Rarefaction Parameter

However, in the transitional regime, i.e., when the rarefaction parameter is intermediate (8 1), the slip solution is not valid. Such a situation is realized when the channel/tube size a is less than 2 X 10 m. In this case, the kinetic Boltzmaiui equation is applied to calculate the coefficients Gp and Gt- hi the free-molecular regime, i.e., when the rarefaction parameter is... 

Gas Flow in Nanochannels, Fig. 2 Poiseuille coefficient versus rarefaction parameter 5 solid lines, kinetic equation solution [11] pointed line, free-molecular value based on Eq. 12 dashed line, Navier-Stokes solution Eq. 16... 

Knudsen number (Kn) is the ratio of mean free path I of fluid molecules to a t3rpical dimension of gas flow a, i.e., Kn = Ija. Rarefaction parameter 8 is the inverse Knudsen number. Velocity distribution function is defined so that the quantity /(f, r, v) dr dv is the number of particles in the phase volume dr dv near the point (r, v) at the time t. 

The main parameter determining the gas rarefaction is the Knudsen number Kn = Ha, where I is the mean free path of fluid molecules and a is a typical dimension of gas flow. If the Knudsen number is sufficiently small, say Kn < 10 , the Navier-Stokes equations are applied to calculate gas flows. For intermediate and high values of the Knudsen number, the Navier-Stokes equations break down, and the implementation of rarefied gas dynamics methods is necessary. In practical calculations usually the rarefaction parameter defined as the inverse Knudsen number, i.e.. 

Typical velocity profiles are presented in Fig. 1. For the large value of the rarefaction parameter (8 = 10), the profile is very close to the parabolic shape. In the transition (8=1) and small (8 = 0.1) values, the profiles are practically flat. 

Micro- and Nanoscale Gas Dynamics, Table 3 Shear stress Pxy versus rarefaction parameter 5... 

Typical temperature profiles are presented in Fig. 3. Their behaviors are quite similar to those of the velocity profiles in the previous example. The heat flux is shown in Table 4. It is negative because the temperature gradient is positive. The magnitude of the heat flux decreases by increasing the rarefaction parameter 5. 

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

Note that if the super-index ch or tb is omitted the quantity is referred to both channel and tube. The coefficient Gj describes a gas flow due to a temperature gradient and it is called the thermal creep coefficient. The coefficients Gp and Gt are introduced so that they are always positive. They are calculated from the kinetic equation and determined by the rarefaction parameter... 

Gas Row in Nanochannels, Rgure 3 Thermal creep coefficient 6 vs rarefaction parameter solid lines - kinetic equation solution [ ], pointed line - free molecular value based on Eqs. (12) and (15), dashed line -Navier- okes solution Eq. (19)... 

Typical velocity profiles are presented in Fig. 2. For the large value of the rarefaction (5 = 10) the velocity Uy varies linearly from —0.5 to 0.5, i. e. it is close to the solution in the hydrodynamic regime. By decreasing the rarefaction parameter S the slope of the velocity profile becomes smaller. The shear stress is shown in Table 3. Note, it is negative because the velocity gradient... 

The magnitude of the heat flux decreases by increasing the rarefaction parameter S. 

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