Momentum-gradient

In the previous section, the molecular basis for the processes of momentum transfer, heat transfer and mass transfer has been discussed. It has been shown that, in a fluid in which there is a momentum gradient, a temperature gradient or a concentration gradient, the consequential momentum, heat and mass transfer processes arise as a result of the random motion of the molecules. For an ideal gas, the kinetic theory of gases is applicable and the physical properties p,/p, k/Cpp and D, which determine the transfer rates, are all seen to be proportional to the product of a molecular velocity and the mean free path of the molecules. 

The analogy for transport processes is readily interpreted from Stokes theory if we consider the generalization that forces or fluxes of a property are proportional to a diffusion coefficient, the surface area of the body, and a gradient in property being transported. In the case of momentum, the transfer rate is related to the frictional and pressure forces on the body. The diffusion coefficient in this case is the kinematic viscosity of the gas (vg = p-g/pg, where pg is the gas density). The momentum gradient is Pjg Uoo/B. 

In these equations 3w/3k represents the momentum gradient of the pseudopotential evaluated at the Fermi energy. 

Let us return to Eq. (6.112) to consider a tractable approximation. We first consider the propagation by the momentum-gradient term, that is,... 

Most of these quantities assure that the present quantization of the nuclear wavefunction in the context of the repeated nonadiabatic transitions work quite accurately, reproducing the full quantum values well. At the same time, the figures of the dispersion of the NVG suggest that it is not an easy task to keep the wavepacket in an appropriate shape. In particular the very sharp peak of the NVG in panel (b4) at about t = 1100 reflects the divergent behavior of the momentum-gradient at the turning points. Nevertheless, we may conclude that the present scheme to generate non-Born-Oppenheimer electronic and nuclear wavepackets works quite well already in this lowest level of approximation. 

First, we determine the momentum gradient of the nonequilibrium part of the distribution function directiy from the approximation for /i and (3.29) as... 

The boundary conditions can often be made to look similar, through the conversion to dimensionless variables. These equations can be used to compute a momentum boundary layer thickness, a thermal boundary thickness, and a concentration boundary layer thickness. The primary difference within these analogies is shown in Table 9.1, where the momentum flux per momentum gradient, the heat flux per heat gradient, and the mass flux per mass gradient are given. This will provide a good comparison if the boundary conditions are made nondimensional to vary between 0 and 1. 

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