Self-similar convergence of a shock wave in a heat conducting gas

The problem of the convergence of a spherical shock wave (SW) to the centre, taking into account the thermal conductivity of the gas in front of the SW, is considered within the limits of a proposed approximate model of a heat conducting gas with an infinitely high thermal conductivity and a small temperature gradient, such that the heat flux is finite in a small region in front of the converging SW. In this model, there is a phase transition in the surface of the SW from a perfect gas to another gas with different constant specific heat and the heat outflow. The gas is polytropic and perfect behind the SW. Constraints are derived which are imposed on the self-similarity indices as a function of the adiabatic exponents on the two sides of the SW. In front of the SW, the temperature and density increase without limit. In the general case, a set of self-similar solutions with two self-similarity indices exists but, in the case of strong SW close to the limiting compression, there are two solutions, each of which is completely determined by the motion of the spherical piston causing the self-similar convergence of the SW.     

Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas

A group theoretic method is used to obtain an exact particular solution to the system of partial differential equations, describing one-dimensional unsteady planar, cylindrically and spherically symmetric motions in an ideal gas, involving shock waves. It is interesting to remark that the exact solution obtained here is precisely the blast wave solution obtained earlier using a different method of approach. Further, the evolution of a discontinuity wave and its interaction with the strong shock are studied within the state characterized by the exact particular solution. The properties of reflected and transmitted waves and the jump in the shock acceleration are completely characterized, and certain observations are noted in respect to their contrasting behavior.     

Propagation of a shock wave in an isentropic gas with small viscosity

A new method is considered for constructing the front and the dynamics of motion of a shock wave propagating in an isentropic gas with small viscosity with a speed close to the speed of sound. The proposed algorithm of solution makes it possible to considerably raise the accuracy and effectiveness of numerical computations on the computer as compared with known universal numerical methods based on the use of difference schemes.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 273–308, 1977.     

Similarity solution for a spherical shock wave in a non‐ideal gas under the influence of gravitational field and monochromatic radiation with increasing energy

The propagation of a spherical shock wave in a non‐ideal gas with or without gravitational effects is investigated under the action of monochromatic radiation. Similarity solutions are obtained for adiabatic flow between the shock and the piston. The numerical solutions are obtained using the Runge‐Kutta method of the fourth order. The density of the gas is assumed to be constant. The total energy of the shock wave is non‐constant and varies with time. The effects of change in values of non‐idealness parameter, gravitational parameter, shock Mach number, radiation parameter, and adiabatic exponent of the gas on shock strength and flow variables are worked out in detail. It is investigated that the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and, therefore, the distance between the inner contact surface and the shock surface is reduced. A comparison is also made between the solutions in the cases of the gravitating and the non‐gravitating media. It is manifested that the gravitational parameter and the radiation parameter have in general opposite behaviour on the flow variables and the shock strength.     

Computation of a shock wave structure in monatomic gas with accuracy control

The structure of a shock wave in a monatomic one-component gas was computed by solving the Boltzmann kinetic equation with accuracy controlled with respect to computational parameters. The hard-sphere molecular model and molecules with the Lennard-Jones potential were considered. The computations were performed in a wide range of Mach numbers with the accuracy no less than 3% for the shock front width and 1% for local values of density and temperature. The shock wave structure was studied in terms of macroscopic gas characteristics and in terms of the molecular velocity distribution function.     

Stability of an imploding spherical shock wave in a van der Waals gas II

The emission of light from a sonoluminescing bubble may dependon whether shock waves launched each acoustic cycle by the implodingsurface of the bubble focus on to a sufficiently small volumeat the bubble centre. This in turn may depend on whether theshock maintains its stability as it travels inwards. With thisapplication in mind, the linear stability of an imploding sphericalshock was studied in Part I, using a van der Waals equationof state for the gas. Conditions for instability were determined,but the subsequent fate of the perturbations of the bubble surfacewas unknown. Would the instabilities grow and persist at finiteamplitude or would they disappear during implosion? The answersto such questions are sought here by integrating the gas dynamicsequations using the finite-difference essentially non-oscillatorymethod of Shu and Osher. The shock is initiated by a nearlyspherical ‘piston’ and its subsequent evolution,including its finite-amplitude deviations from sphericity, isdetermined. Two types of behaviour are found depending on theparameter , where b is thevan der Waals excluded volume and is the initial uncompressed density of gas ahead of the shock.When is sufficiently large, an initially smooth shock front remains smooth as it focusesand, although it is impossible to continue the integrationsup to the moment of implosion, it appears that it will focuson a small volume at the centre of the bubble. This is in sharpcontrast to what happens at smaller values of for which the initial distortion of the shock front,if sufficiently large, becomes and remains polygonal shaped.This is consistent with experimental results for cylindricalimploding shocks as well as with earlier theoretical investigationsof imploding cylindrical and spherical shocks in an ideal gas() that used the Chisnell, Chester and Witham (CCW) approximation or the geometrical shockapproximation of Whitham. Plausibly, the polygonal distortionsreduce the volume on to which the imploding shock in a sonoluminescingbubble focuses.     

Cylindrically and spherically symmetrical rapid intense compression of an ideal perfect gas with adiabatic exponents from 1.001 to 3

The problem of the rapid intense cylindrically or spherically symmetrical compression of an ideal (non-viscous and non-heat-conducting) perfect gas with different adiabatic exponents is considered. We mean by rapid and intense a compression in a time much less than the time taken for the sound wave to propagate through the uncompressed target up to temperatures and densities as high as desired. It is found that the solution previously obtained with a focused non-self-similar compression wave at the point where the shock wave is reflected from the axis or centre of symmetry (henceforth the centre of symmetry) holds for adiabatic exponents not exceeding 1.9092 and 1.8698 respectively in the cylindrical and spherical cases. It was not possible to construct a complete solution with focusing at the centre of symmetry for gases with higher adiabatic exponents. On the other hand, one can focus the compression waves into a cylinder or sphere of as small, but finite, radius as desired at the instant of arrival on them, for example, of a special characteristic or reflected shock wave of the Guderley problem. It is shown that for high degrees of compression, the time dependences of the coordinates of the pistons which produce such focusing, and of the gas density on them are close to power laws.     

A criterion of the linear long wave stability of steady magnetohydrodynamic jet flows of an ideal fluid

The problem of the linear stability of steady axisymmetric shear magnetohydrodynamic jet flows of an inviscid ideally conducting incompressible fluid with a free boundary is investigated. It is assumed that the jet is of unlimited length, there is a longitudinal constant electric current along its surface, and it is directed along the axis of a cylindrical shell with infinite conductivity, such that there is a vacuum layer between its free boundary and the inner surfaces of the shell. The necessary and sufficient condition for the stability of such flows with respect to small axisymmetric long-wave perturbations of special form is obtained by Lyapunov's direct method. Bilateral exponential estimates of the growth of small perturbations are constructed in the case when this stability condition breaks down, where the indices in their exponents are calculated from the parameters of the steady flows and the initial data for the perturbations. An example of a steady axisymmetric shear magnetohydrodynamic jet flow and of the initial small axisymmetric long-wave perturbations imposed on it is given, which, at the linear stage, will evolve in time and space in accordance with the estimates constructed.     

Computational magneto-hydrodynamic modeling of hypersonic flows with resolved shock wave diffusion

The present model describes magneto-hydrodynamic (MHD) effects on diffusing shock waves in compressible flows. The accurateness and the robustness of these new density-based numerical algorithm in his ideal form was tested on a series of benchmark numerical experiments like the present MHD shock tube [1]. Interpolation schemes are used for interpolation of density, temperature, velocity and the magnetic field. For the discretization of laplacian terms with diffusion coefficient for polyhedral meshes, we make use of the Gauss linear uncorrected schemes to deal with the case that the consecutive cell faces are non-orthogonal. The approach described above and developed in the present project has been coupled with semi-discrete central schemes (e.g. Greenshields et al. [2]). All computational results are validated with verified data and analytical results. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blowup in solutions of a quasilinear wave equation with variable‐exponent nonlinearities

We consider, in this paper, the following nonlinear equation with variable exponents: where a,b>0 are constants and the exponents of nonlinearity m,p, and r are given functions. We prove a finite‐time blow‐up result for the solutions with negative initial energy and for certain solutions with positive energy.     

The method of integral relations for computing blunt body flows with a detached shock wave

Dorodnicyn’s classical work concerning the method of integral relations is given in his original presentation. Developments of the method provided by Dorodnicyn’s students are outlined. Belotserkovskii’s 1956 paper concerning a technique for computing the detached shock wave in flow over a cylinder is presented in full (with comments and numerical results). A technique for the study of flow characteristics for space vehicles of particular shapes is described. The breakthrough character of techniques proposed more than 50 years ago is demonstrated, and their (still important) philosophy is assessed.     

On the shock wave spectrum for isentropic gas dynamics with capillarity

We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities.     

On the mechanics of helical flows in an ideal incompressible nonviscous continuous medium

We find a general solution to the problem on the motion in an incompressible continuous medium occupying at any time a whole domain D ? R ³ under the conditions that D is an axially symmetric cylinder and the motion is described by the Euler equation together with the continuity equation for an incompressible medium and belongs to the class of helical flows (according to I.S. Gromeka’s terminology), in which sreamlines coincide with vortex lines. This class is constructed by the method of transformation of the geometric structure of a vector field. The solution is characterized in Theorem 2 in the end of the paper.     

The dissociation of a pure diatomic gas behind a strong normal shock wave

Zusammenfassung Die Arbeit behandelt das Problem der Dissoziation eines reinen diatomischen Gases durch einen starken geraden Verdichtungsstoss. Mit Hilfe einiger vereinfachender Annahmen und eines theoretischen Ausdruckes für die Dissoziationsgeschwindigkeit wird eine analytische Darstellung der Resultate erzielt. Der Einfluss der Erhitzung und der Kompression durch den Stoss wird eingehend diskutiert.

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The research reported in this document has been sponsored in part by the Air Research and Development Command, United States Air Force, under contract AF 61(514)1124, through the European Office, A.R.D.C.     

The Riemann problem for an isentropic ideal dusty gas flow with a magnetic field

This paper deals with Riemann problem for one-dimensional inviscid, isentropic, and perfectly conducting ideal dusty gas flow with a transverse magnetic field. The explicit expressions of elementary waves are derived in terms of the density, velocity, and transverse magnetic induction of an ideal dusty gas flow. The analytical properties of elementary wave curves and the influence of parameter on the elementary waves are discussed. A new approach is proposed to resolve the Riemann problem. By applying this approach, we obtain 10 kinds of exact solutions and their corresponding criteria.     

The problem of wave momentum,radiation pressure and other quantities in the case of plane motions of an ideal gas

Small vibrations and waves of an ideal fluid gas (a liquid or gas) are considered in the quadratic approximation. Actual values of the momentum density and the pressure and also their averaged values are obtained in a number of specific characteristic problems. It is shown that the momentum of an isolated wave with a mean density equal to the density of the unperturbed medium, and the radiation pressure are due to non-linearity of the system of equations, and that this wave has no momentum if its profile remains unchanged during its motion. The latter assertion is also true for finite-amplitude waves.     

Note on “An efficient approach for solving the lot-sizing problem with time-varying storage capacities”

In a recent paper Gutièrrez et al. [1] show that the lot-sizing problem with inventory bounds can be solved in O(TlogT) time. In this note we show that their algorithm does not lead to an optimal solution in general.