粘性物质中正激波稳定性分析

用线性稳定性理论,分析粘性物质中的正激波稳定性问题.粘性物质中任意强度的一维激波,其稳定性问题可归结为处理复数范围内的特征值问题,该特征值问题由两个一阶常微分方程及一个二阶常微分方程构成.这些常微分方程的系数依赖于流动的基本流场的物理量及其梯度.所获得的特征值问题由一个四阶精度的有限差分离散求解.分析考虑物质粘性的金属铝中的正激波稳定性,可以看出,正激波运动是稳定的,并且激波速度对波前和波后的小扰动量的衰减有相反的作用,而物质粘性有致稳的作用.     

Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion

The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet ^– (for some>0) ast, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.     

Viscous flow and ablation pressure phenomena in nanosecond UV laser irradiation of polymers

Acceleration and expulsion of a laser-induced melt layer in laser ablation of polymers is studied based on a combination of a quantitative theoretical modeling of ablation pressure and viscous melt flow with an experimental technique of a precise nanoscale measurement of the resulting surface profile. For two particular examples corresponding to so-called stationary and non-stationary liquid layer flows the following results are obtained: (i) the kinematic viscosity of the laser-induced melt layer on the surface of poly(ethylene terephthalate) at extreme conditions of KrF laser ablation is found for the first time and (ii) a new form of material removal in laser ablation is explained – expulsion of long (up to 1 mm) nanofibers with a radius of about 150–200 nm when a poly(methyl methacrylate) target is irradiated with a single pulse of a KrF excimer laser. PACS 42.62.Cf; 61.80.Ba; 83.80.Ab     

Dynamic Scaling in Miscible Viscous Fingering

We consider dynamic scaling in gravity driven miscible viscous fingering. We prove rigorous one-sided bounds on bulk transport and coarsening in regimes of physical interest. The analysis relies on comparison with solutions to one-dimensional conservation laws, and new scale-invariant estimates. Our bounds on the size of the mixing layer are of two kinds: a naive bound that is sharp in the absence of diffusion, and a more careful bound that accounts for diffusion as a selection criterion in the limit of vanishingly small diffusion. The naive bound is simple and robust, but does not yield the experimental speed of transport. In a reduced model derived by Wooding [20], we prove a sharp upper bound on the size of the mixing layer in accordance with his experiments. Woodings model also provides an example of a scalar conservation law where the entropy condition is not the physically appropriate selection criterion.     

Propagation of elastoplastic waves in thin rods

The problem of the propagation of a perturbation in a thin rod is solved using the dynamic gauge theory of crystalline media with dislocations. When a shock is applied to the rod an elastic forerunner is generated with propagates with a velocity of (where E is Young's modulus and is the density of the medium) and, in addition, unclamping of the end of the rod occurs. Depending on the parameters of the medium, this unclamping may take the form of both purely viscous flow and wave flow. There is no volume elastic compression wave in the rod characteristic of an unbounded medium.Institute of Physics of the Strength and Study of Materials, Siberian Branch, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 39–42, June, 1994.     

Non-invasive determination of shock wave pressure generated by optical breakdown

Shock waves generated by a laser-induced plasma were investigated using a pump-and-probe technique. Both 7-ns and 40-ps laser pulses at 1.06 m were employed to initiate breakdown in water. Two He-Ne laser beams were used as a velocity probe, allowing the accurate measurement of the shock velocity around the plasma. The maximum shock pressure was determined from the measured shock velocities, the jump condition and the equation of state for water. The conservation of the total momentum of the shock front was used to derive expressions for the shock velocity, particle velocity and shock pressure vs. the distance (r) from the center of the plasma. For a shock wave of spherical symmetry, the shock pressure is proportional to 1/r ². Our work shows that the expanding plasma initially induces a shock wave; the shock wave dissipates rapidly becoming an acoustic wave within 300–500 m.     

Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile

We study the nonlinear stability of shock waves for viscous conservation laws. Our approach is based on a new construction of a fundamental solution for a linearized system around a shock profile. We obtain, for the first time, the pointwise estimates of nonlinear wave interactions across a shock wave. Our results apply to all ranges of weak shock waves and small perturbations. In particular, our results reduce to the time-asymptotic behavior of constant state perturbation, uniformly as the strength of the shock wave tends to zero. The research of the first author was partially supported by NSC Grant 96-2628-M-001-011 and NSF Grant DMS-0709248. The research of the second author was partially supported by NSF Grant DMS-0207154 and UAB Advance Program, sponsored by NSF.     

Behavior of point defects under impulsive dynamical loading

The behavior of point defects (interstitial atoms and vacancies) is examined in the front of a plane shock wave. The wave is modeled by a soliton pulse. It is shown that in high-pressure shock waves with a narrow loading wave front an unactivated motion of the point defects is possible, and for a pressure P_o 50 GPa and x_o 10^–8 m, a dragging of the interstitial atoms by the front of the shock wave can be observed. The vacancies are displaced in the field of the shock pulse in the opposite direction to the motion of the pulse. Here the mobility of the vacancies is significantly lower than that of the interstitial atoms.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 49–52, June, 1985.     

On the Formation of Centered Shock Waves in Gas Bubbles under the Conditions of Acoustic Cavitation

The possibility of the formation of centered shock waves in collapsing gas bubbles under the conditions of acoustic cavitation is considered. In this case, the overturning of the front of compression waves occurs at the instant the waves reach the center of the cavitation bubble, resulting in the highest possible temperatures and pressures inside the bubble. Examination of the magnetohydrodynamic equations has shown that the law of the motion of the wall of a bubble at the final stage of compression, described by the Rayleigh–Plesset equation, has a universal form and coincides with the condition of the formation of a spherically symmetric centered shock wave with the adiabatic constant = 5/3. For < 5/3, the collapse of a bubble occurs within a shorter time than it takes for a spherically symmetric centered shock wave to form. In this case, the overturning of the front of compression waves occurs earlier than they reach the center of the bubble, and shock waves are formed inside the bubble at different points. The most appropriate condition for the detection of centered shock waves is the cavitation in cryogenic fluids, such as helium, for which 5/3.     

Scalar conservation laws with discontinuous flux function: I. The viscous profile condition

The equation , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular 2×2 non-strictly hyperbolic system. This augmentation is non-unique and a natural definition is given by means of viscous profiles. By a viscous profile we mean a stationary solution ofu _t +(F ) _x =u _xx , whereF is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. In terms of the 2×2 system, the discontinuity atx=0 is either a regular Lax, an under-or overcompressive, a marginal under- or overcompressive or a degenerate shock wave. In some cases, depending onf andg, there is a unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). The main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx=0 and the viscous profile condition.     

Thermal conductivity of high temperature gas mixture for lighting engineering

Paper presents experimental values of the coefficient of thermal conductivity of a ternary gaseous mixture (0·15 N₂-0·1 Ar-0·75 Kr) in the temperature range between approx. 1500 and 4000 K at about a mospheric pressure. The unsteady method based on the analysis of the heat flux from gas heated by the reflected shock wave to the end wall of a shock tube was used. Presented values agree well with rigorous kinetic theory calculations and with a combined Dulnev's model.Notation (in SI units) a thermal diffusivity - c _p isobaric heat capacity - E voltage - p pressure - T thermodynamic temperature - t time - x coordinate - temperature coefficient of resistance - mass density - coefficient of thermal conductivity Indexes 1 gas state before the experiment - 2 gas state behind the primary shock wave - 5 gas state behind the reflected shock wave - f relates to the solid wall - F relates to the solid wall on the solid-gas phase boundary - g relates to the test gas - G relates to the gas on the gas-solid phase boundary     

Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compressible Navier-Stokes Equation

In this paper we investigate the asymptotic stability of a composite wave consisting of two viscous shock waves for the full compressible Navier-Stokes equation. By introducing a new linear diffusion wave special to this case, we successfully prove that if the strengths of the viscous shock waves are suitably small with same order and also the initial perturbations which are not necessarily of zero integral are suitably small, the unique global solution in time to the full compressible Navier-Stokes equation exists and asymptotically tends toward the corresponding composite wave whose shifts (in space) of two viscous shock waves are uniquely determined by the initial perturbations. We then apply the idea to study a half space problem for the full compressible Navier-Stokes equation and obtain a similar result. Research is supported in part by NSFC Grant No. 10471138, NSFC-NSAF Grant No. 10676037 and 973 project of China, Grant No. 2006CB805902, in part by Japan Society for the Promotion of Science, the Invitation Fellowship for Research in Japan (Short-Term). Research is supported in part by Grant-in-Aid for Scientific Research (B) 19340037, Japan.     

A Hybrid Lattice Boltzmann Flux Solver for Simulation of Viscous Compressible Flows

In this paper, a hybrid lattice Boltzmann flux solver (LBFS) is proposed for simulation of viscous compressible flows. In the solver, the finite volume method is applied to solve the Navier-Stokes equations. Different from conventional Navier-Stokes solvers, in this work, the inviscid flux across the cell interface is evaluated by local reconstruction of solution using one-dimensional lattice Boltzmann model, while the viscous flux is still approximated by conventional smooth function approximation. The present work overcomes the two major drawbacks of existing LBFS [28–31], which is used for simulation of inviscid flows. The first one is its ability to simulate viscous flows by including evaluation of viscous flux. The second one is its ability to effectively capture both strong shock waves and thin boundary layers through introduction of a switch function for evaluation of inviscid flux, which takes a value close to zero in the boundary layer and one around the strong shock wave. Numerical experiments demonstrate that the present solver can accurately and effectively simulate hypersonic viscous flows.     

On nonlinear stability of general undercompressive viscous shock waves

We study the nonlinear stability of general undercompressive viscous shock waves. Previously, the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation, using new pointwise methods. By examining time invariants associated with the linearized equations, we can now overcome a new difficulty in the general case, namely, nonlinear movement of the shock. We introduce a coordinate transformation suitable to treat this new aspect, and demonstrate our method by analyzing a model system of generic type. We obtain sharp pointwise bounds andL ^p behavior of the solution for allp, 1p.     

Asymptotic regularizations and shocks in noninductive heat-current-free relativistic fluids

We have developed elsewhere (cf. [1]) a method, which we call asymptotic regularizations, designed to give shock and infinitesimal shock wave equations for systems of partial differential equations. What we intend to do here is to illustrate its usefulness when applied to a specific system of partial differential equations, namely, that of noninductive, heatcurrent-free perfect relativistic fluids.     

Time-resolved measurements of picosecond optical breakdown

Picosecond optical breakdown was investigated in order to assess its potential for performing highly localized incisions for laser surgery. Measurements of breakdown were performed using single 40-ps Nd: YAG laser pulses in distilled water. Novel optical pump-probe techniques were developed to characterize the transient spatial and temporal dynamics of the plasma, shock wave, and cavitation phenomena which are associated with the breakdown. The maximum cavity radius and the shock wave zone are shown to scale as the cube root of the pump pulse energy over almost three orders of magnitude. For pulse energies close to the threshold energy of 8 J, the shock range was 100–200 m and the cavity radius was 140 m. Complementary experiments were performed with 10-ns pulse durations. Since picosecond pulses have high peak intensities with low pulse energies, a significant enhancement in localizability may be achieved. The implications for ophthalmic microsurgery are discussed.     

Global existence and exponential stability of small solutions to nonlinear viscoelasticity

The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.     

Stability of Viscous Shocks in Isentropic Gas Dynamics

In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier–Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there are no unstable spectra in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected. This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.