Existence of homoenergetic affine flows for the Boltzmann equation

An existence theorem is proved for homoenergetic affine flows described by the Boltzmann equation. The result complements the analysis of Truesdell and of Galkin on the moment equations for a gas of Maxwellian molecules. Existence of the distribution function is established here for a large class of molecular models (hard sphere and angular cut-off interactions). Some of the data lead to an implosion and infinite density in a finite time, in agreement with the physical picture of the associated flows; for the remaining set of data, global existence is shown to hold.     

On the properties of elastic-viscous fluid flow past an obstacle

Fundamental equations for the analysis of plane flow of elastic-viscous fluid are established.On such a basis,a perturbed-weighted residual finite element model for small Deborah number situations is formulated.The model is further incorporated for investigations on the behavioral characteristics of the elastic-viscous fluid flow when passing an obstacle,which include the mechanisms of the retardation of separation point,and the reduction of drag forces and so forth.The numerical investigations demonstrate the favorable advantages of the present model in its remarkable simplicity and reasonable accuracy attained in plane flow analysis.     

Separation zone in fluctuating flow past an obstacle in a channel

The results of an experimental investigation of hydrodynamic processes in separated turbulent flows in the presence of superimposed flow-rate fluctuations are presented. A sharp shortening of the separation zone in the fluctuating flow is found to exist in the vicinity of Sh = 1. The dependence of the separation zone length on the superimposed fluctuation frequency is the same in the cases in which the obstacle is located in the regions of antinodes of both the flow velocity and the pressure.     

Computer-aided analysis of flow past a surface-mounted obstacle

The incompressible, laminar, isothermal flow of a Newtonian fluid at steady state past a surface-mounted obstacle (flow over a step) is studied in a two-dimensional numerical experiment using the Galerkin finite element method. The dimensionless Navier–Stokes equations are solved in the whole range of the laminar flow regime. The numerical predictions are compared with available experimental data. The emphasis in the discussion of the results is on the presentation of the streamlines for various Reynolds numbers, the pressure distribution over and downstream of the step, the shear stress distribution along the surface of the step and the length of the recirculation region as a function of the Reynolds number. This analysis may be used in numerous applications from agricultural to civil, mechanical and chemical engineering. © 1997 John Wiley & Sons, Ltd.     

Complementary modes of impulse generation for flow past a three-dimensional obstacle

A Lagrangian model for viscous incompressible flow past a stationary solid obstacle is developed in terms of the creation over successive instants of time of fluid impulse at the obstacle’s bounding surface. Upon creation this compactly supported vector field actively evolves into the flow interior according to its equation of motion. At the wall, distinct creation modes can be identified by specifying two exterior Poisson problems, distinguished from each other by complementary gradient boundary conditions. We demonstrate how in each case the relevant Poisson problem is to be solved numerically. Impulse generation is modeled in Lagrangian terms for each mode in turn for the case of flow past a sphere. (In the spherical geometry we exploit image theory to simplify some aspects of the computation.) We argue that the concurrent existence of these two distinct modes provides an insight into the complex phenomenology of turbulent wake formation.     

Steady flow around a plane cascade by an ideal gas

The steady separation-free flow around a flat cascade by an ideal gas is discussed. Most of the attention is devoted to blocking regimes with a supersonic velocity in the entire flow and its subsonic component normal to the front of the cascade. A directing action of the cascade (the direction of the velocity and the Mach number of the advancing flow turn out to be related) is exhibited in these regimes which is a consequence of an independence of the flow in front of the cascade of the conditions behind it [1–5]. The most widespread method of their calculation [3, 4, 6] is based on the method of characteristics with establishment of the flow outside the cascade in a timelike coordinate. Although the integrated conservation laws also permit finding the parameters at infinity, the numerical construction of as long-range fields as desired with periodic sequences of attenuating discontinuities is practically impossible. The approximation of nonlinear acoustics (ANA) [7, 8] is justified here, as it is very effective in such problems [8–12]. A combination of ANA, the integrated conservation laws, and establishment in a calculation according to [13, 14] with isolation of the discontinuities has been realized in [5] for the construction of a solution on the entrance section of a cascade and everywhere in front of it. Below the method of [5] is extended to the entire flow and simplified even more. The flow on the entrance section of the cascade is, just as in [3], found in the approximation of a simple wave, in the rest of it and in a finite strip behind it-the flow is found with the help of the straight-through version of the scheme of [13, 14], and in the long-range field-in the ANA. A simpler version is proposed. In it ANA is applied outside the cascade and the linear theory is applied inside the cascade. Examples of the calculations are given. Similarity laws are formulated for all the regimes of streamline flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 35–43, November–December, 1984.     

New results on the semidiscrete Boltzmann equation for a binary gas mixture

Summary A plane semidiscrete model of the Boltzmann equation for a binary gas mixture with molecular collisions ruled by the hard-spheres interaction potential is described. After establishing a model, a theorem demostrating the global existence of mild solutions of the initial-value problem is given and the propagation of unidimensional shock waves examined.

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Sommario Si propone un modello semidiscreto piano dell'equazione di Boltzmann per una miscela binaria con collisioni molecolari soggette al potenziale di interazione delle sfere rigide. Costruito il modello, si dà un teorema di esistenza globale di soluzioni generalizzate per il problema di Cauchy, e si analizza la propagazione di onde d'urto unidimensionali.

     

Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation

In this paper, we consider a dissipative-dispersive nonlinear equation appliable to many physical phenomena. Using the geometric singular perturbation method based on the theory of dynamical systems, we investigate the existence of its traveling wave solutions with the dissipative terms having sufficiently small coefficients. The results show that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ordinary differential equations （ODEs）. Then, we use the Melnikov method to establish the existence of a homoclinic orbit in this manifold corresponding to a solitary wave solution of the equation. Furthermore, we present some numerical computations to show the approximations of such wave orbits.     

Nonlinearity caused by nonstationary perturbations of ideal jet flow past an obstacle

The problem is analyzed in the small perturbation theory approximation. It is shown that the dynamic boundary condition nonlinearity related to the presence of an inertial pressure force results in the flow velocity field becoming discontinuous and, as a consequence, in the appearance of infinite-sheeted spirals on free boundaries. The moment of onset of the discontinuity and its propagation law are found. It is shown that hydrodynamic forces can be calculated using formulas derived on the basis of a linear approximation.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 72–78, September–October, 1995.     

Construction of solutions of the Boltzmann kinetic equation in a Knudsen layer

We consider the moment equation method for solving the Boltzmann equation in a Knudsen layer; the calculation of one of the moments of the collision integral is presented.     

Nonlinear nonequilibrium kinetic model of the Boltzmann equation for a gas with power-law interaction

A model kinetic equation approximating the Boltzmann equation on a wide range of the intensities of nonequilibrium states of gases is derived to describe rarefied gas flows. The kinetic model is based on a distribution function dependent on the absolute velocity of gas particles. Themodel kinetic equation possesses a high computational efficiency and the problem of shock wave structure is solved on its basis. The calculated and experimental data for argon are compared.     

Similarity solutions for the flow behind an exponential shock in a non-ideal gas

Similarity solutions for the flow of a non-ideal gas behind a strong exponential shock driven out by a piston (cylindrical or spherical) moving with time according to an exponential law are obtained. Similarity solutions exist only when the surrounding medium is of constant density. Solutions are obtained, in both the cases, when the flow between the shock and the piston is isothermal or adiabatic. It is found that the assumption of zero temperature gradient brings a profound change in the density distribution as compare to that of the adiabatic case. Effects of the non-idealness of the gas on the flow-field between the shock and the piston are investigated. The variations of density-ratio across the shock and the location of the piston with the parameter of non-idealness of the gas are also obtained.     

Existence and regularity for solutions of the Korteweg-de Vries equation

The construction suggested by an inverse-scattering analysis establishes the existence of solutions u(x, t) of the Korteweg-de Vries equation subject to an initial condition u(x, 0)=U(x), where U has certain regularity and decay properties. It is assumed that UC³(), that U is piecewise of class C ⁴, and that U ^(j) decays at an algebraic rate for j4. The faster the decay of U ^(j) the smoother the solution will be for t0. If U and its first four derivatives decay faster than ¦x¦^–n for all n, then the solution will be infinitely differentiable for t0. For t>0, the decay rate of u(x, t) as x + increases with the decay rate of U; but the decay rate as x - depends on the regularity of U. A solution u ₁ of the Korteweg-de Vries equation such that u ₁(·, 0)C() may fail to remain in class C for all time if u ₁(x, 0) does not decay fast enough as ¦x¦.This research was performed in part as a Visiting Member of the Courant Institute of Mathematical Science.     

Numerical solutions for the flow of a fluid of grade three past an infinite porous plate

This paper presents a numerical study of the flow of an incompressible fluid of grade three past an infinite porous flat plate, subject to suction at the plate. This flow is governed by a non-linear differential equation that is particularly well suited to demonstrate the power and usefulness of different numerical techniques. In this work, the numerical solutions are obtained using a Runge-Kutta method of fourth order. The accuracy of the method for this problem is demonstrated.     

Change of regime in supersonic flow past an obstacle preceded by a thin channel of reduced density

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 146–151, September–October, 1989.     

Similarity analysis and exact solutions for a general discrete two-velocity model of Boltzmann equation

Summary This paper concerns with the similarity analysis for a general discrete two-velocity model of the Boltzmann equation introduced by Illner [12]. We find the general groups of invariance and we get some exact solutions, recovering general results which contain either solutions extensively described in the literature or undiscovered ones.

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Sommario In questa nota si applica l'analisi dei gruppi infinitesimi di trasformazione ad un modello generale discreto a due velocità dell'equazione di Boltzmann introdotto da Illner [12]. Si trovano i più generali gruppi di invarianza e si ottengono alcune soluzioni esatte, ritrovando risultati generali che contengono sia soluzioni ampiamente descritte in letteratura che nuove soluzioni.

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Work supported by the C.N.R. through the G.N.F.M.