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Introduction
Published in Mohamed Gad-el-Hak, MEMS, 2005
The Chapman-Enskog theory attempts to solve the Boltzmann equation by considering a small perturbation of fˆ from the equilibrium Maxwellian form. For small Knudsen numbers, the distribution function can be expanded in terms of Kn in the form of a power series fˆ=fˆ(0)+Knfˆ(1)+Kn2fˆ(2)+⋯
Presenting a new predictive viscosity model based on virial-like equations of state for monatomic fluids
Published in Chemical Engineering Communications, 2018
Viscosity models are considerable tools to illustrate the viscosity of a fluid as a function of temperature, pressure, and composition. The literature includes many diverse viscosity models (Poling et al., 2001). Some of them are very simple and only show one empirical relation. However, there are models that are completely theoretical (Viswanath et al., 2007; Monnery et al., 1995; Mehrotra et al., 1996; Burgessa et al., 2013; Bair, 2014; Umlaa and Vesovicc, 2014; Scalabrin et al., 2002). A lot of these models are only appropriate for predicting either the liquid- (Dutt et al., 2013; Qureshi et al., 1995; Miadonye et al., 1993; Krone, 1983) or the gas-phase viscosity (Martins et al., 2003; Novak, 2013). The cornerstone of theoretical models of viscosity is statistical mechanics, and viscosity is correlated to the intermolecular potential functions. For example, according to the Chapman–Enskog theory, a theoretical model (Chapman and Cowling, 1970; Hirschfelder et al., 1954) is derived from gas kinetic theory. According to statistical mechanic concepts, description of dense fluids due to short and long range attractive forces is difficult.
Beyond direct simulation Monte Carlo (DSMC) modelling of collision environments
Published in Molecular Physics, 2019
O. Schullian, B. R. Heazlewood
To evaluate the performance of the SCMFD model in calculating the viscosity coefficient of argon, we must first consider the conditions of the system in more detail. Chapman–Enskog theory allows the viscosity of a gas to be derived from the Boltzmann equation, providing a continuum model of the system as a perturbed ideal flow [6]. Chapman–Enskog theory is best applied to ideal gas systems at high densities; the further a system is from such a state, the less reliable the predicted viscosity becomes. The Knudsen number for the conditions in the simulated Couette flow is 0.00925 [2], corresponding to a density where Chapman's continuum model begins to break down. Lowering the density of the gas increases the Knudsen number, and the system transitions into a regime where the principles of free molecular flow apply: collisions occur far more frequently with the wall than with other gas particles, leading to the limiting case where the properties of the system are fully determined by the properties of the wall. Under the conditions of molecular free flow, the viscosity coefficient is proportional to the number density. As such, we expect the viscosity of argon in the Couette flow simulation to be directly proportional to the number of gas particles in the system at low densities. When the density of the system becomes sufficiently high, we expect the viscosity to transition from the molecular free flow behaviour to follow the Chapman–Enskog trend line. As Figure 6 shows, the viscosity coefficient of argon as calculated from the SCMFD simulation follows the molecular free flow prediction at low densities, and exhibits Chapman–Enskog behaviour at high densities. In the transition region, corresponding to densities of approximately – particles per m−3 in this system, the viscosity calculated by SCMFD is lower than that predicted by both molecular free flow and Chapman–Enskog. However, it is not immediately clear whether either model (molecular free flow or Chapman–Enskog) can be accurately applied in this regime. It is possible that SCMFD might underestimate correlations between velocity components, because these correlations are calculated by following the collisions of single particles. Alternatively, the traditional DSMC approach might overcorrelate these quantities, as the collisions of relatively few particles are taken as being representative of the whole system. Further work is needed to establish what the expected behaviour is in this transition region before we can quantitatively assess the performance of SCMFD for simulating Couette flow over a vast range of densities. For the moment, we can confidently assert that SCMFD accurately simulates the Couette flow of argon at both low densities and high densities.