Maxwell density

In order to get this expression into a more familiar form (equation 9.7), we now consider the zeroth-order approximation to /. We assume that / is locally a Maxwell-Boltzman distribution, and treat the density p, temperature T[x,t) = < V — u p> (where k is Boltzman s constant), and average velocity u all as slowly changing variables with respect to x and t. We can then write... 

Maxwell, James Clark, 121,459 Measurement, 20q. See also Units density, 14-15 expression, 6-7 gases, 103-105,126q heat flow, 200-203... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Ionic current density maps can be recorded with the aid of the pulse sequence shown in Figure 2.9.2. The principle of the technique [48-52] is based on Maxwell s fourth equation for stationary electromagnetic fields,... 

E and B are the fundamental force vectors, while P and H are derived vectors associated with the state of matter. J is the vector current density. The Maxwell equations in terms of E and B are... 

By interpreting the term in brackets as the total current density the inhomogeneous Maxwell equation (2) is also written as... 

To determine the rate of transfer a large set of systems, each with an atom either to the left or near the saddle point is considered. Once an atom has moved to the right of LL the process is regarded as having occurred. The rate R is then defined as the net flux of atoms across LL, subject to several assumptions, introduced to ensure forward reaction. For n atoms in equilibrium near A, moving with mean Maxwell velocity v = y/2kT/irm towards the right, the required flux density is (1/2)vn, and the total flux is... 

In the region of a first-order transition ip has equal minima at volumes V and V2, in line with the Maxwell construction. The mixed phase is the preferred state in the volume range between V and V2. It follows that the transition from vapour to liquid does not occur by an unlikely fluctuation in which the system contracts from vapour to liquid at uniform density, as would be required by the maximum in the Van der Waals function. Maxwell construction allows the nudeation of a liquid droplet by local fluctuation within the vapour, and subsequent growth of the liquid phase. 

Maxwell s equations, as well as the Lorentz force, can be derived from the Lagrangian density... 

A fluid composed of a single species is described by five fields the three components of the velocity, the mass density, and the temperature. This is a drastic reduction of the full description in terms of all the degrees of freedom of the particles. This reduction is possible by assuming the local thermodynamic equilibrium according to which the particles of each fluid element have a Maxwell-Boltzmann velocity distribution with local temperature, velocity, and density. This local equilibrium is reached on time scales longer than the intercollisional time. On shorter time scales, the degrees of freedom other than the five fields manifest themselves and the reduction is no longer possible. 

The dielectric constant of the pure cyanurate network under dry nitrogen atmosphere at 20 °C is 3.0 (at 1 MHz). For the macroporous cyanurate networks, the dielectric constant decreases with the porosity as shown in Fig. 57, where the solid and dotted lines represent experimental dielectric results together with the prediction of the dielectric constant from Maxwell-Garnett theory (MGT) [189]. The small discrepancies between experimental results and MGT might be due to the error in estimated porosities, which are calculated from the density of the matrix material and cyclohexane assuming that the entire amount of cyclohexane is involved in the phase separation. It is supposed that a small level of miscibility after phase separation would result in closer agreement of dielectric constants measured and predicted. Dielectric constant values as low as 2.5 are measured for macroporous cyanurates prepared with 20 wt % cyclohexane. 

An important equation of electrostatics, which follows directly from Maxwell s equations (Jackson 1975) is Poisson s equation. It relates the divergence of the gradient of the potential charge density at that point ... 

In order to understand the diffuse layer in detail, we need to go back to the fnndamental eqnations of electrostatics due to J.C. Maxwell. The equation of interest relates the local electric field E(r) at the position vector r to the net local electric charge density p(r) ... 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

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