Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic <Emphasis Type="Italic">S</Emphasis>-Model

The nonisothermal steady rarefied gas flow driven by a given pressure gradient (Poiseuille flow) or a temperature gradient (thermal creep) in a long channel (pipe) of an arbitrary cross section is studied on the basis of the linearized kinetic S-model. The solution is constructed using a high-order accurate conservative method. The numerical computations are performed for a circular pipe and for a cross section in the form of a regular polygon inscribed in a circle. The basic characteristic of interest is the gas flow rate through the channel. The solutions are compared with previously known results. The flow rates computed for various cross sections are also compared with the corresponding results for a circular pipe.     

Kinetic analysis of the isothermal flow in a long rectangular microchannel

The linearized kinetic BGK model is used to study the steady Poiseuille flow of a rarefied gas in a long channel of rectangular cross section. The solution is constructed using the finite-volume method based on a TVD scheme. The basic computed characteristic is the mass flow rate through the channel. The effect of the relative width of the cross section is examined, and the difference of the solution from the one-dimensional flow between infinite parallel plates is analyzed. The numerical solution is compared to available results and to the analytical solution of the Navier-Stokes equations with no-slip and slip boundary conditions. The limits of applicability of the hydrodynamic solution are established depending on the degree of rarefaction of the flow and on the ratio of the side lengths of the channel cross section.     

Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.     

Unsteady rarefied gas flow in a microchannel driven by a pressure difference

The kinetic S-model is used to study the unsteady rarefied gas flow through a plane channel between two parallel infinite plates. Initially, the gas is at rest and is separated by the plane x = 0 with different pressure values on opposite sides. The gas deceleration effect of the channel walls is studied depending on the degree of gas rarefaction and the initial pressure drop, assuming that the molecules are diffusely reflected from the boundary. The decay of the shock wave and the disappearance of the uniform flow region behind the shock wave are monitored. Special attention is given to the gas mass flux through the cross section at x = 0, which is computed as a function of time. The asymptotic behavior of the solution at unboundedly increasing time is analyzed. The kinetic equation is solved numerically by applying a conservative finite-difference method of second-order accuracy in space.     

Expansion of a Rarefied Gas Cloud in a Vacuum: Asymptotic Treatment

The unsteady expansion of a rarefied gas of finite mass in an unlimited space is studied. The long-time asymptotic behavior of the solution is examined at Knudsen numbers tending to zero. An asymptotic analysis shows that, in the limit of small Knudsen numbers, the behavior of the macroscopic parameters of the expanding gas cloud at long times (i.e., for small density values) has nothing to do with the free-molecular or continuum flow regimes. This conclusion is unexpected and not obvious, but follows from a uniformly suitable solution constructed by applying the method of outer and inner asymptotic expansions. In particular, the unusual temperature behavior is of interest as applied to remote sensing of rocket exhaust plumes.     

Rarefied gas flow through a pipe of variable square cross section into vacuum

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.     

Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall

The paper deals with a fluid-structure interaction problem. A non steady-state viscous flow in a thin channel with an elastic wall is considered. The problem contains two small parameters: one of them is the ratio of the thickness of the channel to its length (i.e., to the period in the case of periodic solution); the second is the ratio of the linear density to the stiffness of the wall. For various ratios of these two small parameters, an asymptotic expansion of a periodic solution is constructed and justified by a theorem on the error estimates. To this end we prove the auxiliary results on existence, uniqueness, regularity of solution and some a priori estimates. The leading terms of the asymptotic solution are compared to the Poiseuille flow in a channel with absolutely rigid walls. In critical case a non-standard sixth order equation for the wall displacement is obtained.     

Asymptotic decay of solutions of the liouville equation under perturbations

We consider the Cauchy problem for a perturbed Liouville equation. An asymptotic solution is constructed with respect to the perturbation parameter by the two-scale expansion method; this construction can be applied over long time intervals. The main result is the definition of a deformation of the leading term of the asymptotic expansion within a slow time scale. Translated frommatematicheskie Zametki, Vol. 68, No. 2, pp. 195–209, August, 2000.     

Computation of rarefied diatomic gas flows through a plane microchannel

A numerical method based on a model kinetic equation was developed for computing diatomic rarefied gas flows in two dimensions. Nitrogen flows through a plane microchannel were computed, and the gas flow rate was constructed as a function of the Knudsen number for various channel lengths.     

Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe

The nonsteady Navier–Stokes equations are considered in a thin infinite pipe with the small diameter ? in the case of the Reynolds number of order ?. The time-dependent flow rate is a given function. The complete asymptotic expansion is constructed and justified. The error estimate of order O(? ^J ) for the difference of the exact solution and the J-th asymptotic approximation is proved for any real J.     

An analytical method for solving the problem of unshocked conical gas compression

The two-dimensional unsteady self-similar problem of unlimited unshocked conical compression of a gas is investigated. A solution is constructed in the form of a characteristic series in the domain bounded by a weak discontinuity and the sonic perturbation front. A recursion system of ordinary differential equations is obtained for the coefficients. A boundary-value problem corresponding to the next approximation is investigated in detail, a fundamental system of solutions is found by analytical methods and its asymptotic behaviour is investigated. Essentially independent solutions are determined and different methods are used to seek a solution of the inhomogeneous equation with the required asymptotic behaviour. An algorithm is constructed to compute gas flows induced by the motion of a piston taking the first terms of the series into consideration. The results are compared with those of computations carried out using the method of characteristics.     

Continuum models of rarefied gas flows in problems of hypersonic aerothermodynamics

The limits of applicability of continuum flow models in the problem of the hypersonic rarefied gas flow over blunt bodies are determined by an asymptotic analysis of the Navier–Stokes equations, the numerical solution of the viscous shock layer equations and the numerical and asymptotic solution of the thin viscous shock layer equations for low Reynolds numbers. It is shown that the thin viscous shock layer model gives correct values of the skin friction coefficient and the heat transfer coefficient in the transitional to free-molecule flow regime. The asymptotic solutions, the numerical solutions obtained within the framework of different continuum models, and the results of a calculation by Direct Simulation Monte Carlo method are compared.     

Gas-dynamic equations for spatially inhomogeneous gas mixtures with internal degrees of freedom. I. General theory

A general algorithm for building a uniform asymptotic solution of the kinetic equations for spatially inhomogeneous reactive gas mixtures is proposed. It solves the problem of irregular asymptotic solution arising in the ordinary Chapman–Enskog method, providing expressions for chemical reaction rates that agree with the mono-molecular reaction theory. We study a quasi-stationary behavior of the system, characterized by the slowly varying gas-dynamic variables which number is greater than the number of integral invariants of the collision operator. The gas-dynamic equations for reacting and relaxing gas mixtures are derived in general form. It is shown that accurate treatment of non-equilibrium processes gives rise to additional terms caused by the strong influence of small perturbations of quasi-equilibrium distribution functions on the kinetics of high-threshold physical and chemical processes. These terms are describing the influence of inelastic collisions, expansion/compression processes and spatial non-uniformity of gas-dynamic variables.     

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

The similarity solution for a bore on a beach

A similarity solution is found for the asymptotic behavior of a bore as it approaches the shoreline on a sloping beach. This gives direct confirmation of earlier results on the motion of the bore and adds details of the associated flow field. It also makes explicit the analogy with Guderley's implosion problem in gas dynamics; the solution is constructed closely following Guderley's arguments.     

Weak Nonlinear Long Waves in Channel Flow with Internal Dissipation

This article concerns the evolution of long waves ( O (ε^−1/2) wavelength) of small [ O (ε)] amplitude in channel flow with internal dissipation. We use multiple scale expansions to derive a generalized Kuramoto–Sivashinsky (GKS) equation that governs the dominant asymptotic solution in the limit of small disturbances and marginal linear instability. We compare this solution with numerical integrations of the full quasilinear system, and show that the error is consistent with an asymptotic solution to ε^3/2 over a time interval of order ε^−3/2.     

Asymptotic expansion of the solution to the convective diffusion problem in the wake of a spherical particle

The exterior boundary value problem of steady-state diffusion around a spherical particle placed in a Stokes flow is considered at high Peclet numbers. A complete asymptotic expansion of the solution in the wake of the particle is constructed by the method of matched asymptotic expansions.     

Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials

Solutions exhibiting an internal layer structure are constructed for a class of nonlinear singularly perturbed boundary value problems with translation invariant potentials. For these problems, a routine application of the method of matched asymptotic expansions fails to determine the locations of the internal layer positions. To overcome this difficulty, we present an analytical method that is motivated by the work of Kath, Knessl and Matkowsky [4]. To construct a solution having n internal layers, we first linearize the boundary value problem about the composite expansion provided by the method of matched asymptotic expansions. The eigenvalue problem associated with the homogeneous form of this linearization is shown to have n exponentially small eigenvalues. The condition that the solution to the linearized problem has no component in the subspace spanned by the eigenfunctions corresponding to these exponentially small eigenvalues determines the internal layer positions. These “near” solvability conditions yield algebraic equations for the internal layer positions, which are analyzed for various classes of nonlinearities.     

Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations

The main purpose of this paper is to study the asymptotic equivalence of the Boltzmann equation for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t=0) as ?→0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. In this setting, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat conductivity (depending on the viscosity) is also obtained.     

Singularly perturbed boundary value problems for a class of second order turning point on infinite interval

This article solved the asymptotic solution of a singularly perturbed boundary value problem with second order turning point, encountered in the dissipative equilibrium vector field of the coupled convection disturbance kinetic equations under the constrained filed and the gravity. Using the matching of asymptotic expansions, the formal asymptotic solution is constructed. By using the theory of differential inequality the uniform validity of the asymptotic expansion for the solution is proved.