Shocks and finite-time singularities in Hele-Shaw flow

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that this ill-defined problem admits a weak dispersive solution when singularities give rise to a graph of shock waves propagating into the viscous fluid. The graph of shocks grows and branches. Velocity and pressure have finite discontinuities across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2, 3)-cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions.     

Weak solution for the Hele-Shaw problem: Viscous shocks and singularities

In Hele-Shaw flows, a boundary of a viscous fluid develops unstable fingering patterns. At vanishing surface tension, fingers evolve to cusp-like singularities preventing a smooth flow. We show that the Hele-Shaw problem admits a weak solution where a singularity triggers viscous shocks. Shocks form a growing, branching tree of a line distribution of vorticity where pressure has a finite discontinuity. A condition that the flow remains curl-free at a macroscale uniquely determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2, 3)-cusp singularity—an elementary branching event of a branching shock graph.     

Do solitonlike self-similar waves exist in nonlinear optical media?

We show analytically that bright and dark spatial self-similar waves can propagate in graded-index amplifiers exhibiting self-focusing or self-defocusing Kerr nonlinearities. The intensity profiles of the novel waves are identical with those of fundamental bright or dark spatial solitons supported by homogeneous passive waveguides with the same type of nonlinearity. Thus, we reveal a previously unnoticed connection between spatial solitons and self-similar waves. We also suggest that the discovered self-similar waves can be used in a promising scheme for the amplification and focusing of spatial solitons in future all-optical networks.     

Numerical modelling of free interaction of shock waves in corner flows

The inviscid supersonic flows in corners between intersecting compression wedges were studied numerically. Under usual conditions, the flows in such corner configurations are conically self-similar. Besides, shock waves formed by wedges are plane and they interact with one another in these flows and the downstream development of the shock interaction structure at that occurs in a region which is bounded in cross-sectional directions by the walls and contracts towards the corner rib, and in this sense the above interaction of shocks may be subjected to influence of the rib. The corner flows of another type with an interaction of shocks not subjected to the “rib effect”, in other words free, were considered. This was ensured by that the corner configurations were designed with a gradually expanding gap along the rib in a way that the reflected shocks arising as a result of interaction passed through the gap without falling onto the walls. This eliminated the influence of a local flow near the corner rib on the interaction of plane shock waves formed by the wedges. The inviscid flows symmetric with respect to the bisector plane of the corner dihedral angle were considered. The computations showed that, in the gapped corner configurations with the free interaction of shocks, if these interactions were irregular, the cross-flow patterns were practically the same as in the original corner configurations without a gap. The conditions for the flows over corner configurations were also considered under which the interaction of shocks corresponded to the Neumann criterion of mechanical equilibrium. These conditions are equivalent to those, which take place in two-dimensional steady flows, when a hysteresis phenomenon is possible, which manifests itself in changing the moment of transition from a regime of regular reflection of shocks to an irregular one and conversely at a direct and inverse courses of varying the flow parameters—the Mach number and the inclination angle of the wedges. The conducted computations showed that a hysteresis is possible for considered corner flows in gapped configurations, but it must be followed by a detachment of shocks from the corner apex and a breakdown of the conical flow structure.     

Cavitation induced by explosion in an ideal fluid model

We discuss the problem of an explosion in the cubic-quintic superfluid model, in relation to some experimental observations. We show numerically that an explosion in such a model might induce a cavitation bubble for large enough energy. This gives a consistent view for rebound bubbles in superfluid and we identify the loss of energy between the successive rebounds as radiated waves. We compute self-similar solution of the explosion for the early stage, when no bubbles have been nucleated. The solution also gives the wave number of the excitations emitted through the shock wave.     

Generation of flexural waves in plates by laser-initiated airborne shock waves

Results of the semi-analytical modelling of the interaction of laser-initiated airborne shock waves with an infinite horizontally positioned elastic plate are presented. The impact of the airborne shock waves on the plate is approximated by a cylindrically diverging surface force resulting from the pressure of the incident and reflected shock waves. This force is then represented in the wavenumber-frequency domain by means of Hankel and Fourier transforms that are carried out numerically so that the interaction problem can be solved using the Green’s function method. The resulting frequency spectra and time histories of generated flexural wave pulses are calculated for different values of laser pulse energy and for different heights of the laser beam focusing above the plate surface. The theoretical results obtained are compared with the results of laboratory measurements of the interaction of laser-generated acoustic shocks with a large plastic plate. The comparison shows reasonably good agreement between the semi-analytical predictions and data.     

Feshbach resonance induced shock waves in Bose-Einstein condensates

We propose a method for generating shock waves in Bose-Einstein condensates by rapidly increasing the value of the nonlinear coefficient using Feshbach resonances. We show that in a cigar-shaped condensate there exist primary (transverse) and secondary (longitudinal) shock waves. We analyze how the shocks are generated in multidimensional scenarios and describe the related phenomenology.     

Temperature overshoots for a 4-velocity unidimensional discrete Boltzmann model

We propose a 4-velocity unidimensional discrete Boltzmann model with two different speeds 2, 1 and two different masses 1, 2. With the three conservation laws of mass, momentum, and energy satisfied, we can introduce a nontrivial temperature. First, we determine the similarity shock waves satisfying physical properties: positivity, shock stability, inequalities of the subsonic and supersonic flows, increase or decrease of both mass and temperature across the shock. It results that either the speed of the shock front is higher than the speed 1 of the slow particles and the shocks are compressive or less than 1 and the shocks are rarefactive. We observe overshoots of the temperature, across the shock, with bumps higher and higher as the shock front speed increases. Second, we study the (1+1)-dimensional shock waves. They represent the superposition and collision of two compressive shocks traveling in opposite directions and we observe temperature overshoots for not too large times.     

Dispersive nature of high mach number collisionless plasma shocks: Poynting flux of oblique whistler waves

Whistler wave trains are observed in the foot region of high Mach number quasiperpendicular shocks. The waves are oblique with respect to the ambient magnetic field as well as the shock normal. The Poynting flux of the waves is directed upstream in the shock normal frame starting from the ramp of the shock. This suggests that the waves are an integral part of the shock structure with the dispersive shock as the source of the waves. These observations lead to the conclusion that the shock ramp structure of supercritical high Mach number shocks is formed as a balance of dispersion and nonlinearity.     

Imploding shocks in laser-driven fusion

The role of a sequence of imploding spherical and cylindrical shocks is investigated in the context of laser-driven fusion in deuterium-tritium pellets. An approximate analytical treatment of a convergent sequence of shocks is presented within the framework of gas-dynamic equations for self-similar motion. These analytical solutions are compared with the exact numerical solutions. The solutions display an explicit dependence on the relative strength between the successive shocks and the ratio of the final to the initial pressure in the shocks. These solutions are employed to estimate the fusion yield for a given input shock energy.     

Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid in the shallow water approximation

The system of the magnetohydrodynamic equations for a heavy fluid has been analyzed in the shallow water approximation. All discontinuous self-similar solutions and all continuous centered self-similar solutions have been found. It has been shown that magnetogravity compression waves are broken with the formation of a magnetogravity shock wave. The initial decay discontinuity problem for the magnetohydrodynamic equations has been solved in the explicit form in the shallow water approximation. The existence of five different configurations implementing the solution of the decay of an arbitrary discontinuity has been demonstrated. The conditions necessary and sufficient for the implementation of each configuration have been found.     

Planar and Nonplanar Shock Waves in a Degenerate Quantum Plasma

The nonlinear propagation of modified electron‐acoustic (mEA) shock waves in an unmagnetized, collisionless, relativistic, degenerate quantum plasma (containing non‐relativistic degenerate inertial cold electrons, both nonrelativistic and ultra‐relativistic degenerate hot electron and inertial positron fluids, and positively charged static ions) has been investigated theoretically. The well‐known Burgers type equation has been derived for both planar and nonplanar geometry by employing the reductive perturbation method. The shock wave solution has also been obtained and numerically analyzed. It has been observed that the mEA shock waves are significantly modified due to the effects of degenerate pressure and other plasma parameters arised in this investigation. The properties of planar Burgers shocks are quite different from those of nonplanar Burgers shocks. The basic features and the underlying physics of shock waves, which are relevant to some astrophysical compact objects (viz. non‐rotating white dwarfs, neutron stars, etc.), are briefly discussed. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shock shells in Coulomb explosions of nanoclusters

We predict that Coulomb explosion of a nanoscale cluster, which is ionized by high-intensity laser radiation and has a naturally occurring spatial density profile, will invariably produce shock waves. In most typical situations, two shocks, a leading and a trailing one, form a shock shell that eventually encompasses the entire cluster. Being the first example of shock waves on the nanometer scale, this phenomenon promises interesting effects and applications, including high-rate nuclear reactions inside each individual cluster.     

Orientation dependence in molecular dynamics simulations of shocked single crystals

We use multimillion-atom molecular dynamics simulations to study shock wave propagation in fcc crystals. As shown recently, shock waves along the <100> direction form intersecting stacking faults by slippage along 111 close-packed planes at sufficiently high shock strengths. We find even more interesting behavior of shocks propagating in other low-index directions: for the <111> case, an elastic precursor separates the shock front from the slipped (plastic) region. Shock waves along the <110> direction generate a leading solitary wave train, followed (at sufficiently high shock speeds) by an elastic precursor, and then a region of complex plastic deformation.     

Nonlinear effects and shock formation in the focusing of a spherical acoustic wave

The focusing of acoustic waves is used to study nucleation phenomena in liquids. At large amplitude, nonlinear effects are important so that the magnitude of pressure or density oscillations is difficult to predict. We present a calculation of these oscillations in a spherical geometry. We show that the main source of nonlinearities is the shape of the equation of state of the liquid, enhanced by the spherical geometry. We also show that the formation of shocks cannot be ignored beyond a certain oscillation amplitude. The shock length is estimated by an analytic calculation based on the characteristics method. In our numerical simulations, we have treated the shocks with a WENO scheme. We obtain a very good agreement with experimental measurements which were recently performed in liquid helium. In addition, the comparison between numerical and experimental results allows us to calibrate the vibration of the ceramic used to produce the wave, as a function of the applied voltage.Received: 11 July 2003, Published online: 24 October 2003PACS: 67.40.-w Boson degeneracy and superfluidity of 4He - 43.25. + y Nonlinear acoustics - 62.60. + v Acoustical properties of liquids     

隔板深度对激波聚焦起爆影响的数值模拟研究

考虑几何结构参数对激波聚焦触发爆轰波的复杂影响,对H2/Air预混气的环形射流激波聚焦起爆现象开展了数值模拟研究,详细分析了不同隔板深度条件下的激波聚焦过程、流场演化特征以及爆轰波参数变化规律.研究结果表明,凹腔内激波聚焦诱导的局部爆炸以及隔板前缘处射流形成"卷吸涡"是引起爆轰波触发的两个重要机制,而隔板深度是影响环形...     

Self-Action of Wave Beams Containing Shock Fronts

We briefly review the effects of nonlinear self-action of beams of strongly distorted waves containing steep shock fronts. The features of inertial self-actions of periodic sawtooth waves in quadratic nonlinear media without dispersion are discussed. These phenomena can be caused by an acoustic wind or thermal lens formed as a result of the nonlinear dissipation at the shock fronts. Instantaneous self-actions are analyzed on the examples of periodic trapezoidal waves, which are formed in cubic nonlinear media and contain alternating compression and rarefaction shocks, and a single-pulse signal containing a shock front. Mathematical models and solutions to the corresponding nonlinear equations are given. A qualitative comparison with optical self-action phenomena and with available experimental data is performed.     

Waves in interplanetary shocks: a wind/WAVES study

We describe results from the first statistical study of waveform capture data during 67 interplanetary (IP) shocks with Mach numbers ranging from approximately 1-6. Most of the waveform captures and nearly 100% of the large amplitude waves were in the ramp region. Although solitary waves, Langmuir waves, and ion acoustic waves (IAWs) are all observed in the ramp region of the IP shocks, large amplitude IAWs dominate. The wave amplitude is correlated with the fast mode Mach number and with the shock strength. The observed waves produced anomalous resistivities from approximately 1-856 Omega.m (approximately 10(7) times greater than classical estimates.) The results are consistent with theory suggesting IAWs provide the primary dissipation for low Mach number shocks.     

Shock profile solutions of the Boltzmann equation

Shock waves in gas dynamics can be described by the Euler Navier-Stokes, or Boltzmann equations. We prove the existence of shock profile solutions of the Boltzmann equation for shocks which are weak. The shock is written as a truncated expansion in powers of the shock strength, the first two terms of which come exactly from the Taylor tanh (x) profile for the Navier-Stokes solution. The full solution is found by a projection method like the Lyapunov-Schmidt method as a bifurcation from the constant state in which the bifurcation parameter is the difference between the speed of soundc ₀ and the shock speeds.Research supported in part by the National Science Foundation, the Army Research Office, the Air Force of Scientific Research, the Office of Naval Research, and the Department of Energy     

Particle acceleration at astrophysical shocks: A theory of cosmic ray origin

The theory of first order Fermi acceleration at collisionless astrophysical shock fronts is reviewed. Observations suggest that shock waves in different astrophysical environments accelerate cosmic rays efficiently. In the first order process, high energy particles diffuse through Alfvén waves that scatter them and couple them to the background plasma. These particles gain energy, on the average, every time they cross the schock front and bounce off approaching scattering centers. Calculations demonstrate that the distribution function transmitted by a plane shock is roughly a power law in momentum with slope similar to that inferred in galactic cosmic ray sources. The generation of the scattering Alfvén waves by the streaming cosmic rays is described and it is argued that the wave amplitude is probably non-linear within sufficiently strong astrophysical shocks. Hydromagnetic scattering can operate on the thermal particles as well, possibly establishing the shock structure. This suggests a model of strong collisionless shocks in which high energy particles are inevitably produced very efficiently. Observable consequences of this model, together with its limitations and some alternatives, are described. Cosmic ray origin and astrophysical shocks can no longer be considered separately.