Equivalent Mean Free Path

For hard sphere collisions, v(v) would be proportional to v9 and the mean free path independent of v A(v) is an equivalent mean free path for a- general force law. Cf. S. Chapman and T. G. Cowling, The Mathematical Theory of Non- Uniform Oases, pp. 91 and 348, Cambridge University Press, 1958. 

Equivalent mean free path I has the order of the mean free path and is related to the shear viscosity jx, most probable molecular speed vq and pressure P d i = fivolP. 

Rarefaction parameter is the ratio of a typical size of gas flow a to the equivalent mean free path (, i.e., d = alt. 

The equivalent mean-free-path technique has been applied to several systems of engineering interest, and the agreement with more accurate methods is surprisingly good. Table 7.2 lists some values for the age to thermal of fission neutrons in stainless steel-water mixtures at 450 F for various metal-water ratios. Column 2 gives the age as computed... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

Thus even if the mean free path is small compared to the cell length, particle (or equivalently grid) shifting will cause particles to collide with molecules in nearby cells, thereby reducing the effects of locally correlated collision events in the same cell. 

Nucleation. In the presence of adsorbed additives, the mean free path for lateral diffusion of adions is shortened, which is equivalent to a decrease in the diffusion coefficient D (diffusivity) of adions. This decrease in D can result in an increase in adion concentration at steady state and thus an increase in the frequency of the two-dimensional nucleation between diffusing adions. 

Obviously the motions of encounter and of mean free path do not necessarily have any meaning here nor are they needed. On the other hand, the relaxation time x is easy lo define, e.g., as the time taken by sound to travel unit distance, as computed in the equivalent body in equilibrium (equivalent, in having the same total mass and energy in "V"). 

In some cases, particularly in the growth of aerosol particles, the assumption of equilibrium at the interface must be modified. Frisch and Collins (F8) consider the diffusion equation, neglecting the convective term, and the form of the boundary condition when the diffusional jump length (mean free path) becomes comparable to the radius of the particle. One limiting case is the boundary condition proposed by Smoluchowski (S7), C(R, t) = 0, which presumes that all molecules colliding with the interface are absorbed there (equivalent to zero vapor pressure). A more realistic boundary condition for the case when the diffusion jump length, (z) R, has been shown by Collins and Kimball (Cll) and Collins (CIO) to be... 

The relative frequency of the intermolecular collisions and collisions between molecules and the pore wall can be characterized by the ratio of the length of the mean free path A and the equivalent pore diameter dt = 2rf This ratio determines the flow regime and is called the Knudsen number Kn ... 

Wavelength, ionic equivalent conductance, mean free path... 

We have made the additional approximation of assuming that the number of collisions Z at any point is independent of d, the distance between plates. This is justifiable if the mean speed 5 F L/d, where F L/d is the difference in velocity between two layers of gas separated by a mean free path. Und( r such conditions the molecular density in each layer is constant and most collisions then take place between molecules that have essentially the same relative Maxwellian distribution. When this condition is not satisfied, there will be important density gradients and thermal gradients, so that the entire analysis does not apply. This condition is the equivalent of saying that the velocity of the moving plate is small compared to the velocity of sound. 

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

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