Momentum flux density

While the general form of the generalized Euler s equation (equation 9.9) allows for dissipation (through the term Hifc) expression for the momentum flux density as yet contains no explicit terms describing dissipation. Viscous stress forces may be added to our system of equations by appending to a (momentarily unspecified) tensor [Pg.467]

We begin by describing the HPP model, which satisfies all of the above requirements except for the isotropy of the momentum flux density tensor. As we shall, however, this early model nonetheless has some very interesting and suggestive properties, despite not being able to reproduce Navier-Stokes-like behavior exactly. 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

We make two additional comments. First, notice that when u 0, the momentum flux density tensor reduces to the diagonal term p5ij, where the pressure p = Cgp and Cg is the speed of sound. We thus conclude that the speed of sound in the FHP-I LG is given by... 

Isotropy of the Momentum Flux Density Tensor If we trace back our derivation of the macroscopic LG Euler s and Navier-Stokes equations, we see that the only place where the geometry of the underlying lattice really enters is through the form for the momentum flux density tensor, fwhere cp = x ) + y ), k = 1,..., V... 

Now, in order for us to recover standard hydrodynamical behavior, we require that the momentum flux density tensor be isotropic i.e. invariant under rotations and reflections. In particular, from the above expansion we see that must be isotropic up to order... 

The stress tensor a (r), which is the momentum current or momentum flux density [9], is given by... 

Schweitz s representation of the quantum stress tensor a (r) in terms of the flux density operator acting on the momentum in Equation (19) makes clear its interpretation as a momentum flux density. Schweitz does not, however, consider how the surface flux virial in the quantum case, Zs or i ,s, may be related to the pv product. This, as demonstrated in the following section, has been accomplished using the atomic statement of the virial theorem [12]. 

Multiplying the mass flow M = AM/dt with the velocity w yields the momentum flux M-w. The momentum flux density is the momentum flux based on the cross-sectional area / ... 

The scalar product of the vectors m and w yields a volumetric woik or volumetric energy which has the dimension of a pressure. Let us assume a horizontal flow of an ideal ( 7 = 0) fluid. According to the law of conservation of momentum, the differential of the momentum flux density is equal to the differential of the pressure ... 

He considered that the rapid flame propagation could be achieved with the same mechanism as vortex breakdown. Figure 4.2.2 schematically shows his vortex bursting mechanism [4,5]. When a combustible mixture rotates, Ihe pressure on the axis of rotation becomes lower than the ambient pressure. The amount of pressure decrease is equal to max in Rankine s combined vor-fex, in which p denotes fhe unburned gas density and Vg denotes the maximum tangential velocity of the vortex. However, when combustion occurs, the pressure on the axis of rofafion increases in the burned gas owing to the decrease in the density, and becomes close to the ambient pressure. Thus, there appears a pressure jump AP across the flame on fhe axis of rotation. This pressure jump may cause a rapid movement of the hot burned gas. By considering the momentum flux conservation across the flame, fhe following expression for the burned gas speed was derived ... 

In addition, Turner and Trimble defined a slip equation of state combination as the specification of mass flux, momentum flux, energy density, and energy flux as single-valued functions of the geometric parameters (area, equivalent diameter, roughness, etc.) at any z location, and of mass flux, pressure, and enthalpy,... 

It has been assumed that the flow is incompressible so that there are no fluctuations of the density. Equation 1.91 shows that the momentum flux consists of a part due to the mean flow and a part due to the velocity fluctuation. The extra momentum flux is proportional to the square of the fluctuation because the momentum is the product of the mass flow rate and the velocity, and the velocity fluctuation contributes to both. The extra momentum flux is equivalent to an extra apparent stress perpendicular to the face, ie a normal stress component. As (v x)2 is always positive it produces a compressive stress, which is positive in the negative sign convention for stress. 

In the case of the flux of mass, the result is the normal component of pua. But for the flux of momentum and energy, in general the flux density is not the normal component of a vector or tensor function of (t, x), since it will depend on the extended shapes of if and Y. But in the case of short-range forces and slowly varying p, ua, E, it can be shown to have this form with sufficient approximation. Thus one is led to the familiar pressure tensor and heat flow vector Qa, both as functions of (t, x). It is to be emphasized that the general expression of these quantities involves not only expected values of products of momenta (or velocities), but the effect of intermolecular forces. 

When placed in a static magnetic field of flux density B0, a nucleus may undergo nuclear magnetic resonance (NMR) [1-5] if it possesses an angular momentum p. This angular momentum is referred to as nuclear spin. The component of p in the direction of B0 (Fig. 1.1), denoted as p0, can only take on values which are half-integral or integral multiples m of hj2 n ... 

This can be extended straightforwardly to angular momentum operators and infinitesimal magnetic field generators. Therefore, a commutator such as Eq. (818) is equivalent to a vector cross-product. If we write Bm> as the scalar magnitude of magnetic flux density, the commutator (818) becomes the vector cross-product... 

In Table IV, we see that established techniques for velocity measurement allow us to determine the average momentum flux, average velocity, turbulent intensities, and shear stress. Next on the list, to complete the flow field description, is the fluctuation mass flux, and first on the combustion field list is the temperature and major species densities of the flame gases. 

Equations (8.10)—(8.13) are merely Stokes equations rewritten in a suggestive form chosen to emphasize transport of the momentum tracer density pv, as well as to exploit the analogy between Eq. (8.10) for momentum transport and the comparable equation (Brenner, 1980b) for transport of the scalar probability density P, which is equivalent to the material tracer density. The absence of a convective term vp from the flux expression in Eq. (8.12)... 

The forces exerted by the pressure, the friction forces at the channel wall and gravity all act on a volume element of length dz, as illustrated in the picture. We are presuming the flow to be steady and one-dimensional and that the cross section of the channel is constant. The pressure, density and momentum flux over a cross section are only dependent on the flow path z. The momentum balance implies that the sum of the pressure, friction and gravitational forces are equal to the change in momentum... 

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