Initial-boundary value problems for gas dynamics

Archive for Rational Mechanics and Analysis -     

Some geometric acoustic problems of one-dimensional unsteady and two-dimensional steady flows

The behavior of discontinuities (weak shocks) of the parameters of a disturbed flow and their interaction with the discontinuities of the basic flow in the geometric acoustics approximation, when the variation of the intensity of such shocks along the characteristics or the bicharacteristics is described by ordinary differential equations, has been investigated by many authors. Thus, Keller [1] considered the case when the undisturbed flow is three-dimensional and steady, and the external inputs do not depend on the flow parameters. An analogous study was made by Bazer and Fleischman for the MGD isentropic flow of an ideal conducting medium [2], while Lugovtsov [3] studied the three-dimensional steady flow of a gas of finite conductivity for small magnetic Reynolds numbers and no electric field. Several studies (for example, [4]) have considered the behavior of discontinuities of the solutions from the general positions of the theory of hyperbolic systems of quasilinear equations. Finally, the interaction of weak shocks (or the equivalent continuous disturbances) with shock waves was studied in [5–11].In what follows we consider one-dimensional (with plane, cylindrical, and spherical waves) and quasi-one-dimensional unsteady flows, and also plane and axisymmetric steady flows. Two problems are investigated: the variation of the intensity of weak shocks in the presence of inputs which depend on the stream parameters, and the interaction of weak shocks with strong discontinuities which differ from contact (tangential) discontinuities.The thermodynamic properties of the gas are considered arbitrary. We note that the resulting formulas for the interaction coefficients of the weak and strong discontinuities are also valid for nonequilibrium flow.     

A variational problem in one-dimensional unsteady gas dynamics

The problem of finding the optimal motion of a piston that confines a certain volume of gas at the initial time is solved. The motion with which the piston performs the maximum work subject to constraints on its motion and the duration of this motion is found. Cases of plane, cylindrical, and spherical symmetry are considered. Numerical examples are given. It is noted that there is some analogy between the solution obtained and the known solutions for two-dimensional supersonic optimal nozzles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 171–175, July–August, 1984.I thank A. N. Kraiko for helpful discussions of the work.     

The general and basic boundary value problems of plane steady jet flows and the corresponding nonlinear equations

The general boundary value problem, including known plane steady jet flows of an ideal incompressible fluid, is formulated. The simplest problem retaining all the specific features of the general problem, known as the basic problem, is separated from the general problem. The solution of the basic problem is reduced to solving a non-linear integro-differential equation and also to solving nonlinear integral equations. Examples of flows whose determination is reduced to' solving the basic problem are cited.     

Second-order effects in unsteady laminar compressible three-dimensional stagnation-point boundary layers

The flow, heat and mass transfer at the stagnation point of a three-dimensional body in unsteady laminar compressible fluid with variable properties have been studied using a second-order boundary-layer theory when the basic potential flow admits selfsimilarity. Both nodal and saddle point regions have been considered. The equations governing the flow have been solved numerically using an implicit finite-difference scheme. It is observed that the enhancement or reduction in the skin friction and heat transfer due to the second-order boundary layers depends upon the values of the parameter characterizing the unsteadiness in the free-stream velocity, the nature of the stagnation point, the variation of the density-viscosity product across the boundary layer, mass transfer and the wall temperature. The suction increases the skin friction and heat transfer whereas injection does the opposite.     

Specific gasdynamical features of spontaneous condensation in an unsteady rarefaction wave

On the basis of numerical modeling, the formation of an unsteady shock wave induced by a condensation shock in a rarefaction wave moving in the high-pressure channel of a shock tube filled with moist air is demonstrated. It is shown that in a fairly long channel a periodic structure consisting of an alternating sequence of condensation shocks and the shock waves they generate may be formed. This structure is a linear unsteady analog of the self-oscillation regime of type IV in the classification [1] for condensing medium flows in the subsonic section of a Laval nozzle. The specific features detected are important for planning and interpreting experiments aimed at investigating spontaneous condensation using a “condensation shock tube”.     

Extremum problems of boundary control for steady equations of thermal convection

An inverse extremum problem of boundary control for steady equations of thermal convection is considered. The cost functional in this problem is chosen to be the root-mean-square deviation of flow velocity or vorticity from the velocity or vorticity field given in a certain part of the flow domain; the control parameter is the heat flux through a part of the boundary. A theorem on sufficient conditions on initial data providing the existence, uniqueness, and stability of the solution is given. A numerical algorithm of solving this problem, based on Newton’s method and on the finite element method of discretization of linear boundary-value problems, is proposed. Results of computational experiments on solving extremum problems, which confirm the efficiency of the method developed, are discussed.     

Positive solutions of three-point boundary value problems

In this paper, we consider existence of single or multiple positive solutions of three-point boundary value problems involving one-dimensional p-Laplacian. We then study existence of solutions when the problems are in resonance cases. The proposed approach is based on the Krasnoselskii＇s fixed point theorem and the coincidence degree.     

TWO POINTS BOUNDARY VALUE PROBLEMS IN BANACH SPACES

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Initial and boundary value problems of hyperbolic heat conduction

This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux , where e and are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density e is solved by a new kinetic approach that was introduced and evaluated by Dreyer and Kunik in [1], [2] and Pertame [3]. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in [4]. The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields as well as integrals that need knowledge on energy and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by continuity conditions, it turns out that after some very short time the energy density and heat flux are related to the initial data according to the Rankine Hugoniot relation. Received October 6, 1998     

Quasi-one-dimensional approximation in two-dimensional problems of gas dynamics

A useful means of constructing approximate flow models is the hydraulic (for two-dimensional problems quasi-one-dimensional) approach, based on averaging the initial nonuniform flows over some direction or cross section [1]. In this case, at the expense of a rougher model it is possible to reduce the dimensionality of the problem. Here, this approach is extended to unsteady two-dimensional gas-dynamic processes; certain problems (flow around a cone or a blunt body, jet flows) are considered in the framework of the quasi-one-dimensional model obtained, and results are compared with the solutions of the corresponding two-dimensional problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 136–143, March–April, 1989.     

Fractional boundary value problems with singularities in space variables

We are concerned with the existence of solutions for the singular fractional boundary value problem $^{c}\kern-1pt D^{\alpha}u = f(t,u)$ , u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(??{0})) and lim _x→0 f(t,x)=∞ for all t∈[0,1]. Here, $^{c}\kern-1pt D$ is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of (0,1), that is, they “pass through” the singularity of f inside of (0,1). The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.     

Nonlinear oscillations and boundary value problems for Hamiltonian systems

We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms $$\begin{gathered} \dot x(t) = H'_p (t, x(t), p(t)) + g(t), \hfill \\ \dot p(t) = - H'_x (t, x(t), p(t)) - f(t). \hfill \\ \end{gathered} $$ Among these problems is the existence of T-periodic solutions, namely those satisfying x(t+T)=x(t) and p(t+T)+p(t). A special study is made of the classical case, where H(x, p)=1/2 |p|²+V(x). In the case of parametric oscillations, where (f, g)=(0, 0) and t ? H(t, x, p) is T-periodic, we give a lower bound for the true (minimal) period of the T-periodic solution (x, p) produced by our method, and we prove the existence of an infinite number of subharmonics.     

On the final boundary value problems in linear thermoelasticity

In the present study we derive some uniqueness criteria for solutions of the Cauchy problem for the standard equations of dynamical linear thermoelasticity backward in time. We use Lagrange-Brun identities combined with some differential inequalities in order to show that the final boundary value problem associated with the linear thermoelasticity backward in time has at most one solution in appropriate classes of displacement-temperature fields. The uniqueness results are obtained under the assumptions that the density mass and the specific heat are strictly positive and the conductivity tensor is positive definite.     

The asymptotic expansions of singularly perturbed boundary value problems

In this paper we study the singularly perturbed boundary value problem:εy″=f(t,y,ε),y(0)=ξ(ε),y(1)=η(ε).where εis a positive small parameter.In the conditions:f_(?)(0,y,0)≥m_0 ,f_(?)(l,y,0)≥m_0 and f_(?)f(t,y,ε)≥0 ,we prove the existences,and uniformly valid asymptotic expansions of solutions for the given boundary value problems,and hence we improve the existing results.