Peculiar velocity

It is crucial to understand that, even assuming a zero peculiar velocity, it is only rarely possible, even in principle, for this process to yield a cosmological redshift to better than 10 km/s accuracy—and, because astronomers have no particular need for highly accurate redshift determinations, the effort to obtain them is rarely made. 

In our development and notation we mainly follow Truesdell [10] and use a subscript to indicate a constituent and a prime to denote material time derivative following the motion of that constituent thus, vt = x[ and at = x ( are the 2th peculiar velocity and acceleration, respectively. / ), Li = grad vt and I), = Li — Lf) are the peculiar gradient of deformation, velocity gradient and rate of deformation, respectively. 

Figure 5.4. Left. The overall SZ effect in Coma produced by the combination of various electron populations thermal hot gas with ksT = 8.2 keV and r = 4.9 10-3 (solid blue curve) which best fits the available SZ data relativistic electrons which best fit the radio-halo spectrum (yellow curve) provide a small additional SZ effect (Colafrancesco 2004a) warm gas with ksT v 0.1 keV and n 10-3 cm-3 (cyan curve) provides a small SZ effect due to its low pressure (Colafrancesco 2004c) DM produced secondary electrons with Mx = 10 (black dotted curve), 20 GeV (black solid curve) and 30 GeV (dashed solid curve). A pure-gaugino x reference model is assumed in the computations. The relative overall SZ effect is shown as the dotted, solid and dashed red curves, respectively. A zero peculiar velocity of Coma is assumed consistently with the available limits. SZ data are from OVRO (magenta), WMAP (cyan) and MITO (blue). Right. The constraints on the

Although these are the equations that we would right down from first principles in a Newtonian analysis, they are also the small-scale and small-velocity limit of General Relativity. To account for the expansion of the Universe, we change to comoving coordinates, r and peculiar velocity, u, defined from physical coordinates, x, and velocities, v as... 

The equation for q, (Eq. [123]) can be thought of as the velocity of the particle in the laboratory frame. This can be understood more completely by noting that the laboratory velocity, q,- is a sum of ajhermal (or peculiar) velocity P,/ot, and a component due to the velocity field iy y,. At the steady state, the laboratory velocity profile will be linear as in Figure 10, with the slope (the... 

However, it is clear that for a general tensor Vu, trajectory analysis based on the SLLOD dynamics in Eqs. [129] will yield incorrect results. Equation [132] has an extra term in the force, which is equivalent to saying that the momenta in Eqs. [129] are not peculiar with respect to a general flow (indeed, Eqs. [129] yield peculiar velocities for the case of planar Couette flow), and therefore the flow profile produced will not be q Vu as expected. Equations [129] also lead to problems when one is considering definitions of pressure... 

Since the energies of photons are directly proportional to their frequencies, as the universe expands photon energies redshift to smaller values E7 = hv ==> E7 oc (l+z)-1. For all particles, massless or not, de Broglie told us that wavelength and momentum are inversely related, so that p oc A-1 => p oc (1 + z) 1. All momenta redshift for non-relativistic particles (e.g., galaxies) this implies that their peculiar velocities redshift v = p/M oc (1 + z)-1. 

It follows from the above definitions that the average peculiar velocity equals zero. 

In the gas we consider an infinitesimal element of surface area dA as sketched in Fig. 2.1. The orientation of the surface area is defined by a unit vector n normal to the surface, and u> is the angle between C and n. Imagine further that the element of surface area moves along with the fluid having the velocity v(r, t). The collection of molecules will then move back and forth across this element area with their peculiar velocities C about the mean velocity V, in accordance with (2.59). 

Fig. 2.1. A cylinder containing all molecules with peculiar velocity C which cross the surface element dA during the time interval dt.

The trace of the pressure tensor can be expressed in terms of the peculiar velocity ... 

The peculiar velocity of a molecule of species s is defined in terms of the mass average velocity ... 

Thus, the diffusion velocity for a given species equals the average of the corresponding peculiar velocity and can be written in the form ... 

FYom this formula it is seen that to calculate /i we need to determine the mean molecular speed ( ci )m- For real systems the average molecular speed is difficult to determine. Assuming that the system is sufficiently close to equilibrium the velocity distribution may be taken to be Maxwellian. For molecules in the absolute Maxwellian state the peculiar velocity equals the microscopic molecular velocity, i.e., Ci = ci, because the macroscopic velocity... 

The kinetic pressure term pkin can be deduced from the second term in the momentum balance after introducing the peculiar velocity (2.59). The second moment term is reformulated as follows ... 

Introducing the peculiar velocity (2.59) we can re-write the terms to obtain more common forms. We first note that ... 

For multiphase flows perturbed by the presence of particles to obtain a turbulence like behavior the local instantaneous velocity of the continuous phase can for example be decomposed adopting the Reynolds averaging procedure (i.e., other methods including time-, volume-, ensemble-, and Favre averaging have been used as well) and expressed as Vg = Vg- - < v >g, where v(. is the fluctuating component of the continuous phase velocity. Introducing the peculiar velocity for the dispersed phase this relation can be re-arranged as ... 

In this section an alternative derivation of the governing equations for granular flow is examined. In this alternative method the peculiar velocity C, instead of the microscopic particle velocity c, is used as the independent variable in the particle property and distribution functions. The transformation of these functions and the governing equation follows standard mathematical procedures for changing the reference frame. The translational motion of an individual particle may be specified either by its microscopic velocity c relative to a fixed or Galilean frame of reference, or by its velocity relative to a frame of reference moving with the local velocity of the granular material Yd-... 

Experience has shown that one might benefit substantially from deriving the governing transport equations after having transformed the property and distribution functions so that they are dependent on the peculiar velocity variable (2.59) instead of the microscopic particle velocity c. This means that the... 

The chain rule provides a relation between the partial derivative of / with respect to the individual particle velocity c and the partial derivative of fc with respect to the peculiar velocity C. To understand the forthcoming transformation it might be informative to specify explicitly the meaning of the partial derivatives. 

To complete the reformulation of the Boltzmann equation replacing the microscopic particle velocity with the peculiar velocity, the collisional rate of change term has to be modified accordingly. Jenkins and Richman [32] proposed the following approximate formula ... 

In the derivation of this expression, several manipulations are made [60]. The chain rule is applied, and the integral operator and the derivative operator in (4.17) are commuted. The order in which these operations are performed can be inverted because the Taylor series expansion is written for a fixed point in space. The third term in (4.72) occurs after the peculiar velocity is introduced. The symbol corresponds to the tensor equivalent of valid for the particular case in which the argument is a vector, both calculated in accordance with (4.20). 

In the particular case in which ip C) is a function of the peculiar velocity only, the given relation reduces to ... 

To explain the basic problem we use the Reynolds decomposition and averaging procedure, as an example. Introducing the peculiar velocity for the dispersed phase (4.103) can be re-arranged as ... 

The state of a gas is characterized by the distribution function / which depends on the peculiar velocity, the radius vector and the time t. For a stationary gas, the distribution function is Maxwellian and is denoted by f , which is given by... 

The diffusion velocity or peculiar velocity of the constituent a is its velocity relative to the mean velocity ... 

Here e. is the internal energy of a molecule at the fth vibrational and /th rotational levels, k is the Boltzmann constant, = u — v is the peculiar velocity. The zero-order distribution function for atomic species reads... 

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