Showing papers on "Shock wave published in 2001"

Nonlinear theory of diffusive acceleration of particles by shock waves

Abstract: Among the various acceleration mechanisms which have been suggested as responsible for the nonthermal particle spectra and associated radiation observed in many astrophysical and space physics environments, diffusive shock acceleration appears to be the most successful. We review the current theoretical understanding of this process, from the basic ideas of how a shock energizes a few reactionless particles to the advanced nonlinear approaches treating the shock and accelerated particles as a symbiotic self-organizing system. By means of direct solution of the nonlinear problem we set the limit to the test-particle approximation and demonstrate the fundamental role of nonlinearity in shocks of astrophysical size and lifetime. We study the bifurcation of this system, proceeding from the hydrodynamic to kinetic description under a realistic condition of Bohm diffusivity. We emphasize the importance of collective plasma phenomena for the global flow structure and acceleration efficiency by considering the injection process, an initial stage of acceleration and, the related aspects of the physics of collisionless shocks. We calculate the injection rate for different shock parameters and different species. This, together with differential acceleration resulting from nonlinear large-scale modification, determines the chemical composition of accelerated particles. The review concentrates on theoretical and analytical aspects but our strategic goal is to link the fundamental theoretical ideas with the rapidly growing wealth of observational data.

Particle acceleration by ultrarelativistic shocks: theory and simulations

Abstract: We consider the acceleration of charged particles near ultra-relativistic shocks, with Lorentz factor Gamma_s >> 1. We present simulations of the acceleration process and compare these with results from semi-analytical calculations. We show that the spectrum that results from acceleration near ultra-relativistic shocks is a power law, N(E) \propto E^{-s}, with a nearly universal value s \approx 2.2 - 2.3 for the slope of this power law. We confirm that the ultra-relativistic equivalent of Fermi acceleration at a shock differs from its non-relativistic counterpart by the occurence of large anisotropies in the distribution of the accelerated particles near the shock. In the rest frame of the upstream fluid, particles can only outrun the shock when their direction of motion lies within a small loss cone of opening angle theta_c \approx 1/Gamma_s around the shock normal. We also show that all physically plausible deflection or scattering mechanisms can change the upstream flight direction of relativistic particles originating from downstream by only a small amount: Delta theta ~ 1/Gamma_s. This limits the energy change per shock crossing cycle to Delta E ~ E, except for the first cycle where particles originate upstream. In that case the upstream energy is boosted by a factor ~ Gamma_s^2 for those particles that are scattered back across the shock into the upstream region.

Observation of quantum shock waves created with ultra- compressed slow light pulses in a Bose-Einstein condensate.

Abstract: We have used an extension of our slow light technique to provide a method for inducing small density defects in a Bose-Einstein condensate. These sub- resolution, micrometer-sized defects evolve into large-amplitude sound waves. We present an experimental observation and theoretical investigation of the resulting breakdown of superfluidity, and we observe directly the decay of the narrow density defects into solitons, the onset of the “snake” instability, and the subsequent nucleation of vortices.

Cosmic ray acceleration to very high energy through the non-linear amplification by cosmic rays of the seed magnetic field

Abstract: The maximum energy for cosmic ray acceleration at supernova shock fronts is usually thought to be limited to around 10 14 ±10 15 eV by the size of the shock and the time for which it propagates at high velocity. We show that the magnetic field can be amplified non-linearly by the cosmic rays to many times the pre-shock value, thus increasing the acceleration rate and facilitating acceleration to energies well above 10 15 eV. A supernova remnant expanding into a uniform circumstellar medium may accelerate protons to 10 17 eVand heavy ions, with charge Ze ,t oZ � 10 17 eV: Expansion into a pre-existing stellar wind may increase the maximum cosmic ray energy by a further factor of 10.

Collapse and rebound of a laser-induced cavitation bubble

Abstract: A strong laser pulse that is focused into a liquid produces a vapor cavity, which first expands and then collapses with subsequent rebounds. In this paper a mathematical model of the spherically symmetric motion of a laser-induced bubble is proposed. It describes gas and liquid dynamics including compressibility, heat, and mass transfer effects and nonequilibrium processes of evaporation and condensation on the bubble wall. It accounts also for the occurrence of supercritical conditions at collapse. Numerical investigations of the collapse and first rebound have been carried out for different bubble sizes. The results show a fairly good agreement with experimental measurements of the bubble radius evolution and the intensity of the outgoing shock wave emitted at collapse. Calculations with a small amount of noncondensable gas inside the bubble show its strong influence on the dynamics.

Principles of shock wave therapy.

Abstract: A shock wave is a transient pressure disturbance that propagates rapidly in three-dimensional space. It is associated with a sudden rise from ambient pressure to its maximum pressure. A significant tissue effect is cavitation consequent to the negative phase of the wave propagation. The current authors summarize the basic physics of shock waves and the physical parameters involved in assessing the amount of energy delivered to the target tissue and in comparing the various high- and low-energy devices being evaluated clinically for musculoskeletal applications.

Internal shocks in the jets of radio-loud quasars

Abstract: The central engine causing the production of jets in radio sources may work intermittently, accelerating shells of plasma with different mass, energy and velocity. Faster but later shells can then catch up slower earlier ones. In the resulting collisions shocks develop, converting some of the ordered bulk kinetic energy into magnetic field and random energy of the electrons which then radiate. We propose that this internal shock scenario, which is the scenario generally thought to explain the observed gamma-ray burst radiation, can also work for radio sources in general, and for blazars in particular. We investigate in detail this idea, simulating the birth, propagation and collision of shells, calculating the spectrum produced in each collision, and summing the locally produced spectra from those regions of the jet which are simultaneously active in the observer's frame. We can thus construct snapshots of the overall spectral energy distribution, time-dependent spectra and light curves. This allows us to characterize the predicted variability at any frequency, study correlations between the emission at different frequencies, specify the contribution of each region of the jet to the total emission, and find correlations between flares at high energies and the birth of superluminal radio knots and/or radio flares. The model has been applied to reproduce qualitatively the observed properties of 3C 279. Global agreement in terms of both spectra and temporal evolution is found. In a forthcoming work, we will explore the constraints that this scenario sets on the initial conditions of the plasma injected in the jet and the shock dissipation for different classes of blazars.

An Origin of Supersonic Motions in Interstellar Clouds

Abstract: The propagation of a shock wave into an interstellar medium is investigated by two-dimensional numerical hydrodynamic calculation with cooling, heating, and thermal conduction. We present results of the high-resolution, two-dimensional calculations to follow the fragmentation that results from thermal instability in a shock-compressed layer. We find that the geometrically thin cooling layer behind the shock front fragments into small cloudlets. The cloudlets have supersonic velocity dispersion in the warm neutral medium, in which the fragments are embedded as cold condensations. The fragments tend to coalesce and become larger clouds.

Conditions for shock revival by neutrino heating in core-collapse supernovae

Abstract: Energy deposition by neutrinos can rejuvenate the stalled bounce shock and can provide the energy for the supernova explosion of a massive star. This neutrino-heating mechanism, though investigated by numerical simulations and analytic studies, is not finally accepted or proven as the trigger of the explosion. Part of the problem is that different groups have obtained seemingly discrepant results, and the complexity of the hydrodynamic models often hampers a clear and simple interpretation of the results. This demands a deeper theoretical understanding of the requirements of a successful shock revival. A toy model is developed here for discussing the neutrino heating phase analytically. The neutron star atmosphere between the neutrinosphere and the supernova shock can well be considered to be in hydrostatic equilibrium, with a layer of net neutrino cooling below the gain radius and a layer of net neutrino heating above. Since the mass infall rate to the shock is in general different from the rate at which gas is advected into the neutron star, the mass in the gain layer varies with time. Moreover, the gain layer receives additional energy input by neutrinos emitted from the neutrinosphere and the cooling layer. Therefore the determination of the shock evolution requires a time-dependent treatment. To this end the hydrodynamical equations of continuity and energy are integrated over the volume of the gain layer to obtain conservation laws for the total mass and energy in this layer. The radius and velocity of the supernova shock can then be calculated from global properties of the gain layer as solutions of an initial value problem, which expresses the fact that the behavior of the shock is controlled by the cumulative effects of neutrino heating and mass accumulation in the gain layer. The described toy model produces steady-state accretion and mass outflow from the nascent neutron star as special cases. The approach is useful to illuminate the conditions that can lead to delayed explosions and in this sense supplements detailed numerical simulations. On grounds of the model developed here, a criterion is derived for the requirements of shock revival. It confirms the existence of a minimum neutrino luminosity that is needed for shock expansion, but also demonstrates the importance of a sufficiently large mass infall rate to the shock. If the neutrinospheric luminosity or accretion rate by the shock are too low, the shock is weakened because the gain layer loses more mass than is resupplied by inflow. On the other hand, very high infall rates damp the shock expansion and above some threshold, the development of positive total energy in the neutrino-heating layer is prevented. Time-dependent solutions for the evolution of the gain layer show that the total specific energy transferred to nucleons by neutrinos is limited by about 1052 erg (~5 MeV per nucleon). This excludes the possibility of very energetic explosions by the neutrino-heating mechanism, because the typical mass in the gain layer is about 0.1 and does not exceed a few tenths of a solar mass. The toy model also allows for a crude discussion of the global effects of convective energy transport in the neutrino-heating layer. Transfer of energy from the region of maximum heating to radii closer behind the shock mainly reduces the loss of energy by the inward flow of neutrino-heated matter through the gain radius.

The approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction

Abstract: A formulation of the approximate deconvolution model (ADM) for the large-eddy simulation (LES) of compressible flows in complex geometries is detailed. The model is applied to supersonic compression ramp flow where shock-turbulence interaction occurs. With the ADM approach an approximation to the unfiltered solution is obtained from the filtered solution by a series expansion involving repeated filtering. Given a sufficiently good approximation of the unfiltered solution at a time instant, the flux terms of the underlying filtered transport equations can be computed directly, avoiding the need to explicitly compute subgrid-scale closures. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation and a dynamically estimated relaxation parameter. Results of the large-eddy simulation of the turbulent supersonic boundary layer along a compression ramp compare well with filtered DNS data. The filtered shock solution is correctly predicted by the ADM pr...

Nonideal Effects Behind Reflected Shock Waves in a High-Pressure Shock Tube

Abstract: . Shock tubes often experience temperature and pressure nonuniformities behind the reflected shock wave that cannot be neglected in chemical kinetics experiments. Because of increased viscous effects, smaller tube diameters, and nonideal shock formation, the reflected-shock nonidealities tend to be greater in higher-pressure shock tubes. Since the increase in test temperature ( $\Delta T_5$ ) is the most significant parameter for chemical kinetics, experiments were performed to characterize $\Delta T_5$ in the Stanford High Pressure Shock Tube using infrared emission from a known amount of CO in argon. From the measured change in vibrationally equilibrated CO emission with time, the corresponding d $T_5/$ dt (or $\Delta T_5$ for a known time interval) of the mixture was inferred assuming an isentropic relationship between post-shock temperature and pressure changes. For a range of representative conditions in argon (24–530 atm, 1275–1900 K), the test temperature 2 cm from the endwall increased 3–8 K after 100 $\mu$ s and 15–40 K after 500 $\mu$ s, depending on the initial conditions. Separate pressure measurements using a shielded piezoelectric transducer confirmed the isentropic assumption. An analytical model of the reflected-shock gas dynamics was also developed, and the calculated $\Delta T_5$ 's agree well with those obtained from experiment. The analytical model was used to estimate the effects of temperature and pressure nonuniformities on typical chemical kinetics measurements. When the kinetics are fast ( $<300\mu$ s), the temperature increase is typically negligible, although some correction is suggested for kinetics experiments lasting longer than 500 $\mu$ s. The temperature increase, however, has a negligible impact on the measured laser absorption profiles of OH (306 nm) and CH $_3$ (216 nm), validating the use of a constant absorption coefficient. Infrared emission experiments are more sensitive to temperature and density changes, so $T_5$ nonuniformities should be taken into account when interpreting ir-emission data.

Cosmic-Ray Protons Accelerated at Cosmological Shocks and Their Impact on Groups and Clusters of Galaxies

Abstract: We investigate the production of cosmic ray (CR) protons at cosmological shocks by performing, for the first time, numerical simulations of large scale structure formation that include directly the acceleration, transport and energy losses of the high energy particles. CRs are injected at shocks according to the thermal leakage model and, thereafter, accelerated to a power-law distribution as indicated by the test particle limit of the diffusive shock acceleration theory. The evolution of the CR protons accounts for losses due to adiabatic expansion/compression, Coulomb collisions and inelastic p-p scattering. Our results suggest that CR protons produced at shocks formed in association with the process of large scale structure formation could amount to a substantial fraction of the total pressure in the intra-cluster medium. Their presence should be easily revealed by GLAST through detection of gamma-ray flux from the decay of neutral pions produced in inelastic p-p collisions of such CR protons with nuclei of the intra-cluster gas. This measurement will allow a direct determination of the CR pressure contribution in the intra-cluster medium. We also find that the spatial distribution of CR is typically more irregular than that of the thermal gas because it is more influenced by the underlying distribution of shocks. This feature is reflected in the appearance of our gamma-ray synthetic images. Finally, the average CR pressure distribution appears statistically slightly more extended than the thermal pressure.

Multidimensional Stability of Planar Viscous Shock Waves

Abstract: Physical and mathematical considerations warrant the inclusion of regularizing effects such as diffusion, dissipation, and/or relaxation in the study of stability of shock waves, particularly in MHD, combustion, and multiphase flow. Indeed, in this generality, the “ideal shock” approximation has relevance only in the long-wave limit, that is, in the large, but not in the small scale. Likewise, multidimensional effects are known both from experiment and from the study of the inviscid (hyperbolic) case to play a key role in determining stability. Yet, until very recently, there were no rigorous analyses for systems taking into account both effects simultaneously.

Stability of Multidimensional Shocks

Abstract: This series of lectures is devoted to the study of shock waves for systems of multidimensional conservation laws. In sharp contrast with one-dimensional problems, in higher space dimensions there is no general existence theorem for solutions which allow discontinuities. Our goal is to study the existence and the stability of the simplest pattern of a single wave front ∑, separating two states u + and u -, which depend smoothly on the space-time variables x. For example, our analysis applies to perturbations of planar shocks. They are special solutions given by constant states separated by a planar front. Given a multidimensional perturbation of the initial data or a small wave impinging on the front, we study the following stability problem. Is there a local solution with the same wave pattern? Similarly, a natural problem is to investigate the multidimensional stability of one-dimensional shock fronts. However, the analysis applies to much more general situations and the main subject is the study of curved fronts.

Anomalous elastic response of silicon to uniaxial shock compression on nanosecond time scales.

Abstract: We have used x-ray diffraction with subnanosecond temporal resolution to measure the lattice parameters of orthogonal planes in shock compressed single crystals of silicon (Si) and copper (Cu). Despite uniaxial compression along the (400) direction of Si reducing the lattice spacing by nearly 11%, no observable changes occur in planes with normals orthogonal to the shock propagation direction. In contrast, shocked Cu shows prompt hydrostaticlike compression. These results are consistent with simple estimates of plastic strain rates based on dislocation velocity data. Although the response of materials to uniaxial shock compression has been a field of study for more than a century, our understanding at the lattice level of the response of crystals to rapid loading is still far from complete. While constitutive models are useful, a full description of phenomena such as shock-induced elastic-plastic flow and polymorphic phase transitions requires a knowledge of the atomic positions and the history of their rearrangement during the passage of the shock wave. In principle, one of the most direct methods of obtaining such information is the technique of in situ time-resolved x-ray diffraction (TXRD). Indeed, the TXRD experiments of Johnson and co-workers over three decades ago gave the first direct evidence of the retention of crystallinity under shock compression [1,2]. TXRD yields information about the interatomic spacings within the crystal. The change in Bragg angle due to the shock-induced alteration of the lattice parameter for monochromatic radiation is given, for small compressions, by simple differentiation of Bragg’s law: D2dhkl2dhkl 2 cotubDu. TXRD also provides information about the degree of plastic flow within the crystal: in the limit of purely hydrostatic response, an initially cubic lattice remains cubic under shock compression (at least in the hard sphere approximation). Thus diffraction from planes with reciprocal lattice vectors orthogonal to the shock propagation direction also exhibits angular shifts: a feature that has been confirmed by TXRD for certain crystals such as LiF and KCl under sufficiently intense loading [3‐5]. We provide here similar evidence for shocked single-crystal copper, which responds approximately hydrostatically to shocks of order 180 kbar on nanosecond time scales. However, plastic flow relies on dislocation generation and transport, a process that takes a characteristic time

Rayleigh-Taylor Instabilities in Young Supernova Remnants Undergoing Efficient Particle Acceleration

Abstract: We employ hydrodynamic simulations to study the effects of high shock compression ratios, as expected for fast shocks with efficient particle acceleration, on the convective instability of driven waves in supernova remnants. We find that the instability itself does not depend significantly on the compression ratio, σ, with the growth rates and the width of the mixing region at saturation being comparable for the range of ratios we studied; 4 ≤ σ ≤ 21. However, because the width of the interaction region between the forward and reverse shocks can shrink significantly with increasing σ, we find that convective instabilities can reach all the way to the forward shock front if compression ratios are high enough. Thus, if supernova blast waves accelerate particles efficiently, we expect the forward shock to be perturbed with small-amplitude, small-wavelength bumps and to find clumps and filaments of dense ejecta material in the vicinity of the shock. In addition and in contrast to situations in which σ ≤ 4, any enhancement of the radial magnetic field from Rayleigh-Taylor instabilities will also extend all the way to the shock front, and this may help explain the slight dominance of radial fields long seen in polarization measurements of young remnants like Tycho.

Analytic adjoint solutions for the quasi-one-dimensional Euler equations

Abstract: The analytic properties of adjoint solutions are examined for the quasi-one-dimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green's function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging–diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock.

Dust-acoustic shocks in a strongly coupled dusty plasma

Abstract: It is shown that the nonlinear propagation of the dust-acoustic waves in a strongly coupled dusty plasma is governed by a Korteweg-de Vries-Burgers (K-dV-Burgers) equation. The latter is derived from a set of generalized hydrodynamic equations for strongly correlated dust grains, as well as the Boltzmann distribution for electrons and ions. Possible stationary solutions of the K-dV-Burgers equation are represented in terms of monotonic/oscillatory shock profiles.

Shocks and sonic booms in the intracluster medium: x-ray shells and radio galaxy activity

Abstract: Motivated by hydrodynamic simulations, we discuss the X-ray appearance of radio galaxies embedded in the intracluster medium (ICM) of a galaxy cluster. We distinguish three regimes. In the early life of a powerful source, the entire radio cocoon is expanding supersonically and hence drives a strong shock into the ICM. Eventually, the sides of the cocoon become subsonic and the ICM is disturbed by the sonic booms of the jet’s working surface. In both of these regimes, X-ray observations would find an X-r ay shell. In the strong shock regime, this shell will be hot and relatively thin. However, in the weak shock (sonic-boom) regime, the shell will be approximately the same temperature as the undisturbed ICM. If a cooling flow is present, the observed shell may even be cooler than the undisturbed ICM due to the lifting of cooler material into the shell from the inner (cooler) regions of the cluster. In the third and final regime, the cocoon has collaps ed and no well-defined X-ray shell will be seen. We discuss ways of estimating the power and age of the source once its regime of behavior has been determined. Subject headings: galaxies: jets, galaxies: clusters: general, hydrodynami cs, shock waves

Transverse waves in numerical simulations of cellular detonations

Abstract: In this paper the structure of strong transverse waves in two-dimensional numerical simulations of cellular detonations is investigated. Resolution studies are performed and it is shown that much higher resolutions than those generally used are required to ensure that the flow and burning structures are well resolved. Resolutions of less than about 20 numerical points in the characteristic reaction length of the underlying steady detonation give very poor predictions of the shock configurations and burning, with the solution quickly worsening as the resolution drops. It is very difficult and dangerous to attempt to identify the physical structure, evolution and effect on the burning of the transverse waves using such under-resolved calculations. The process of transverse wave and triple point collision and reflection is then examined in a very high-resolution simulation. During the reflection, the slip line and interior triple point associated with the double Mach configuration of strong transverse waves become detached from the front and recede from it, producing a pocket of unburnt gas. The interaction of a forward facing jet of exploding gas with the emerging Mach stem produces a new double Mach configuration. The formation of this new Mach configuration is very similar to that of double Mach reflection of an inert shock wave reflecting from a wedge.

Numerical testing of the stability of viscous shock waves

Abstract: A new theoretical Evans function condition is used as the basis of a numerical test of viscous shock wave stability. Accuracy of the method is demonstrated through comparison against exact solutions, a convergence study, and evaluation of approximate error equations. Robustness is demonstrated by applying the method to waves for which no current analytic results apply (highly nonlinear waves from the cubic model and strong shocks from gas dynamics). An interesting aspect of the analysis is the need to incorporate features from the analytic Evans function theory for purposes of numerical stability. For example, we find it necessary, for numerical accuracy, to solve ODEs on the space of wedge products.

On the Formation of Cluster Radio Relics

Abstract: (abridged) We present detailed 3-dimensional magneto-hydrodynamical simulations of the passage of a radio plasma cocoon filled with turbulent magnetic fields through a shock wave. Taking into account synchrotron, inverse Compton and adiabatic energy losses and gains we evolved the relativistic electron population to produce synthetic polarisation radio maps. On contact with the shock wave the radio cocoons are first compressed and finally torn into filamentary structures, as is observed in several cluster radio relics. In the synthetic radio maps the electric polarisation vectors are mostly perpendicular to the filamentary radio structures. If the magnetic field inside the cocoon is not too strong, the initially spherical radio cocoon is transformed into a torus after the passage of the shock wave. Very recent, high-resolution radio maps of cluster radio relics seem to exhibit such toroidal geometries in some cases. This supports the hypothesis that cluster radio relics are fossil radio cocoons that have been revived by a shock wave. For a late-stage relic the ratio of its global diameter to the filament diameter should correlate with the shock strength. Finally, we argue that the total radio polarisation of radio relic should be well correlated with the 3-dimensional orientation of the shock wave that produced the relic.

Blast waves and how they interact with structures.

Abstract: The paper defines and describes blast waves, their interaction with a structure and its subsequent response. Explosions generate blast waves, which need not be due to explosives. A blast wave consists of two parts: a shock wave and a blast wind. The paper explains how shock waves are formed and their basic properties. The physics of blast waves is non-linear and therefore non-intuitive. To understand how an explosion generates a blast wave a numerical modelling computer code, called a hydrocode has to be employed. This is briefly explained and the cAst Eulerian hydrocode is used to illustrate the formation and propagation of the blast wave generated by a 1 kg sphere of TNT explosive detonated 1 m above the ground. The paper concludes with a discussion of the response of a structure to a blast wave and shows that this response is governed by the structures natural frequency of vibration compared to the duration of the blast wave. The basic concepts introduced are illustrated in a second simulation that introduces two structures into the blast field of the TNT charge.

Gamma-ray burst phenomenology, shock dynamo, and the first magnetic fields

Abstract: A relativistic collisionless shock propagating into an unmagnetized medium leaves behind a strong large-scale magnetic field. This seems to follow from two assumptions: (1) Gamma-ray burst (GRB) afterglows are explained by synchrotron emission of a relativistic shock. (2) The magnetic field cannot exist on microscopic scales only; it would decay by phase-space mixing. Assumption 1 is generally accepted because of an apparent success of the shock synchrotron phenomenological model of GRB afterglows. Assumption 2 is confirmed in this work by a low-dimensional numerical simulation. One may hypothesize that relativistic shock velocities are not essential for the magnetic field generation and that all collisionless shocks propagating into an unmagnetized medium generate strong large-scale magnetic fields. If this hypothesis is true, the first cosmical magnetic fields could have been generated in shocks of the first virialized objects.

Mechanical haemolysis in shock wave lithotripsy (SWL): I. Analysis of cell deformation due to SWL flow-fields

Abstract: This work analyses the interaction of red blood cells (RBCs) with shock-induced and bubble-induced flows in shock wave lithotripsy (SWL), and calculates, in vitro, the lytic effects of these two flows. A well known experimentally observed fact about RBC membranes is that the lipid bilayer disrupts when subjected to an areal strain (ΔA/A)c of 3%, and a corresponding, critical, isotropic tension, Tc, of 10 mN m-1 (1 mN m-1 = 1 dyne cm-1). RBCs suspended in a fluid medium tend to deform in accordance with the deformation of the surrounding fluid medium. The fluid flow-field is lytically effective if the membrane deformation exceeds the above threshold value. From kinematic analysis, motion of an elementary fluid particle can always be decomposed into a uniform translation, an extensional flow (e.g. vec u∞(x,y,z) = (k(t)x,-k(t)y,0)) along three mutually perpendicular axes, and a rigid rotation of these axes. However, only an extensional flow causes deformation of a fluid particle, and consequently deforms the RBC membrane. In SWL, a fluid flow-field, induced by a non-uniform shock wave, as well as radial expansion/implosion of a bubble, has been hypothesized to cause lysis of cells. Both the above flow-fields constitute an unsteady, extensional flow, which exerts inertial as well as viscous forces on the RBC membrane. The transient inertial force (expressed as a tension, or force/length), is given by Tiner~ρrc3k/τ, where τ is a timescale of the transient flow and rc is a characteristic cell size. When the membrane is deformed due to inertial effects, membrane strain is given by ΔA/A~kτ. The transient viscous force is given by Tvisc~ρ(ν/τ)1/2rc2k, where ρ and ν are the fluid density and kinematic viscosity. For the non-uniform shock, the extensional flow exerts an inertial force, Tinerapprox64 mN m-1, for a duration of 3 ns, sufficient to induce pores in the RBC membrane. For a radial flow-field, induced by bubble expansion/implosion, the inertial forces are of a magnitude 100 mN m-1, which last for a duration of 1 µs, sufficient to cause rupture. Bubble-induced radial flow is predicted to be lytically more effective than shock-induced flow in typical in vitro experimental conditions.