We present a review of recent results obtained jointly with Carlo Cercignani. The review is based... more We present a review of recent results obtained jointly with Carlo Cercignani. The review is based on series of our publications in “Journal of Statistical Physics” in 2002–2003. We consider the spatially homogeneous non‐linear Boltzman equation (BE) for elastic and inelastic Maxwell models. Then it is eas to show at the formal level that the equation admits a class of self‐similar solutions of the form f(v,t)=A(t)F(vexp(−at)), a=const. Two main questions are: (a)does such non‐negative solutions really exist? and (b) do they describe a large time asymptotics for some classes of the initial data (self‐similar asymptotics)? The answer to both questions is “yes”. Moreover, we constructed two such solutions for the classical (elastic) BE in explicit form.For the same equation, we proved the existence of such solutions for any a>0 and that they are asymptotic states for some (different for different a) classes of the initial data having infinite second moment (energy). We note that the...
The methods of qualitative theory of dynamical systems are used to provide new information of the... more The methods of qualitative theory of dynamical systems are used to provide new information of the Navier–Stokes solutions for gas flows driven by evaporation and condensation at interphase surfaces. It is shown that these solutions correspond to separatrixes of the saddle point in the (u,T) plane. The classification of solutions is given, and some special cases are studied in detail. The qualitative methods are applied to the problem of evaporation/condensation between two plates. It is shown that the topology of the saddle point implies the following situation: one of the functions u(x) (velocity) or T(x) (temperature) is always non-monotone for sufficiently small Knudsen number. This explains some previously published numerical results. The case of finite Reynolds number is considered separately, and relationship with the kinetic-theory results is discussed throughout.
We propose a kinetic model of the aggregation process in a system consisting of two different typ... more We propose a kinetic model of the aggregation process in a system consisting of two different types of particles. Aggregating particles (cells) are polyvalent and bear on the surface a huge number of binding sites for the other type of particles, ligands. The ligand is bivalent and has two identical active sites for binding to cells. The cross-linking of the cells by the ligands causes the aggregation phenomenon called agglutination. We obtained the analytical solution of this model task describing the time dependence of the aggregate mean size versus the composition of the system. The comparison of the analytical solution with the experimental data for the agglutination of bacterial cells by bivalent antibodies shows that the main factors affecting agglutination were correctly taken into account.
It is well-known that the classical Chapman-Enskog procedure does not work at the level of Burnet... more It is well-known that the classical Chapman-Enskog procedure does not work at the level of Burnett equations (the next step after the Navier-Stokes equations). Roughly speaking, the reason is that the solutions of higher equations of hydrodynamics (Burnett's, etc.) become unstable with respect to short-wave perturbations. This problem was recently attacked by several authors who proposed different ways to deal with it. We present in this paper one of possible alternatives. First we deduce a criterion for hyperbolicity of Burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases. Then we discuss in more detail the problem of truncation of the Chapman-Enskog expansion and show that the way of truncation is not unique. The general idea of changes of coordinates (based on analogy with the theory of dynamical systems) leads finally to nonlinear Hyperbolic Burnett Equations (HBEs) without using any information beyond the classical Burnett equations. It is proved that HBEs satisfy the linearized H-theorem. The linear version of the problem is studied in more detail, the complete Chapman-Enskog expansion is given for the linear case. A simplified proof of the Slemrod identity for Burnett coefficients is also given.
ABSTRACT We consider the general problem of the construction of discrete kinetic models (DKMs) wi... more ABSTRACT We consider the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem was first stated by Gatignol in connection with discrete models of the Boltzmann equation (BE) and it has been addressed in the last decade by several authors. Even though a practical criterion for the non-existence of spurious conservation laws has been devised, and a method for enlarging existing physical models by new velocity points without adding non-physical invariants has been proposed, a general algorithm for the construction of all normal (physical) discrete models with assigned conservation laws, in any dimension and for any number of points, is still lacking in the literature. We introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal DKMs. In particular, it is proved that for any given dimension d≥2 and for any sufficiently large numberN of velocities (for example, N≥6 for the planar case d=2) there exists just a finite number of distinct classes of DKMs. We apply the general method in the particular cases of discrete velocity models (DVMs) of the inelastic BE and elastic BE. Using our general approach to DKMs and our results on normal DVMs for a single gas, we develop a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures. We call such models supernormal models (SNMs) (they have the property that by isolating the velocities of single gases involved in the mixture, we also obtain normal DVMs).
We present a review of recent results obtained jointly with Carlo Cercignani. The review is based... more We present a review of recent results obtained jointly with Carlo Cercignani. The review is based on series of our publications in “Journal of Statistical Physics” in 2002–2003. We consider the spatially homogeneous non‐linear Boltzman equation (BE) for elastic and inelastic Maxwell models. Then it is eas to show at the formal level that the equation admits a class of self‐similar solutions of the form f(v,t)=A(t)F(vexp(−at)), a=const. Two main questions are: (a)does such non‐negative solutions really exist? and (b) do they describe a large time asymptotics for some classes of the initial data (self‐similar asymptotics)? The answer to both questions is “yes”. Moreover, we constructed two such solutions for the classical (elastic) BE in explicit form.For the same equation, we proved the existence of such solutions for any a>0 and that they are asymptotic states for some (different for different a) classes of the initial data having infinite second moment (energy). We note that the...
The methods of qualitative theory of dynamical systems are used to provide new information of the... more The methods of qualitative theory of dynamical systems are used to provide new information of the Navier–Stokes solutions for gas flows driven by evaporation and condensation at interphase surfaces. It is shown that these solutions correspond to separatrixes of the saddle point in the (u,T) plane. The classification of solutions is given, and some special cases are studied in detail. The qualitative methods are applied to the problem of evaporation/condensation between two plates. It is shown that the topology of the saddle point implies the following situation: one of the functions u(x) (velocity) or T(x) (temperature) is always non-monotone for sufficiently small Knudsen number. This explains some previously published numerical results. The case of finite Reynolds number is considered separately, and relationship with the kinetic-theory results is discussed throughout.
We propose a kinetic model of the aggregation process in a system consisting of two different typ... more We propose a kinetic model of the aggregation process in a system consisting of two different types of particles. Aggregating particles (cells) are polyvalent and bear on the surface a huge number of binding sites for the other type of particles, ligands. The ligand is bivalent and has two identical active sites for binding to cells. The cross-linking of the cells by the ligands causes the aggregation phenomenon called agglutination. We obtained the analytical solution of this model task describing the time dependence of the aggregate mean size versus the composition of the system. The comparison of the analytical solution with the experimental data for the agglutination of bacterial cells by bivalent antibodies shows that the main factors affecting agglutination were correctly taken into account.
It is well-known that the classical Chapman-Enskog procedure does not work at the level of Burnet... more It is well-known that the classical Chapman-Enskog procedure does not work at the level of Burnett equations (the next step after the Navier-Stokes equations). Roughly speaking, the reason is that the solutions of higher equations of hydrodynamics (Burnett's, etc.) become unstable with respect to short-wave perturbations. This problem was recently attacked by several authors who proposed different ways to deal with it. We present in this paper one of possible alternatives. First we deduce a criterion for hyperbolicity of Burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases. Then we discuss in more detail the problem of truncation of the Chapman-Enskog expansion and show that the way of truncation is not unique. The general idea of changes of coordinates (based on analogy with the theory of dynamical systems) leads finally to nonlinear Hyperbolic Burnett Equations (HBEs) without using any information beyond the classical Burnett equations. It is proved that HBEs satisfy the linearized H-theorem. The linear version of the problem is studied in more detail, the complete Chapman-Enskog expansion is given for the linear case. A simplified proof of the Slemrod identity for Burnett coefficients is also given.
ABSTRACT We consider the general problem of the construction of discrete kinetic models (DKMs) wi... more ABSTRACT We consider the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem was first stated by Gatignol in connection with discrete models of the Boltzmann equation (BE) and it has been addressed in the last decade by several authors. Even though a practical criterion for the non-existence of spurious conservation laws has been devised, and a method for enlarging existing physical models by new velocity points without adding non-physical invariants has been proposed, a general algorithm for the construction of all normal (physical) discrete models with assigned conservation laws, in any dimension and for any number of points, is still lacking in the literature. We introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal DKMs. In particular, it is proved that for any given dimension d≥2 and for any sufficiently large numberN of velocities (for example, N≥6 for the planar case d=2) there exists just a finite number of distinct classes of DKMs. We apply the general method in the particular cases of discrete velocity models (DVMs) of the inelastic BE and elastic BE. Using our general approach to DKMs and our results on normal DVMs for a single gas, we develop a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures. We call such models supernormal models (SNMs) (they have the property that by isolating the velocities of single gases involved in the mixture, we also obtain normal DVMs).
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Papers by A. Bobylev