Browsing by Author "Benilov, E. S."
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- Energy conservation and H theorem for the Enskog-Vlasov equationPublication . Benilov, E. S.; Benilov, M. S.The Enskog-Vlasov (EV) equation is a widely used semiphenomenological model of gas-liquid phase transitions. We show that it does not generally conserve energy, although there exists a restriction on its coefficients for which it does. Furthermore, if an energy-preserving version of the EV equation satisfies an H theorem as well, it can be used to rigorously derive the so-called Maxwell construction which determines the parameters of liquid-vapor equilibria. Finally, we show that the EV model provides an accurate description of the thermodynamics of noble fluids, and there exists a version simple enough for use in applications.
- Existence and stability of regularized shock solutions, with applications to rimming flowsPublication . Benilov, E. S.; Benilov, M. S.; O’Brien, S. B. G.This paper is concerned with regularization of shock solutions of nonlinear hyperbolic equations, i.e., introduction of a smoothing term with a coefficient ε, then taking the limit ε → 0. In addition to the classical use of regularization for eliminating physically meaningless solutions which always occur in non-regularized equa tions (e.g. waves of depression in gas dynamics), we show that it is also helpful for stability analysis. The general approach is illustrated by applying it to rimming flows, i.e., flows of a thin film of viscous liquid on the inside of a horizontal rotating cylinder, with or without surface tension (which plays the role of the regularizing effect). In the latter case, the spectrum of available linear eigenmodes appears to be continuous, but in the former, it is discrete and, most importantly, remains discrete in the limit of infinitesimally weak surface tension. The regularized (discrete) spectrum is fully determined by the point where the velocity of small perturbations vanishes, with the rest of the domain, including the shock region, being unimportant.
- Kinetic approach to condensation: diatomic gases with dipolar moleculesPublication . Benilov, E. S.; Benilov, M. S.We derive a kinetic equation for rarefied diatomic gases whose molecules have a permanent dipole moment. Estimating typical parameters of such gases, we show that quantum effects cannot be neglected when describing the rotation of molecules, which we thus approximate by quantum rotators. The intermolecular potential is assumed to involve an unspecified short-range repulsive component and a long-range dipole-dipole Coulomb interaction. In the kinetic equation derived, the former and the latter give rise, respectively, to the collision integral and a self-consistent electric field generated collectively by the dipoles (as in the Vlasov model of plasma). It turns out that the characteristic period of the molecules’ rotation is much shorter than the time scale of the collective electric force and the latter is much shorter than the time scale of the collision integral, which allows us to average the kinetic equation over rotation. In the averaged model, collisions and interaction with the collective field affect only those rotational levels of the molecules that satisfy certain conditions of synchronism. It is then shown that the derived model does not describe condensation; i.e., permanent dipoles of molecules cannot exert the level of intermolecular attraction necessary for condensation. It is argued that an adequate model of condensation must include the temporary dipoles that molecules induce on each other during interaction, and that this model must be quantum, not classical.
- Peculiar property of noble gases and its explanation through the Enskog--Vlasov modelPublication . Benilov, E. S.; Benilov, M. S.An observation is presented that the ratio of the critical and triple-point temperatures Tcr/Ttp of neon, argon, krypton, and xenon fit within a narrow interval, Tcr/Ttp = 1.803 ± 0.5%, and the same applies to the density ratio, ncr/ntp = 0.3782 ± 1.7% (of the two remaining noble gases, helium does not have a triple point and, for radon, ntp is unknown). We explain this peculiar property by the fact that the molecules of noble gases are nearly spherical, as a result of which they satisfy the Enskog-Vlasov (EV) kinetic model based on the approximation of hard spheres. The EV model has also allowed us to identify two more parameter combinations which are virtually the same for all noble gases.
- Semiphenomenological model for gas-liquid phase transitionsPublication . Benilov, E. S.; Benilov, M. S.We examine a rarefied gas with inter-molecular attraction. It is argued that the attraction force amplifies random density fluctuations by pulling molecules from lower-density regions into high-density regions and thus may give rise to an instability. To describe this effect, we use a kinetic equation where the attraction force is taken into account in a way similar to how electromagnetic forces in plasma are treated in the Vlasov model. It is demonstrated that the instability occurs when the temperature T is lower than a certain threshold value Ts depending on the gas density. It is further shown that, even if T is only marginally lower than Ts, the instability generates clusters with density much higher than that of the gas. These results suggest that the instability should be interpreted as a gas-liquid phase transition, with Ts being the temperature of saturated vapor and the high-density clusters representing liquid droplets.
- Steady rimming flows with surface tensionPublication . Benilov, E. S.; Benilov, M. S.; Kopteva, N.We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the ‘rising’ side of the cylinder and, for large ones, pools at the cylinder’s bottom. The paper examines rimming flows with a pool affected by weak surface tension. Using the lubrication approximation and the method of matched asymptotics, we find a solution describing the pool, the ‘outer’ region, and two transitional regions, one of which includes a variable (depending on the small parameter) number of asymptotic zones.
- The Enskog–Vlasov equation: a kinetic model describing gas, liquid, and solidPublication . Benilov, E. S.; Benilov, M. S.The Enskog–Vlasov (EV) equation is a semi-empiric kinetic model describing gas–liquid phase transitions. In the framework of the EV equation, these correspond to an instability with respect to infinitely long perturbations, developing in a gas state when the temperature drops below (or density rises above) a certain threshold. In this paper, we show that the EV equation describes one more instability, with respect to perturbations with a finite wavelength and occurring at a higher density. This instability corresponds to fluid-solid phase transition and the perturbations’ wavelength is essentially the characteristic scale of the emerging crystal structure. Thus, even though the EV model does not describe the fundamental physics of the solid state, it can ‘mimic’ it—and, thus, be used in applications involving both evaporation and solidification of liquids. Our results also predict to which extent a pure fluid can be overcooled before it definitely turns into a solid.