Precursor vacuum-ultraviolet radiation ahead of strong shock waves in a shock tube

The profile and excitation mechanism of vacuum-ultraviolet radiation emitted from shock wave is investigated in a shock tube. For shock wave in argon, the rdiation is due to resonant transition excited by argon-argon collision in the shock front with excitation cross section coefficientS ^*=1.0×10⁻¹⁷ cm²·ev⁻¹ and activation energyE ^*=11.4 ev. For shock wave in air the radition is emitted from a very thin shock layer in which the mechanism ofX ¹∑→b ¹∑ of N₂ is excited with excitation cross sectionQ=2×10⁻¹⁶cm² and activation energyE ^*=12.1 ev. Institute of Mechanics, Academia Sinica     

Reflection, transmission and excitation of SH-surface waves by a discontinuity in mass-loading on a semi-infinite elastic medium

The scattering of an SH-wave by a discontinuity in mass-loading on a semi-infinite elastic medium is investigated theoretically. The incident wave is either a plane body wave or a plane SH-surface wave. The problem is reduced to a Wiener-Hopf problem for the scattered wave. In this problem the amplitude spectral density of the particle displacement occurs as unknown function. Special attention is given to the numerical values of the surface wave contributions to the scattered field.Nomenclature x ₁, x ₂, x ₃ Cartesian coordinates - , polar coordinates in x ₁, x ₃-plane - volume mass density - surface mass density of mass-loading - , Lamé constants - U scalar wave function, defined by (2.1) - c _S speed of propagation of uniform shear waves in bulk medium (c _S=(/)^1/2) - angular frequency - t time - k _S wave number of uniform shear waves (k _S=/c _S) - reduced specific acoustic impedance of mass-loading (=k _S /) - k _m wave number of SH-surface wave (k _m=k _S(1+ ²)^1/2) - _1,2,3 partial differentiation with respect to x _1,2,3 - ⁱ angle between x ₃-axis and direction of propagation of incident body wave - ⁱ wave number in horizontal direction of incident body wave ( ⁱ=k _S sin( ⁱ)) - ⁱ wave number in vertical direction of incident body wave ( ⁱ=k _S cos( ⁱ)) - C _1,2 complex amplitude of surface wave excited by a body wave - R reflection factor of surface wave, when surface wave is incident - T transmission factor of surface wave, when surface wave is incident - S particle displacement vector The research presented in this paper has been carried out with partial financial support from the Delfts Hogeschoolfonds.     

Forerunning mode transition in a continuous waveguide

We have discovered a forerunning mode transition as the periodic wave changing the state of a uniform continuous waveguide. The latter is represented by an elastic beam initially rested on an elastic foundation. Under the action of an incident sinusoidal wave, the separation from the foundation occurs propagating in the form of a transition wave. The critical displacement is the separation criterion. Under these conditions, the steady-state mode exists with the transition wave speed independent of the incident wave amplitude. We show that such a regime exists only in a bounded domain of the incident wave parameters. Outside this domain, for higher amplitudes, the steady-state mode is replaced by a set of local separation segments periodically emerging at a distance ahead of the main transition point. The crucial feature of this waveguide is that the incident wave group speed is greater than the phase speed. This allows the incident wave to deliver the energy required for the separation. The analytical solution allows us to show in detail how the steady-state mode transforms into the forerunning one. The latter studied numerically turns out to be periodic. As the incident wave amplitude grows the period decreases, while the transition wave speed averaged over the period increases to the group velocity of the wave. As an important part of the analysis, the complete set of solutions is presented for the waves excited by the oscillating or/and moving force acting on the free beam. In particular, an asymptotic solution is evaluated for the resonant wave corresponding to a certain relation between the load's speed and frequency.     

Scattering by penetrable acoustic targets

An acoustic target of constant density ?_t and variable index of refraction is imbedded in a surrounding acoustic fluid of constant density ?_a. A time harmonic wave propagating in the surrounding fluid is incident on the target. We consider two limiting cases of the target where the parameter ε ≡ ?_a/?_t → 0 (the nearly rigid target) or ε → ∞ (the nearly soft target). Wh en the frequency of the incident wave is bounded away from the ‘in-vacuo’ resonant frequencies of the target, the resulting scattered field is essentially the field scattered by the rigid target for ε = 0 or the soft target if ε → ∞. However, when the frequency of the incident wave is near a resonant frequency,the target oscillates and its interaction with the surrounding fluid produces peaks in the scattered field amplitude. In this paper we obtain asymptotic expansions of the solutions of the scattering problems for the nearly rigid and the nearly soft targets as ε → 0 or ε → ∞, respectively, that are uniformly valid in the incident frequency. The method of matched asymptotic expansions is used in the analysis. The outer and inner expansions correspond to the incident frequencies being far or near to the resonant frequencies, respectively. We have applied the method only to simple resonant frequencies, but it can be extended to multiple resonant frequencies. The method is applied to the incidence of a plane wave on a nearly rigid sphere of constant index of refraction. The far field expressions for the scattered fields, including the total scattering cross-sections, that are obtained from the asymptotic method and from the partial wave expansion of the solution are in close agreement for sufficiently small values of ε.     

Self-similar motion of a gas heated by a nonequilibrium continuous radiation spectrum

In this paper we discuss the motion of the vapor formed during the evaporation of a solid by a continuous radiation spectrum. The vapor is assumed to be heated by this radiation to a temperature T much higher than the phase-transition temperature T_v and much higher than the temperature T_i at which significant ionization of the vapor begins. in the case, T_v and T_i can be neglected (as can the heat of evaporation Q_v and the energy Q_i expended on ionization). As a result of this motion, the vapor has a density ? much lower than the densityρ ₀ of the solid. It can therefore be assumed that the heating wave moves through an absolutely cold and infinitely dense gas. At the same time, the vapor temperature is assumed low enough that reradiation can be neglected. The radiation-absorption coefficient η for the ionized vapor can be described by a power-law dependence on T and ? for certain ranges of T, ε, and the photon energy ε. In this case, the motion of the gas is a self-similar problem. The spectrum and angular distribution of the incident radiation [φ (ε, θ)] and the η and ε dependences can be arbitrary. A system of ordinary differential equations is found and solved. Intense radiation incident on a solid surface will evaporate the solid. If the absorption coefficient η of the vapor and the flux density q of the radiation are high enough, the escaping vapor will be heated to a high temperature in a relatively short time. This temperature will not only be much higher than the evaporation temperature T_v, but it will also be higher than the “ionization temperature” T_i. If the internal energy per unit mass of the vapor is much higher than the heat of evaporation Q_v and the energy Q_i expended on ionization, and if the vapor density ? is much lower than the initial densityρ ₀ as a result of its escape, then the problem of the motion and heating of the vapor can be simplified through the assumptions. (0.1) $$T_v = T_i = Q_v = Q_i = 0,\rho _0 = \infty (v_0 = 1/\rho _0 = 0)$$ (here and below, v is the specific volume). We can therefore assume that the heating wave moves through an infinitely dense and absolutely cold gas. In the region of multiple and complete ionization, the ionized-vapor absorption coefficient η, associated with free-free electron transitions in the field of ions, and bound-free transitions from the higherlying states of atoms and ions, has an approximately power-law dependence on T and ? [1], or on p and ? (p is the pressure): Here k and K are numerical coefficients which depend on the substance and on the ranges of T, ?, and ε in which (0.2) is used. For a completely ionized gas, we have α=3/2, β=1,a=?5/2, b=?3/2, and \( - \bar 1/2\) when ε?T; or α=3/2, β=1,a=3/2, and b=?1/2 when when ε ? T. We assume that (0.2) holds for any T, for approximation (0.1). We assume the ratio of specific heats γ to be constant for a certain temperature range in the range of multiple and complete ionization. With these simplifying approximations, the problem of the planar, transient flow of a gas heated by a beam of monochromatic radiation is a self-similar problem. It has been studied in [2,3]. It is shown below that the analogous problem of the motion of a gas heated by a nonequilibrium continuous radiation spectrum is also self-similar. For a partially ionized gas, approximation (0.2) is usualy satisfied only for the long-wavelength part of the incident spectrum. For the short-wavelength part of the spectrum (that is, for photons whose energy is close to or greater than the ionization potential characteristics of the ions for the given temperature range, and which are capable of direct photoionization of these ions from the ground or first excited status), the absorption coefficient is usually much smaller (by several orders of magnitude). This “hard” radiation penetrates a short distance into the solid, causing intense heating of a thin surface layer of small mass. An afterionization wave propagates through the substance, moving under the influence of the radiation flux in the hard part of the spectrum; if the temperature of the surface layer is close to the source temperature T_e, and reradiation becomes important, there will also be a thermal wave [1]. Since the energy expended in heating is large in these waves, their propagation velocity is small (in comparison with that of the wave of evaporation, initial ionization, and heating of the plasma by the long-wavelength part of the spectrum), even if the hard and soft parts of the incidence spectrum have comparable energies (E_h and E_s). Also, the intense reradiation by the thermal wave in the hard part of the spectrum increases its propagation velocity. Finally, the energy in the short-wavelength part of the spectrum may in general be small because of self-adsorption in the source itself (for example, adsorption of the short-wavelength radiation in the cold working gas ahed of a shock wave front in an explosive source [4]). Accordingly, the heating waves for the various parts of the source spectrum may propagate differently Since the mass of the surface layer heated by the short-wave-length part of the spectrum is small, the pressure produced as a result of of the disintegration of the surface layer is small when E_h is of the order order of E_s or, especially, when E_h?E_s; that is, the hydrodynamic effects of the heating and surface-layer disintegration on the motion and and heating of the deep layers heated by the “basic” part of the spectrum can also be neglecred. The high temperature and low density of this layer only facilitate the penetration of the long-wavelength part of the spectrum into the deeper layers; however, because of the small mass of this layer, even this phenomenon has little effect on the hydrodynamic processes in the deeper layers. Accordingly, Eq. (0.2) can frequently be assumed valid for the basic part of the spectrum in the case of a partially ionized gas, also; the rest of the spectrum may simply be neglected. These restrictions on the applicability of the self-similar problem are generally removed in the case of a completely ionized gas. A state close to that of complete ionization arises when two ionization potentials typical of a given temperature range are greatly different (this occurs, for example, in the case of the alkaline metals, and also when one atomic shell has been essentially ionized, while another has not yet started to be ionized; e. g., the L- and K-shells or the M- and L-shells). We consider here the case in which the heating is caused by nonequilibrium radiation, that is, radiation such that the intrinisic radiation of the vapor may be neglected. This is a valid assumption when the vapor temperature is considerably below the source temperature T_e, or, more accurately, when the following condition holds (for a Planckian source spectrum): (0.3) $$W\sigma T_e^4 \chi \left( {\frac{{\varepsilon _1 }}{{T_e }},\frac{{\varepsilon _2 }}{{T_e }}} \right) \gg \sigma \Upsilon ^4 \chi \left( {\frac{{\varepsilon _1 }}{T},\frac{{\varepsilon _2 }}{T}} \right)$$ Here W is the source-radiation dilution coefficient due to geometric factors, σ is the Stefan-Boltzmann constant, ?₁ and ?₂ are the boundaries of the “basic part” of the spectrum, and χ is the fraction of the spectral energy of a Planckian source with a temperature T_e or T for photous with energies ?₁≤?≤?₂. We note that the boundaries ?₁ and ?₂ for the source and vapor-radiation spectra are sometimes slightly differnt, but condition (0.3) can be easily modified for this situation or for a non-Planckian source spectrum. For our problem, the radiation intensity J=J (m, t, ε, θ) is a function of four variables: the time t, the Lagrangian mass coordinate m, the photon energy ε, and the angle θ between the direction of motion and the beam direction. The intensity J_o=J (o, t, ε, θ) of the radiation incident on the boundary m=0 is assumed to be a given function. In the self-similar problem, J can be represented as (0.4) $$J = t^\lambda J(mt^{ - n} ,\varepsilon ,\theta )$$ . This can be done (when conditions (0.1)–(0.3) are satisfied) when J_o can be represented by (0.5) $$J_0 = t^\lambda \psi (\varepsilon ,\theta )(\varepsilon _1 \leqslant \varepsilon \leqslant \varepsilon _2 ,\theta _1 \leqslant \theta \leqslant \theta _2 )$$ If the source spectrum is Planckian, condition (0.5) requires that T_e=const. In this case, the power-law time dependence of the intensity J_o may reflect, for example, motion of the radiation source toward the irradiated surface; in this case, however, the limiting angle θ₂ of the incident radiation also changes (usually, θ₁=0). As before, the problem is self-similar if these angles θ₂(t) are always small; that is, if the radiation is almost completely unidirectional. The arbitrary nature of the function ψ(ε, ч), which shows the spectrum and angular distribution of thesource radiation, and the arbitrary nature of the function φ (ε), which shows the dependence of the absorption coefficient on the photon energy, permit us to analyze the effects of these functions on the heating and motion of the substance for the case of the self-similar solution.     

Interaction of a gasdynamic shock with small disturbances

The stability of shock wave based on the definition of Landau and Lifschitz^[1] is treated in this paper. This is tantamount to solving the problem of interaction of small disturbances with a shock wave. Small disturbances are introduced on both sides of a steady, non-dissipative, plane shock wave. Landau et al.^[1] obtained the stability criterionM ₁>1,M ₂<1 for small disturbances which are travelling in the direction perpendicular to the shock wave. In the present paper, we assume that the small disturbances may be two dimensional, i.e. they may be propagating in the direction inclined to the shock wave. The conclusions obtained are: regardless of whether the incident wave and diverging wave are defined according to the direction of the phase velocity or the group velocity, the shock wave is unstable for some frequencies and longitudinal wave lengths of the disturbances, even if the conditionsM ₁>1,M ₂<1 are fulfilled. Then several experiments are proposed, and the problem of ways to define the incident wave and diverging wave is discussed. The meaning of this problem is illustrated. The same results can be obtained for the steady shock wave in a tube.     

Experimental study of the shock–bubble interaction with reshock

The interaction of a planar shock wave with a spherical density inhomogeneity is studied experimentally under reshock conditions. Reshock occurs when the incident shock wave, which has already accelerated the spherical bubble, reflects off the tube end wall and reaccelerates the inhomogeneity for a second time. These experiments are performed at the Wisconsin Shock Tube Laboratory, in a 9m-long vertical shock tube with a large square cross section (25.4×25.4 cm²). The bubble is prepared on a pneumatically retracted injector and released into a state of free fall. Planar diagnostic methods are used to study the bubble morphology after reshock. Data are presented for experiments involving two Atwood numbers (A = 0.17 and 0.68) and three Mach numbers (1.35 < M < 2.33). For the low Atwood number case, a secondary vortex ring appears immediately after reshock which is not observed for the larger Atwood number. The post-reshock vortex velocity is shown to be proportional to the incident Mach number, M, the initial Atwood number, A, and the incident shock wave speed, W _i.     

Extended Lagrangian formalism and the corresponding energy relations

In this paper an extended Lagrangian formalism for the rheonomic systems with the nonstationary constraints is formulated, with the aim to examine more completely the energy relations for such systems in any generalized coordinates, which in this case always refer to some moving frame of reference. Introducing new quantities, which change according to the law τ_a=φ_a(t), it is demonstrated that these quantities determine the position of this moving reference frame with respect to an immobile one. In the transition to the generalized coordinates qⁱ they are taken as the additional generalized coordinates q^a=τ_a, whose dependence on time is given a priori. In this way the position of the considered mechanical system relative to this immobile frame of reference is determined completely.Based on this and using the corresponding d'Alembert–Lagrange's principle, an extended system of the Lagrangian equations is obtained. It is demonstrated that they give the same equations of motion qⁱ=qⁱ(t) as in the usual Lagrangian formulation, but substantially different energy relations. Namely, in this formulation two different types of the energy change law dE/dt and the corresponding conservation laws are obtained, which are more general than in the usual formulation. So, under certain conditions the energy conservation law has the form E=T+U+P=const, where the last term, so-called rheonomic potential expresses the influence of the nonstationary constraints.Afterwards, a detailed analysis of the obtained results and their connection with the usual formulation of mechanics are given. It is demonstrated that so formulated energy relations are in full accordance with the corresponding ones in the usual vector formulation, when they are expressed in terms of the rheonomic potential. Finally, the obtained results are illustrated by several simple, but characteristic examples.     

An optical analysis of an induced flow ejector using light polarization properties

This paper presents an optical analysis of an induced flow ejector by means of plane laser sheets. The visualization method, which is developed here, takes advantage of the polarization properties of the light scattered by the fine droplets produced by condensation within the flow. This optical analysis shows that the droplets scatter near the Rayleigh scattering regime, thereby proving that their mean radius does not exceed 0.05 m. Furthermore, the injection of depolarizing tracers into the induced stream makes it is possible to distinguish visually between the supersonic primary jet and the subsonic induced stream, and to obtain information about the mixing of the two streams.List of symbols A polarization angle - D inner diameter of the mixing tube - d exit diameter of the primary nozzle - d _c throat diameter of the primary nozzle - E electric field - F transformation matrix - I intensity - L _m length of the mixing tube - m mass flow rate - M Mach number - P static pressure - r radius - S Stokes vector - U entrainment ratio - X penetration of the primary nozzle - (I, Q, U, V) Stokes parameters - OU _o propagating direction of the incident wave - (r _o, l _o) vibrating plane - (U _o, l _o) scattering plane - ejector throat area ratio - wavelength - scattering (or observation) angle Indices a atmospheric value - i stagnation value - 0 incident value - 1 primary (or central) jet - 2 secondary (or induced) jet     

On a Volume‐Constrained Variational Problem

. Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = ò_W W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where L^N({u = z_i}) = a_i, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which z_iz_i are extremal points of a compact, convex set in \Bbb R^d\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|²W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.     

Mach reflection of a plane shock wave passing a mountain of 45° inclination

The transition from regular reflection (RR) to Mach reflection (MR) as a plane shock wave diffracts around a triangular mountain of 45° inclination is analysed in this paper, both by optical measurement in a shock tube and by numerical simulation the numerical method developed by Li Yingfan^[1] is of the FLIC type with triangular mesh. The dependence of the critical transition point Lk ofRR→MR on shock Mach numberM _i is analyzed and the variations of the incidence angle ω _i of the impinging shock and the reflection angle ω _r with the distanceL ^* are investigated. Our experimental and numerical results agree well with the theoretical results of Iton and Italya.     

Axisymmetric resonant disturbances in a homogeneous wave guide

The initial boundary-value linear stability problem for small localised axisymmetric disturbances in a homogeneous elastic wave guide, with the free upper surface and the lower surface being rigidly attached to a half-space, is formally solved by applying the Laplace transform in time and the Hankel transforms of zero and first orders in space. An asymptotic evaluation of the solution, expressed as a sum of inverse Laplace-Hankel integrals, is carried out by using the approach of the mathematical formalism of absolute and convective instabilities. It is shown that the dispersion-relation function of the problem D₀ (κ, ω), where the Hankel parameter κ is substituted by a wave number (and the Fourier parameter) κ, coincides with the dispersion-relation function D₀ (k, ω) for two-dimensional (2-D) disturbances in a homogeneous wave guide, where ω is the frequency (and the Laplace parameter) in both cases. An analysis for localised 2-D disturbances in a homogeneous wave guide is then applied. We obtain asymptotic expressions for wave packets, triggered by axisymmetric perturbations localised in space and finite in time, as well as for responses to axisymmetric sources localised in space, with the time dependence satisfying e^−iω₀t + O(e^−εt) for t → ∞, where Im ω₀ = 0, ε > 0, and t denotes time, i.e. for signalling with frequency ω₀. We demonstrate that, for certain combinations of physical parameters, axisymmetric wave packets with an algebraic temporal decay and axisymmetric signalling with an algebraic temporal growth, as √t, i.e., axisymmetric temporal resonances, are present in a neutrally stable homogeneous wave guide. The set of physically relevant wave guides having axisymmetric resonances is shown to be fairly wide. Furthermore, since an axisymmetric part of any source is L²-orthogonal to its non-axisymmetric part, a 3-D signalling with a non-vanishing axisymmetric component at an axisymmetric resonant frequency will generally grow algebraically in time. These results support our hypothesis concerning a possible resonant triggering mechanism of certain earthquakes, see Brevdo, 1998, J. Elasticity, 49, 201–237.     

The nonlinear theory of electromagnetic wave attenuation in plasmas

The absorption of a circularly polarized electromagnetic wave which propagates in a plasma along a magnetic field is analyzed. The exact equations of particle motion in the resonance region are solved with aid of elliptic functions. It is shown that the nonlinear damping constant has an oscillatory form. For t→0, it coincides with the constant obtained on the basis of linear theory, while for t→∞, in the absence of collisions, it tends to zero. The influence of collisions on wave absorption is studied. It is shown that with allowance for collistions, the damping constant depends on the amplitude of both the H₁ and H₁ ^−3/2 waves. The analysis of slowly decaying waves may be based on a model proposed by Dawson [1] and later modified in [2,3]. According to this model, all plasma particles are grouped into resonant and nonresonant ones. The velocity distribution function of the nonresonant particles is assumed to be the same as in the case of undamped waves. The distribution function of resonant particles at the initial instant is assumed to be Maxwellian. The nonlinear equations of motion of the resonant particles are integrated exactly. The damping constant is defined as the ratio of the energy expended by the wave at the resonant particles to the total energy of the wave. In nonlinear formulation, resonant absorption appears to be nonstationary. After a time lapse on the order of several vibrational period of a particle captured by the wave, nonstationary absorption ceases, and stationary absorption, created by infrequent collisions, becomes essential. It is noteworthy that absorption of this type has been studied by V. E. Zakharov and V. I. Karpman [4] for the case of plasma waves. Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 5, pp. 11–17, 1968.     

Numerical calculation of the propagation of a plane subsonic radiation wave through a gas in opposition to a flow of light radiation

The propagation of a plane heating and ionization wave through a gas is considered; the wave is sustained by a strong flow of monochromatic optical radiation (traveling in the opposite direction) through energy transfer attributable to the emission of a continuous spectrum. In the range of radiation flux densities under consideration, a situation arises in which the expanding hot layer generates a shock wave transparent to the incident radiation. The radiation wave is subsonic. The pressure within the hot layer is smoothly distributed, so that its parameters may be determined by considering the equations of energy and transport of the monochromatic source radiation and the radiative-transfer equations for various frequencies and directions. The true spectral composition and distribution of the radiation are considered in detail, using refined tables of the thermodynamic and optical properties. The results of numerical calculations relating to air are presented; so are certain details of the methods used in averaging the transfer equations, which prove very efficient for the radiation-gasdynamic problem under consideration and greatly reduce the volume of calculations.

     

Interpretation of capillary pressure curves using invasion percolation theory

Invasion percolation was studied on three-dimensional regular lattices of various node numbers. A new model has been developed to obtain the pore-size distribution from capillary pressure measurements. The new model is superior to the conventional percolation model, since it takes into account the physical trapping of the wetting phase. The irreducible wetting phase saturation includes the film of the wall of the pores, the dead-end pore volume, and the main contribution by pores isolated from the outlet of the medium by the nonwetting phase. This has been related to the node number and the sample 3dimensions. Over 100 capillary pressure curves of consolidated media have been collected. Good agreement was obtained between this data set out and our invasion percolation predictions using node numbers of 6–13, as reported by Mishra and Sharma. The pore-throat size distribution function estimated by our new model is broader than from the conventional percolation and the capillary tube models.Nomenclature c constant - D pore throat diameter [m] - D _max maximum pore diameter [m] - f(D) correlation function of pore throat size and pore body size - L a parameter representing the dimension of a sample - n node number - p pressure [N/m²] - S _n the nonwetting phase saturation - x random number ranging from 0 to 1.0 - X ^a X _t ^a /X/ _t - X _e ^a X _t ^a –X _t ⁱ - X ⁱ X _t ⁱ /X _t ^a - X _nw fraction of pore volume occupied by the injected phase - X _t fraction of pores larger thanD - X _t ^a total accessibility of pores larger thanD - X _t ⁱ total isolation of pores larger thanD - contact angle - interfacial tension [N/m] - (D) pore throat size distribution     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

Modification of Whitham's ray shock theory for analyzing the reflection of weak shock waves over small wedge angles

The weak Mach reflection phenomenon has been analyzed by applying both the shock dynamics approach and the disturbance propagation concept. The analysis which is based on modified Whitham's ray shock theory results in analytical expressions for the triple point trajectory angle,, and the shape of the curved Mach stem, which are functions of the incident shock wave Mach number,M _i, and the reflecting wedge angle, _w. The analytical results were found to be in good agreement with experimental results.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

A photoelastic study of friction at multipoint contacts

A new method of measuring the normal and sliding loads associated with multiple-point contact is introduced. A multiple-point contact is modeled with a steel die with a profile that simulates a rough surface. A very large scale factor is used in modeling this surface. The steel die is placed in contact with a photoelastic model of a half plane and is subjected to a normal load. This normal load is partitioned over the multiple points of contact producing an isochromatic fringe pattern that describes the stress distribution in the local neighborhood of the contact points. A sliding load is then imposed on the model which destroys the symmetry of this fringe pattern. The fringe data in this pattern are sufficient to determine the local loadsP _i andQ _i and the local coefficient of frictionf _i =Q _i /P _i at each contact point. An overdeterministic method is introduced which gives the solution forP _i ,Q _i andf _i using many data points taken from the isochromatic pattern in the local neighborhood of the contacts.Paper was presented at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WJ on June 9–13.     

Acoustic scattering from baffled membranes that are backed by elastic cavities

An elastic membrane backed by a fluid-filled cavity in an elastic body is set into an infinite plane baffle. A time harmonic wave propagating in the acoustic fluid in the upper half-space is incident on the plane. It is assumed that the densities of this fluid and the fluid inside the cavity are small compared with the densities of the membrane and of the elastic walls of the cavity, thus defining a small parameter . Asymptotic expansions of the solution of this scattering problem as →0, that are uniform in the wave number k of the incident wave, are obtained using the method of matched asymptotic expansions. When the frequency of the incident wave is bounded away from the resonant frequencies of the membrane, the cavity fluid, and the elastic body, the resultant wave is a small perturbation (the “outer expansion”) of the specularly reflected wave from a completely rigid plane. However, when the incident wave frequency is near a resonant frequency (the “inner expansion”) then the scattered wave results from the interaction of the acoustic fluid with the membrane, the membrane with the cavity fluid, and finally the cavity fluid with the elastic body, and the resulting scattered field may be “large”. The cavity backed membrane (CBM) was previously analyzed for a rigid cavity wall. In this paper, we study the effects of the elastic cavity walls on modifying the response of the CBM. For incident frequencies near the membrane resonant frequencies, the elasticity of the cavity gives only a higher order (in ) correction to the scattered field. However, near a cavity fluid resonant frequency, and, of course, near an elastic body resonant frequency the elasticity contributes to the scattered field. The method is applied to the two dimensional problem of an infinite strip membrane backed by an infinitely long rectangular cavity. The cavity is formed by two infinitely long rectangular elastic solids. We speculate on the possible significance of the results with respect to viscoelastic membranes and viscoelastic instead of elastic cavity walls for surface sound absorbers.     

Effect of incident shock wave strength on the decay of Richtmyer–Meshkov instability-introduced perturbations in the refracted shock wave

The effect of incident shock wave strength on the decay of interface introduced perturbations in the refracted shock wave was studied by performing 20 different simulations with varying incident shock wave Mach numbers (M ~ 1.1? 3.5). The analysis showed that the amplitude decay can be represented as a power law model shown in Eq.7, where A is the average amplitude of perturbations (cm), B is the base constant (cm^?(E?1), S is the distance travelled by the refracted shockwave (cm), and E is the power constant. The proposed model fits the data well for low incident Mach numbers, while at higher mach numbers the presence of large and irregular late time oscillations of the perturbation amplitude makes it hard for the power law to fit as effectively. When the coefficients from the power law decay model are plotted versus Mach number, a distinct transition region can be seen. This region is likely to result from the transition of the post-shock heavy gas velocity from subsonic to supersonic range in the lab frame. This region separates the data into a high and low Mach number region. Correlations for the power law coefficients to the incident shock Mach number are reported for the high and low Mach number regions. It is shown that perturbations in the refracted shock wave persist even at late times for high incident Mach numbers.