Mean free path and viscosity

In the kinetic theory, the gas molecules are represented by hard spheres colliding elastically with each other and with the container walls. Details of this theory are given, for example, in ref. [1], An important parameter that can be calculated by this model is A, the mean free path of a molecule between collisions. The mean free path A of molecules is  

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It is thus seen that the kinematic viscosity, the thermal diffusivity, and the diffusivity for mass transfer are all proportional to the product of the mean free path and the root mean square velocity of the molecules, and that the expressions for the transfer of momentum, heat, and mass are of the same form. 

Collision Cross-Section The model of gaseous molecules as hard, non-interacting spheres of diameter o can satisfactorily account for various gaseous properties such as the transport properties (viscosity, diffusion and thermal conductivity), the mean free path and the number of collisions the molecules undergo. It can be easily visualised that when two molecules collide, the effective area of the target is no1. The quantity no1 is called the collision cross-section of the molecule because it is the cross-sectional area of an imaginary sphere surrounding the molecule into which the centre of another molecule cannot penetrate. 

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. 

Maxwell derived an equation which linked the viscosity of a gas to the density, the mean free path and the average velocity of the molecules. From measurements of viscosity made with the help of his wife he estimated the mean free path of oxygen molecules at 0 C to be 5.6 x 10" cm (modern value 6.6 x 10" cm). Maxwell was also able to estimate the mean free path from diffusion experiments, and the two values were in satisfactory agreement. 

Pressure-Driven Single Phase Gas Rews, Table 1 Mean free path and dynamic viscosity for classic IPL collision models... 

Transport processes in a hard-sphere gas can be analyzed theoretically. A formula for the self-diffusion coefficient was derived in this chapter, and similar formulas for thermal conductivities and viscosity coefficients were presented. Each transport coefficient is proportional to the mean free path and to the mean speed, and thus proportional to the square root of the temperature. 

Extracted from U.S. Standard Atmosphere, 1976, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration and tte U.S. Air Force, Washington, 1976. Z = geometric altitude, T = temperature, P = pressure, g = acceleration of gravity, M = molecular weight, a = velocity of sound, i = viscosity, k = thermal conductivity, X = mean free path, p = density, and H = geopotential altitude. The notation 1.79.—5 signifies 1.79 X 10 . ... 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

The correction of mean free path, hi by the nanoscale effect function results in a smaller mean free path, or a smaller Knudsen number in other word. As a matter of fact, a similar effect is able to be achieved even if we use the conventional definition of mean free path, / = irSn, and the Chapmann-Enskog viscosity equation, /r = (5/16)... 

XlylmnkT/ird ), or h = [TT[i 2RTI2p), as long as substituting the gap-dependent viscosity rather than the bulk viscosity. Because the effective viscosity decreases as the Knudsen number enters the slip flow and transition flow ranges, and thus the mean free path becomes smaller as discussed by Morris [20] on the dependence of slip length on the Knudsen number. 

These expressions for the shear viscosity are compared with simulation results in Fig. 5 for various values of the angle a and the dimensionless mean free path X. The figure plots the dimensionless quantity (v/X)(x/a2) and for fixed y and a we see that (vkin/A,)(x/a2) const A, and (vcol/A)(r/u2) const/A. Thus we see in Fig. 5b that the kinetic contribution dominates for large A since particles free stream distances greater than a cell length in the time x however, for small A the collisional contribution dominates since grid shifting is important and is responsible for this contribution to the viscosity. 

In the relations given earlier, it is assumed that the fluid can be regarded as a continuum and that there is no slip between the wall of the capillary and the fluid layers in contact with it. However, when conditions are such that the mean free path of the molecules of a gas is a significant fraction of the capillary diameter, the flowrate at a given value of the pressure gradient becomes greater than the predicted value. If the mean free path exceeds the capillary diameter, the flowrate becomes independent of the viscosity and the process is one of diffusion. Whereas these considerations apply only at very low pressures in normal tubes, in fine-pored materials the pore diameter and the mean free path may be of the same order of magnitude even at atmospheric pressure. 

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