much larger than tire de Broglie wavelength. 

The second temi is proportional to the optical quadnipole transition moment, and so on. For small values of momentum transfer, only the first temi is significant, thus... 

Most instruments are configured with a fixed value for the radius of curvature, r, so changing the value of B selectively passes ions of particular values of momentum, mv, tlirough tlie magnetic sector. Thus, it is really the momentum that is selected by a magnetic sector, not mass. We can convert this expression to one involving the accelerating potential. 

A well known example of this is obtained by settmg % = p a, a=x, y, z, any component of momentum, giving the equipartition-of-energy relation... 

Now encounters between molecules, or between a molecule and the wall are accompanied by momentuin transfer. Thus if the wall acts as a diffuse reflector, molecules colliding wlch it lose all their axial momentum on average, so such encounters directly change the axial momentum of each species. In an intermolecuLar collision there is a lateral transfer of momentum to a different location in the cross-section, but there is also a net change in total momentum for species r if the molecule encountered belongs to a different species. Furthermore, chough the total momentum of a particular species is conserved in collisions between pairs of molecules of this same species, the successive lateral transfers of momentum associated with a sequence of collisions may terminate in momentum transfer to the wall. Thus there are three mechanisms by which a given species may lose momentum in the axial direction ... 

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

Equations (2.10), (2.18) and (2.24) provide the flux relations in situations where each of the three separate mechanisms of momentum transfer dominates. However, there remains the problem of finding the flux relations in "intermediate" situations where all three mechanisms may be of comparable importance. This has been discussed by Mason and Evans [7], who assumed first that the rates of momentum transfer due to mechanisms (i) and (ii) should be combined additively. If we write equation (2.10) in the form... 

The right hand side represents the rate of momentum transfer from species r by mechanism (i) and, combining this with the rate of transfer by mechanism (ii) as given by equation (2.IS), we obtain... 

The equation of motion is based on the law of conservation of momentum (Newton s second law of motion). This equation is written as... 

When M is an atom the total change in angular momentum for the process M + /zv M+ + e must obey the electric dipole selection mle Af = 1 (see Equation 7.21), but the photoelectron can take away any amount of momentum. If, for example, the electron removed is from a d orbital ( = 2) of M it carries away one or three quanta of angular momentum depending on whether Af = — 1 or +1, respectively. The wave function of a free electron can be described, in general, as a mixture of x, p, d,f,... wave functions but, in this case, the ejected electron has just p and/ character. 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

The relationship between the two conditions is estabUshed by conservation of energy and by conservation of momentum across the shock front. 

The conservation of mass gives comparatively Httle useful information until it is combined with the results of the momentum and energy balances. Conservation of Momentum. The general equation for the conservation of momentum is... 

The creation terms embody the changes in momentum arising from external forces in accordance with Newton s second law (F = ma). The body forces arise from gravitational, electrostatic, and magnetic fields. The surface forces are the shear and normal forces acting on the fluid diffusion of momentum, as manifested in viscosity, is included in these terms. In practice the vector equation is usually resolved into its Cartesian components and the normal stresses are set equal to the pressures over those surfaces through which fluid is flowing. 

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