Boltzmann equation flow regimes

Wang-Chang-Uhlenbeck equation Generalized Boltzmann equation Flow simulations in transitional and rarefied regimes... 

Figure 1 shows the four gas flow regimes and the applicable models. The Boltzmann equation is valid for the whole range of Am, from 0 to infinity. A simplified Boltzmann equation or the collisionless Boltzmann equation, where the right-hand side reduces to zero, is suitable when Kn is very... 

In Figure 2.2 DSMC results of Karniadakis and Beskok [2] and results obtained with the linearized Boltzmann equation are compared for channel flow in the transition regime. The velocity profiles at two different Knudsen numbers are shown. Apparently, the two results match very well. The fact that the velocity does not reach a zero value at the channel walls (Y = 0 and Y = 1) indicates the velocity slip due to rarefaction which increases at higher Knudsen numbers. 

As the flow enters the transition flow regime and eontinues into the free-molecular flow regime, the Knudsen number becomes significant enough that the molecular approach has to be utilized. Thus, the Boltzmann equation... 

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 comoving with the fluid. When gas flow in microreactors at high temperature or low pressure is considered, this assumption may break down. The principle quantity determining the flow regime of gases and deviations from the standard continuum description is the Knudsen number, defined as... 

Physically, gas flows in the transition regime, where Kn is 0(1), are characterized by the formation of narrow, highly nonequilibrium zones (Knudsen layers) of thickness of the order of the molecular mean free path X the flow structure is then determined by the fast kinetic processes. Moreover, in the case of unsteady flows, an initial Knudsen time interval is of the order Tq = X/v, where v is the molecular velocity. Thus, the Knudsen layer can be computed accurately only by directly solving the Boltzmann equation. 

Rarefied Gas Dynamics is a scientific field which aims at describing gas flows on the basis of the kinetic Boltzmann equation considering the whole range of the gas rarefaction covering the free molecular, transitimial, and hydrodynamic regimes. 

In the transition regime, the rarefaction effects dominate and the intermolecular collisions need to be taken into account. For the free-molecular flow, intermolecular collisions can be considered negligible when compared to the probability of the molecule colliding with the wall surface. As the flow enters the transition flow regime and continues into the free-molecular flow regime, the Kn becomes significant enough that the molecular approach has to be utilized. Thus, the Boltzmann equation... 

The Boltzmann equation describes the behavior of gas flows in non-equilibrium due to molecular motion. It is applicable for all mean free paths X between the molecules, that is, for all Knudsen numbers Kn (= X/L, where L is a characteristic dimension in the flow). Thus it can be employed to calculate the flow fields in a variety of applications ranging from hypersonic flow past a space shuttle to flow in microdevices, where all the flow regimes - continuum (Kn 0), transitional (Kn 0(1)) and rarefied (Kn 1) - may be encountered. The classical Boltzmann equation (CBE) describes the flow of... 

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