Spatial stochasticity and non-continuum effects in gas flows

doi:10.1016/j.physleta.2012.01.004

Physics Letters A

Volume 376, Issues 8–9, 6 February 2012, Pages 967-972

https://doi.org/10.1016/j.physleta.2012.01.004 Get rights and content

Abstract

We investigate the relationship between spatial stochasticity and non-continuum effects in gas flows. A kinetic model for a dilute gas is developed using strictly a stochastic molecular model reasoning, without primarily referring to either the Liouville or the Boltzmann equations for dilute gases. The kinetic equation, a stochastic version of the well-known deterministic Boltzmann equation for dilute gas, is then associated with a set of macroscopic equations for the case of a monatomic gas. Tests based on a heat conduction configuration and sound wave dispersion show that spatial stochasticity can explain some non-continuum effects seen in gases.

Highlights

► We investigate effects of molecular spatial stochasticity in non-continuum regime. ► Present a simplify spatial stochastic kinetic equation. ► Present a spatial stochastic macroscopic flow equations. ► Show effects of the new model on sound wave dispersion prediction. ► Show effects of the new approach in density profiles in a heat conduction.

Introduction

The Boltzmann kinetic equation is the standard model for dilute gas flows. However, as this equation lacks an exact solution, rarefied gas models generally consist of approximate solutions [1]. Most of these approximate models are constructed so that they reproduce the Navier–Stokes–Fourier model of continuum fluid mechanics under relevant assumptions. These kinetic models generally predict weakly thermodynamic disequilibrium flows well. However, “in a gas in which finite departures from equilibrium are imposed by forces too strong or too rapid to be overcome by collisions, a satisfactory comparison between kinetic theory and experiments is much harder to achieve” [2].

The Boltzmann kinetic model states an evolution equation of gas media presumed from deterministic equations of molecular motions. Subsequently, most derivations of it start with the deterministic Liouville equation, which enforces conservation of the microscopic phase space probability density [1], [3]. The main characteristics of the Boltzmann equation are: it contains a collision integral describing the exchange of momentum between molecules; position and velocity variables are treated as independent variables; finally, and importantly, it does not involve an explicitly obvious stochastic component in the spatial variable. However, a large number of gaseous molecules exchanging momentum and positions represents a perfect example at the kinetic level of a physical stochastic process in both position and velocity spaces [4].

Recognising the distinction between a physical region in hydrodynamics and a physical region in kinetic theory, Klimontovich introduced a length scale separation, with averaging over kinetic space/volume, to arrive at a generalised kinetic equation that, in fact, is just a stochastic version of the deterministic Boltzmann equation (i.e., the Boltzmann equation with an additional spatial diffusion term) [5]. Zimdahl showed the importance of the Klimontovich version in deriving the relativistic Boltzmann equation [6]. Others, such as Ueyama [7], and Stryjewski et al. [8], have also claimed a spatial stochastic term should be included in the deterministic Boltzmann kinetic model. Note that the stochastic Boltzmann equation is now known to derive from the more general stochastic Liouville equation [9], [10]. While these works on stochastic Liouville and stochastic Boltzmann equations are mostly oriented toward other fields [8], [11], dilute gas flows remain predominantly the field of the deterministic Boltzmann equation. The various approximation solutions to the Boltzmann equation, and associated continuum equations, have their advantages and known limitations [12], [13], [14]. More precisely, the local-equilibrium assumption underlies most of these approximation models.

In this Letter we develop a kinetic equation using strictly stochastic reasoning, without primarily referring directly to the Liouville or Boltzmann (deterministic) equations. The resulting kinetic equation, which is a stochastic version of the Boltzmann equation, is then associated with a set of macroscopic equations for the case of monatomic gases. The derivation is presented in a way that explicitly traces the impact of the additional spatial kinetic stochastic term on the macroscopic conservation equations of mass, momentum and energy. Predictions of thermodynamically non-equilibrium heat conduction and sound wave dispersion are then used to assess the role of the spatial stochastic terms.

Section snippets

The Brownian model in position space

Consider an arbitrary particle moving in physical space, with its position X at a time t. This moving particle performs random walks in the sense that its trajectory is changing randomly. Then the problem is to obtain the spatial evolution in time of the particle, represented by a probability density function, $g (t, X)$ . According to the Einstein–Smoluchowski model of Brownian motion, $g (t, X)$ satisfies the equation: $\frac{\partial g}{\partial t} + \nabla \cdot J = 0,$ with J given by, $J = - κ \nabla g,$ with κ being a diffusivity coefficient. Eq. (1)

Effects of the additional dissipative mass/volume flux

The main result of the approach followed in the preceding section is that there is now in the new kinetic equation a diffusive term related to spatial variables, which leads to additional diffusive terms in the macroscopic fluid equations. To investigate the effects of such dissipative terms, particularly the mass or volume diffusion term in the macroscopic density equation, we investigate two thermodynamically non-equilibrium fluid situations: pure heat conduction, and sound wave dispersion.

Conclusion

Two gas flow configurations have been investigated in the context of non-continuum and non-local-equilibrium behaviour. A stochastic kinetic model and associated set of continuum-fluid equations have been derived in order to assess the relationship between spatial stochasticity and non-continuum gas behaviour that the traditional Navier–Stokes–Fourier model does not account for. In contrast to the conventional Boltzmann kinetic equation, our new kinetic equation incorporates a microscopic

Acknowledgements

The authors would like to thank the Royal Academy of Engineering and the Leverhulme Trust, in the UK for funding through a Senior Research Fellowship that enabled the completion of this research.

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Spatial stochasticity and non-continuum effects in gas flows

Abstract

Highlights

Introduction

Section snippets

The Brownian model in position space

Effects of the additional dissipative mass/volume flux

Conclusion

Acknowledgements

Physics Letters A

Journal of Fluid Mechanics

Theory and Application of the Boltzmann Equation

Journal of Physics A

Stochastic Ordinary and Stochastic Partial Differential Equations

Classical and Quantum Gravity

Journal of Statistical Physics

Journal of Mathematical Physics

Review of Modern Physics

Journal of the Physical Society of Japan