IV. On the dynamical theory of gases

Maxwell, James Clerk

doi:10.1098/rstl.1867.0004

Cited by 1,742 publications

(271 citation statements)

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“…where the collision kernel (∂f k /∂t) coll drives the evolution of f k [7]. Applying the "molecular chaos" assumption -"velocities of colliding particles are uncorrelated, and independent of position" [40] -, the basis of the DSMC method is the ad hoc assumption that particle motion and collisions are decoupled over the small time-step ∆t [7]. During a DSMC "push" step, simulated macro-particles are moved ballistically over ∆t.…”

Section: The Direct Simulation Monte Carlo Methodsmentioning

confidence: 99%

pdFOAM: A PIC-DSMC code for near-Earth plasma-body interactions

et al. 2017

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Understanding the interaction of the near-Earth space environment with orbiting bodies is critical, both from a design and scientific perspective. In Low Earth Orbit (LEO), the interaction between the ionosphere and orbiting objects is well studied from a charging perspective. Not well understood is the effect of the ionosphere on the motion of LEO objects i.e. charged aerodynamics. This paper presents the implementation, validation, and verification of the hybrid electrostatic Particle-in-Cell (PIC) -Direct Simulation Monte Carlo (DSMC) code, pdFOAM, to study both the neutral and charged particle aerodynamics of LEO objects. The 2D aerodynamic interaction of a cylinder with a fixed uniform surface potential of −50 V and mesothermal O + and H + plasmas representative of ionospheric conditions is investigated. New insights into the role of bounded ion jets and their effect on surface forces are presented. O + bounded ion jets are observed to cause a 4.4% increase ion direct Charged Particle Drag (dCPD), while H + ion jets produce a net reduction in H + drag by 23.7% i.e. they cause a thrust force. As a result, we conclude that past work, primarily based on Orbital Motion Limited theory, does not adequately capture the physics of LEO charged aerodynamics. Hence, we recommend a revisit of conclusions regarding the significance of CPD to LEO objects -pdFOAM being an appropriate tool for this purpose.

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Section: The Direct Simulation Monte Carlo Methodsmentioning

confidence: 99%

pdFOAM: A PIC-DSMC code for near-Earth plasma-body interactions

et al. 2017

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“…-a, 05.60.Cd, 47.70. -n Linear flux-force dependences (such as Fick's law) are a special, limiting case of time-delayed equations [1], as first noticed by Maxwell [2]. Such equations arise, e.g., in time-delayed diffusion, which has been considered for many years [3][4][5] and applied to turbulent diffusion [3], optics in turbid media [6], x-ray bursters [7], etc.…”

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confidence: 99%

Time-Delayed Theory of the Neolithic Transition in Europe

1999

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The classical wave-of-advance model of the neolithic transition (i.e., the shift from hunter-gatherer to agricultural economies) is based on Fisher's reaction-diffusion equation. Here we present an extension of Einstein's approach to Fickian diffusion, incorporating reaction terms. On this basis we show that second-order terms in the reaction-diffusion equation, which have been neglected up to now, are not in fact negligible but can lead to important corrections. The resulting time-delayed model agrees quite well with observations. [ S0031-9007(98) [11,12]. Thus HRD equations, instead of the usual PRD equations, should be regarded as the first choice from a conceptual perspective. Hyperbolic reaction-diffusion equations have been very recently applied to the spread of epidemics [13], forest fire models [14], and chemical systems [15].An interesting application of PRD equations arose after archaeological data led to the conclusion that European farming originated in the Near East, from where it spread across Europe. The rate of this spread was measured [16], and a mathematical model was proposed according to which early farming expanded in the form of a PRD wave of advance [17]. Such a model provides a consistent explanation for the origin of Indo-European languages [18], and also finds remarkable support from the observed gene frequencies [19]. However, this PRD model predicts a velocity for the spread of agriculture that is higher than that inferred from archaeological evidence, provided that one accepts those values for the parameters in the model that have been measured in independent observations [17]. Here we will analyze this problem by means of a HRD model.Let p͑x, y, t͒ stand for the population density (measured in number of families per square kilometer), where x and y are Cartesian coordinates and t is the time. We assume that a well-defined time scale t between two successive migrations exists. We begin, as usual [20], noting that, between the values of time t and t 1 t, both migrations and population growth will cause a change in the number of families in an area differential ds dx dy, i.e., ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ds ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ m ds 1 ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ g ds ,where the subindices m and g stand for migrations and population growth, respectively. We denote the coordinate variations of a given family during t by Dx and Dy. The effect of migrations on the evolution of p͑x, y, t͒ will be derived here by means of a simple extension of Einstein's model of Fickian diffusion [21]. The migration term in Eq. (1) can be written as ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ m ds ds Z 12`Z 12`p ͑x 1 Dx, y 1 Dy, t͒f͑Dx, Dy͒dDx dDy 2 dsp͑x, y, t͒ ,where f͑Dx, Dy͒ is the fraction of those families lying at time t in an area ds, centered at ͑x 1 Dx, y 1 Dy͒, such that they are at time t 1 t in an area ds, centered at ͑x, y͒. Therefore, Z 12`Z 12`f ͑Dx, Dy͒dDx dDy 1 ,

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“…A similar relation was obtained by Maxwell (1867) in connection with the kinetic theory of gases, and was elaborated by Grad (1958). This relation, combined with the continuity equation (2.2), yields the well-known dissipating wave equation or telegrapher's equation…”

Section: Generalized Constitutive Equationmentioning

confidence: 84%

On oscillating flows in randomly heterogeneous porous media

Trefry

McLaughlin

Metcalfe

et al. 2010

Phil. Trans. R. Soc. A.

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The emergence of structure in reactive geofluid systems is of current interest. In geofluid systems, the fluids are supported by a porous medium whose physical and chemical properties may vary in space and time, sometimes sharply, and which may also evolve in reaction with the local fluids. Geofluids may also experience pressure and temperature conditions within the porous medium that drive their momentum relations beyond the normal Darcy regime. Furthermore, natural geofluid systems may experience forcings that are periodic in nature, or at least episodic. The combination of transient forcing, nearcritical fluid dynamics and heterogeneous porous media yields a rich array of emergent geofluid phenomena that are only now beginning to be understood. One of the barriers to forward analysis in these geofluid systems is the problem of data scarcity. It is most often the case that fluid properties are reasonably well known, but that data on porous medium properties are measured with much less precision and spatial density. It is common to seek to perform an estimation of the porous medium properties by an inverse approach, that is, by expressing porous medium properties in terms of observed fluid characteristics. In this paper, we move toward such an inversion for the case of a generalized geofluid momentum equation in the context of time-periodic boundary conditions. We show that the generalized momentum equation results in frequency-domain responses that are governed by a second-order equation which is amenable to numerical solution. A stochastic perturbation approach demonstrates that frequency-domain responses of the fluids migrating in heterogeneous domains have spatial spectral densities that can be expressed in terms of the spectral densities of porous media properties.

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IV. On the dynamical theory of gases

Cited by 1,742 publications

References 0 publications

pdFOAM: A PIC-DSMC code for near-Earth plasma-body interactions

pdFOAM: A PIC-DSMC code for near-Earth plasma-body interactions

Time-Delayed Theory of the Neolithic Transition in Europe

On oscillating flows in randomly heterogeneous porous media

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