Momentum kinematic

The connection of the canonical momentum operator p with the effect of external vector potentials A on a moving electron with charge (/g = — e is often simply written as 

It is important to note that the kinematic momentum components do not commute. 

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Having established that these assumptions are reasonable, we need to consider the relationship between the parameters of the actual offset jet and the equivalent wall jet that will produce the same (or very similar) flow far downstream of the nozzle. It can be shown that the ratio of the initial kinematic momentum per unit length of nozzle of the wall jet to the offset jet,, and the ratio of the two nozzle heights,, depend on the ratio D/B, where D is the offset distance betw een the jet nozzle and the surface of the tank, and h, is the nozzle height of the offset jet. The relationship, which because of the assumptions made in the analysis is not valid at small values of D/hj, is shown in Fig 10.72. 

Figures 10.75 and 10.76 show the initial kinematic momentum required to meet this criterion as a function of the buoyancy velocity and the length of the tank, for different values of the allowable concentration Qj. , and critical veloc-ity As we would expect, the required momentum increases both as the length of the tank increases and as the buoyancy of the contaminant increases.

FIGURE 10.75 Required initial kinematic momentum, f/p, as a function of the length of the tank, L, and the buoyancy velocity, v, when the critical contour criterion is applied with the critical concentration, C ,j, equal to 5% and the cross-drafts equal to 0.05 m s". ... 

An alternative procedure is to simplify the transport equation for the kinematic momentum fluxes or Re molds stresses. The contraction of the Re3molds stress transport equation (1.394) (that is, when the 3 equations for the 3 normal stresses, i = k = 1,2,3) are summed up) gives an exact transport equation for the turbulent kinetic energy (e.g., [167] [131] [106]). 

By noting that 7t = p — A the vector product of the kinematic momentum operator with itself can be simplified to... 

The form invariance of the Schrodinger equation will then lead to gauge invariant expectation values of the Hamiltonian. However, this will not be the case for an arbitrary operator. In particular, it turns out that expectation values of the canonical momentum operator, given in Eq. (2.45), are not gauge invariant, whereas expectation values of the mechanical or kinematical momentum operator, given in Eq. (2.97), are gauge invariant [see Exercise 2.15]... 

The mechanical or kinematical momentum operator is therefore sometimes also called the gauge invariant momentum operator. 

Exercise 2.15 Prove equation (2.118) for a one-electron system, i.e. with the kinematical momentum operator in Eq. (2.57) and with a one-electron transformation operator... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

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