A Shock, 20–23, 26, 28 Acoustic Cavitation, 45, 46, 59, 60 Acoustic

Index

A shock, 20–23, 26, 28 bubble rebound, 47, 49 acoustic cavitation, 45, 46, 59, 60 Burgers–KdV equation, 17, 22 acoustic impedances, 37, 58 activation function, 203, 208, 221, 222 C shock, 23–27 adiabatic heating, 60 capacitance technique, 68, 86 air bubble, 37–43, 49, 52, 53, 56, 58–60 Carrol–Holt model, 332 anomalous reflection, 41, 42 cavitating liquid, 68, 73–75, 77, 79, 80, approximate Rankine–Hugoniot 82, 84 relations, 160, 172 cavitation, 35–37, 45, 48, 49, 52, 58, 59, arbitrary discontinuity, 80 99, 101 archetypal relaxation equation, 137 cavitation cloud, 45 asymptotic pressure profile, 160 cavitation erosion, 93 asymptotic solution, 203, 204, 206, cavitation nuclei, 35, 70, 72, 73, 85, 88, 220–223, 226 91, 92, 94 attenuation, 303, 346 cavitation process, 75, 80, 94 avalanche-like population, 73 cavitation threshold, 68, 85, 86, 89 cavitation zone, 68, 73, 75, 76, 78, 80, B shock, 23, 25–27 82, 83, 85–87, 93, 94 back-steepened shock, 118 cell disruption, 59 barriers, 272 classical nucleation theory, 189, 206, Benedict-Webb-Rubin equation, 234 209, 212, 251 blackbody, 59, 60 collapse time, 46 blood analog fluids, 91 combination structure, 70, 72 Bose–Einstein condensation, 103 compaction, 329 Bremsstrahlung, 60 compaction model, 313 bubble aspect ratio, 40 compaction wave, 286, 287 bubble cavitation, 68, 73, 78, 91 comparative magnitudes, 141 bubble cluster, 68, 72, 75, 94 complete evaporation, 161 bubble collapse, 35–37, 45, 46, 48, 49, complete liquefaction shock, 239, 244 51, 53, 54, 58–60 compression of molybdenum, 331 bubble distributions, 22 compression wave, 287 bubble flatness, 40 computer processing, 82 bubble form instability, 67 condensation front, 239 354 Index condensation induced flow oscillations, double exposure holographic 217 interferometry, 52 condensation induced shock waves, 187, double shock, 118, 129 202, 220, 223, 227 droplet growth, 190, 203, 205, 206, 208, condensation models, 190 209, 212, 221–223 condensation nuclei, 188, 203, 205, 208, droplet growth function, 203 211, 220 droplet growth parameter, 203 condensation rate equation, 202, 203, droplet growth zone, 205, 206, 208, 209, 208, 209, 212, 220–222, 226 221–223 condensation zones, 203, 205, 206, 222, droplet size spectrum, 228 223, 226 droplet temperature relaxation time, condensed explosives, 60 141 conservation equations, 143 dynamic branch, 88 convex-shaped boundary, 49 dynamics of free surface, 85–87 cooling rates, 187 copper powder, 339 elastoplastic model, 318 corner vortex, 256 electromagnetic shock tube, 68 corner wave, 38 embedded shock, 224 corrected Tait equation, 53 energy transfer, 140 coupled relaxation processes, 153, 165 entropy change, 236 critical amount of condensation, 168 entropy rise, 184 critical quantity of heat, 167 epoxy resin, 52 cryogenic fluid, 99–101 equations of motion, 299 cryogenic shock tube, 101, 112–114, equilibrated, 142 116, 119, 123 equilibrium, 138 cryostat, 100, 101, 119, 122, 125 equilibrium fully dispersed, 158 cumulative microjets, 89, 93 equilibrium isentropic exponent, 177 equilibrium isentropic index, 146, 171 ∆T shows unstable behaviour, 150 equilibrium Mach number, 147 ∆V becomes unstable, 151 equilibrium partly dispersed, 158 damage pit, 53–57 equilibrium speed of sound, 145 Damköhler’s parameter, 138 equilibrium total pressure, 181, 184 decay of, 80 erosion, 36 deformation, 331 ESWL, 36, 45 deformation rate of cavitating liquid, 85 Eulerian–Lagrangian time-marching densification, 319 technique, 144 density distribution, 82 exact jump condition, 173, 175, 176 density of cavitation zone, 83 expansion fan, 201, 202, 220–222 density of microinhomogeneities, 71 destruction of tissue and cells, 68, 89 fibrinolysis by liquid jet impact, 91 deterioration, 346 finite volume, 144 diffraction effect, 86 finite-volume method, 321 diffraction node, 41 first sound, 104, 109–112 direct mechanism of failure, 92 flows with heat addition, 190–192, 196, discontinuity, 85 202 dislocation, 56 foam, 284, 297 dispersed wave, 293 foam collapses, 307 dispersion, 11–13, 17–19, 22, 28 frame stiffness, 306 dispersion equation, 7, 8, 26 free molecule, 141 Index 355 free-wave zone, 86 incompressible kernel, 335 frequency dispersion, 145 indicatrix petals, 70 front-steepened shock, 118 indirect mechanism, 92 frozen, 138 indium, 53–56 frozen Mach number, 146 inertial relaxation time, 139 frozen profile of mass velocities, 68, 83 influence coefficients in condensing flow, frozen speed of sound, 145, 194, 226 166 frozen total pressure, 181 initial growth zone, 203, 222 frozen total temperature, 184 integral approach, 167 fully dispersed shock wave, 148 internal state variable, 137 fully dispersed wave, 147 interpreting total pressure, 185 fully dispersed with complete ionization, 60 evaporation, 158 irreversible deformation, 88 fundamental derivative, 236 further growth zone, 203, 204, 208, jump conditions, 179 221–223 jump relations, 161 FWHM pulse duration, 47 Keller–Miksis model, 47 gallbladder, 48, 58 Khalatnikov approximate theory, 127 gas–particle mixture, 176, 177, 179 kidney, 48, 58 gas-saturated foam, 303 kidney stone disintegration, 68, 89 gelatin, 44, 58, 59 kinetics of condensation, 188 generation of entropy, 162 Klein–Gordon equation, 74, 75 Gibbs formation energy, 189 Knudsen number, 140 Gilmore model, 43 Grüneisen coefficient, 326 λ-shock, 256 granular media, 325 lambda point, 102 grid, 272 lambda-phase transition, 116 group velocity, 26, 27 Landau two fluid equation, 106 laser surgery, 36 He I, 99, 102, 116, 125, 127, 128 latent heat, 232 He II, 99, 100, 102, 104–107, 112, 114, latent heat of condensation, 192, 202, 116, 117, 120, 124–128 211 heat capacity, 232, 234 Laval-nozzle, 242 heat conduction, 61 Levovist contrast agent bubble, 59 helium bubble, 39, 40 light attenuation, 249 heterogeneous heating, 342 light emission, 59, 61, 62 high heating, 329 limiting piston velocity, 164 high-strength compacts, 345 limiting wetness fraction, 174 hollow sphere model, 334 line of complete evaporation, 159 Hugoniot curve, 45 line of no discontinuity, 159 Hugoniot relations, 10, 17, 18 lipid bilayer, 48 hydrodynamic tube of rarefaction, 80 liquefaction, 232 hydrogen bubble, 39, 43, 44 liquefaction shock, 233 hysteresis effect, 94 liquid jet, 35–37, 42–45, 48, 49, 51–56, hysteresis loop, 219 58–61 liquid strength, 68, 88 IKW-model, 74 lithotripsy, 89, 90 incompressible fluid, 309 lithotripters, 46 356 Index loading time, 88 oblique impact, 289 Lorentz–Lorenz equation, 52 oblique reflection, 292 luminescence, 36, 59–61 oblique shock, 212, 219–223 onset front, 222, 224 Mach line, 42 onset zone, 204, 205, 208, 222, 223 mass transfer, 140 optimum bubble size, 46, 53, 56 MBSL, 59 oscillating front, 80 mechanocaloric effect, 105, 106 medical application, 89 partial liquefaction shock, 239, 244 membrane permeability, 48 particle beds, 292 method of characteristics, 220, 226 particle size, 343 micro-bubble, 94 particle velocity, 37–40 microinhomogeneities, 68, 70, 72, 73, partly dispersed shock wave, 147, 153 75, 84, 94 partly dispersed with complete microjet, 51, 53, 58 evaporation, 158 microkinetic energy, 335, 337 perforated plates, 272 microparticles, 70 permanent liquid, 102, 103 microreactor, 59 permeable, 272, 279 minimum bubble radius, 46 permeable media, 297 model gypsum, 90 phase boundaries, 233 model of instantaneous relaxation, 84 phase change, 60 molecular delivery, 36, 46, 47 phase separation, 228 molecular dynamics, 47 phase velocity, 301, 303 momentum transfer, 139 photon, 61 monodisperse distribution, 84 Pitot tube, 180 multiphase, 67 pk-model, 74, 76 Munroe jet, 44 plot of Rp versus St, 183 mutual friction term, 108 polyurethane foams, 284 pore collapse, 333 natural coordinates, 220 porosity, 325 negative pressures, 49 porous media, 271 non-dimensional total pressure, 182 PP3, 232, 246 non-equilibrium effects, 180 PP9, 246 non-equilibrium heating, 339 Prandtl–Meyer flows, 187, 220–223 non-equilibrium variable, 139 pre-threshold, 87 noninvasive destruction, 94 precursor, 74, 80 nonlinear steepening, 3, 17 pressure amplification, 271, 275, 284, normal fluid component, 103, 104, 106, 287 107, 111 pressure antinode, 45, 46 normal shock, 136 pressure impulse, 45, 46, 53 nucleation, 188, 189, 203–205, 207–215, pseudogas, 284 220–224, 226, 227, 231, pseudogas model, 297 251, 258 Pure Vapour–Droplet Medium, 138 nucleation parameter, 203, 204 nucleation zone with growth, 205, 221, quantized circulation, 109 223 quantized vortex, 108 numerical solution, 144 quantum effect, 102 numerical solution for a partly quasi-one-dimensional nozzle flow, 192, dispersed shock wave, 154 202, 222 Index 357 quasistatic mode, 337 shock tube flows, 225 quench of superconducting magnet, 101 shock wave, 68, 71, 74, 75, 77, 79, 80, 82, 84, 86, 87, 89–93 radical, 61 shock wave energy, 35, 47, 56 rain erosion, 52 shock wave front, 326 Rankine-Hugoniot equations, 136, 244 shock wave splitting, 322 rapid growth zone, 204, 208, 221–223 shock-induced cavitation, 94 rarefaction phase profiles, 77 similarity criterion, 75 rarefaction shocks, 237 similarity parameter, 75 rate process, 137 slip Reynolds number, 140 Rayleigh, 43 solitons, 3, 28, 30, 31 Rayleigh line relations, 191 sonoluminescence, 37, 59, 60, 62 Rayleigh’s spherical bubble collapse, 42 sonoporation, 47, 59 Rayleigh–Plesset equation, 3, 7, 8, 18 sound velocity, 3, 6, 13, 17 real gas dynamics, 231, 258 spall fracture, 88 real liquid state, 68 specific contact area, 341 reduced pressure, 237 specific humidity, 206 reduced temperature, 237 speeds of sound, 145 reduced volume, 237 SPH method, 347 reflectivity, 311 spherical bubble collapse, 43 refractive index, 52 stability limit of stationary shock, 215 regular fluids, 234 stabilization of nuclei, 72 relative humidity, 206, 207, 209–218, stagnation pressure, 56 222, 224, 227 stagnation pressure loss, 274 relative motion, 11, 13, 14, 22, 24, 25 stagnation process, 180 relaxation, 3, 12, 14, 17, 22, 24, 25, 31, standing wave, 45 68, 77, 79, 83, 137, 177 stationary normal shock, 187, 200, 206, relaxation time, 13, 14, 25, 137, 141 213 release wave, 51 steady-state heat transfer, 141 retrograde fluids, 232, 233 steepening coefficient, 117, 118 retrogradicity, 236 Stokes number, 182 streamer, 45, 46 saturation boundary, 238 stress fringes, 52 SBSL, 59, 60 stress–strain relation, 298 scattering indicatrix, 69–71 stresses, 79 schlieren method, 43 structure-time failure criterion, 88 second sound, 104, 109–111, 116, 117, subcritical flow, 197–200, 205–207, 121, 122 211–213, 225, 226 sensitivity to stress, 88 superconducting magnet, 101 shaped charge, 44 superconductive temperature sensor, shear stress, 93, 95 113, 116, 120, 124, 126, 128 shock adiabat, 325 supercritical flow, 197, 199, 200, 205, shock fitting, 209, 223 206, 210, 215, 220, 222 shock focusing, 36, 41, 58, 60 supercritical heat, 197 shock heating, 60 superfluid breakdown, 107 shock splitting, 239 superfluid component, 103, 104 shock thickness, 15, 25 superfluid helium, 99, 102 shock tube, 14, 16, 20, 22, 28, 69, 71, superfluid thermal shock tube, 119 80, 81, 83, 239, 245 superleak, 104, 105, 125 358 Index superleak-tight windows, 124, 125 two-fluid model, 102–104, 106, 143 supersaturated state, 187 type I fully dispersed wave, 148 supersonic nozzle, 233 type II fully dispersed wave, 151 superthermal conduction, 104 type III fully dispersed wave, 152 temperature/entropy diagram, 232 ultimate tensile stress, 79 tensile stress, 73, 76, 77, 79, 84, 88 ultrasound, 36, 45, 46 tensile stress field, 85 underwater explosion, 73, 76, 77 tensile wave, 38 unsteady development of shock waves, textiles, 275, 280 162 theory of shock-wave therapy, 89 unsteady periodic flow, 187, 206, 214 thermal choking, 165 unsteady rarefaction waves, 223 thermal conductivity, 340 thermal counterflow, 104 vapour subcooling, 139 thermal diffusivity, 340 vapour thermal relaxation time, 141 thermal effects, 299 vapour–droplet flow with a carrier gas, thermal equilibrium, 344 168 thermal permeability, 300 vapour–droplet mixture, 139 thermal shock wave, 112, 114, 116, 117, velocity slip, 139 119–121, 127–129 viscous correction factor, 302 thermally choked flow, 201, 202, 209 viscous dissipation, 3, 17 thermodynamics of shock compaction, void fraction, 10 338 volumetric gas concentration, 67 thermomechanical effect, 105–107 vortex line density, 108 thickness of the shock wave, 136 vortex pattern, 274 time cracks grow, 93 vortex ring, 239, 255 time evolution of shock waves, 155 time of relaxation, 79 water hammer pressure, 51, 56 tissue damage, 36, 57, 58 wave diagram, 201, 225 tissue-damage mechanism, 91 wave instabilities, 232 tissue-stimulating gelatin, 94 weak oblique shock waves, 187 total pressure, 180 Wilson point, 204 total temperature, 184 tracks of diffraction spots, 71 x-ray pulse technique, 81 transmitted compression shock wave, X-shock, 187, 188, 209, 211–214 126, 127 transonic nozzle flows, 206, 209 yield strength, 46, 56 Shock Wave Science and Technology Reference Library, Volume 1 Multiphase Flows I

About the Authors

Chapter 1

Leen van Wijngaarden University of Twente Leen van Wijngaarden is an Emeritus Professor at the Physics of Fluids University of Twente, the Netherlands. He has a PhD Enschede degree from the Technical University of Delft. He was at the Maritime Research Institute of the Netherlands from The Netherlands 1962–1966, where he did research in gravity waves and l.vanwijngaarden cavitation. From 1966–1997 he taught ﬂuid dynamics @tnw.utwente.nl in Twente, both in the Dept. of Mechanical Engineering and in the Dept. of Applied Physics. His, still continu- ing, research interests are multiphase ﬂow, acoustics and waves of various kinds. He is a member of the Dutch Academy of Arts and Sciences. He was Associate Edi- tor of the Journal of Fluid Mechanics from 1988–2000. He served as Treasurer, President and Vice-President of IUTAM (International Union of Theoretical and Ap- plied Mechanics), and was involved with the Euromech Society in many ways.

Chapter 2

Yukio Tomita Hokkaido University of Dr. Yukio Tomita is a Professor of the Faculty of Educa- Education tion at the Hokkaido University of Education, Hakodate. Faculty of Education He obtained his PhD degree in Engineering from To- hoku University in 1986. He was a Royal Society-Japan Hakodate Society for Promotion of Science Visiting Research Fel- Japan low at the University of Birmingham in 1996/1997 and [email protected] a Visiting Research Fellow at Imperial College London in 2003 through a Short-Term Research Experience Fel- lowship (MEXT). His research interests include cavitation, especially bubble dynamics, shock wave, jet and drop impact problems in connection with material erosion, medical application and environmental science. Shock Wave Science and Technology Reference Library, Volume 1 Multiphase Flows I

Chapter 3

Valery K. Kedrinskii Lavrentyev Institute of Professor Valery Kedrinskii is head of the Department Hydrodynamics of Physical Hydrodynamics and Vice-Director of the of the Russian Academy Lavrentyev Institute of Hydrodynmaics and lecturer for four-semesters general courses of physics at Novosi- of Science birsk State University. He obtained his PhD degree in Novosibirsk 1968 from the Joint Scientiﬁc Board at the Presidium Russian Federation of the Siberian Branch of the USSR Academy of Sci- [email protected] ences, Novosibirsk, and in 1978 his Doctor of Sciences for Physics and Mathematics. He is a State Prize Win- ner of the USSR (Gold Medal, 1983) for fundamental contributions to multi-phase media mechanics.

Chapter 4

Masahide Murakami University of Tsukuba Dr. Murakami is a Professor at the Graduate School of Graduate School of Systems and Information Engineering at the University Systems and Information of Tsukuba. He obtained his PhD degree in Aeronau- tics from the University of Tokyo in 1974. He has been Engineering working in the fields of superfluid thermo-fluid dynam- Tsukuba, Japan ics, space cryogenics and sports engineering, and re- [email protected] ceived the Russell B. Scott Memorial Award (Best research paper at Cryogenic Engineering Conference) in 2003. He is an international advisory editor of the journal Cryogenics. Shock Wave Science and Technology Reference Library, Volume 1 Multiphase Flows I

Chapter 5

Abhijit Guha Aerospace Engineering Dr. Abhijit Guha’s research interests are in the thermo- Department fluid-dynamics of multiphase flow, computational fluid University of Bristol dynamics, gas turbine, and solar energy. He obtained his PhD in Engineering from Trinity College, Univer- Bristol BS8 1TR sity of Cambridge, as the prestigious Prince of Wales United Kingdom Scholar. He later became a Senior Rouse Ball Scholar [email protected] at Trinity College, and then a Fellow of Gonville & Caius College, Cambridge. While at Cambridge, his research was based at the Whittle Laboratory. In 1995 he joined the University of Bristol. He has published many, key and comprehensive, mostly single-authored, fundamental as well as applied research papers in top-ranking journals, books and conferences on a wide range of in- terdisciplinary topics. He has presented many keynote lectures, short courses and invited seminars, at international conferences and reputed institutions worldwide. In 1995 he delivered the renowned VKI Lecture Series (von Karman Institute, Belgium) on Two-phase Flows with Phase Transition. In 2000 he was elected to the Editorial Board of Journal of Aerosol Science. Dr. Guha taught Two-phase Heat Transfer at University of Cam- bridge, and now teaches Fluid Mechanics, Thermody- namics, and Aircraft Propulsion at Bristol. He received the University of Bristol Teaching Excellence Award in the very year of its inception. Shock Wave Science and Technology Reference Library, Volume 1 Multiphase Flows I

Chapter 6

Can F. Delale Faculty of Aeronautics Dr. Can F. Delale is Professor of Applied Mathematics and Astronautics and Aerospace Engineering at Istanbul Technical Uni- Istanbul Technical versity. He received his PhD from Brown University in 1983, majoring in Fluid Mechanics and Thermodynam- University ics. His research interests include the kinetic theory of 34469 Maslak, Istanbul, gases, gas/liquid transitions, gas dynamics with conden- Turkey sation, wave phenomena and hydrodynamic cavitation. [email protected]

Marinus E.H. van Dongen Applied Physics Professor Marinus (Rini) van Dongen is a physicist who Department has been active in research and education in fluid dy- Eindhoven University of namics, physical gas dynamics, physical transport phenomena, waves in porous media, waves with phase tran- Technology sition, nucleation and condensation in real gases and in PO Box 513 bio-fluid dynamics. He is a member of the J.M. Burger- 5600 MB Eindhoven, scentrum, Research School for Fluid Mechanics. He has Netherlands been affiliated with Eindhoven University of Technol- [email protected] ogy and part-time with Twente University, Department of Mechanical Engineering.

Gunter¨ H. Schnerr Chair of Fluid Mechanics Dr.-Ing.habil. Gunter¨ H. Schnerr is Professor for Fluid Department for Mechanics and Gasdynamics at the Technical Uni- Gasdynamics versity of Munich. He received his Dr.-Ing. and Dr.-Ing.habil. degrees from the University of Karlsruhe Technical University (TH) in 1977 and 1986, with major scopes in Fluid Munich Mechanics and Nonequilibrium Gasdynamics. From D-85747 Garching, 2000–2003 has been affiliated with the J.M. Burgers Germany Centrum for Fluid Dynamics at Delft and as part time schnerr@flm.mw.tu- Professor with the University of Twente, The Nether- muenchen.de lands. His research interests include theory, computa- tion and experiments on transonic flows, gasdynamics with phase transition – condensation and compressible liquid flows with cavitation in macro and microscale systems, especially in fuel injection systems, with more than 170 publications dedicated to these subjects. Shock Wave Science and Technology Reference Library, Volume 1 Multiphase Flows I

Chapter 7

Gerd E.A. Meier Am Menzelberg 6 Professor Meier was the director of the Institute of D-37077 Gottingen,¨ Aerodynamics and Flow Technology in Gottingen,¨ Ger- Germany many from 1990 to 2002. This institute of the DLR – the German Aerospace Organisation – is devoted to exper- [email protected] imental and theoretical flow studies for aerospace ap- plications. From 1964 until 1990 he headed the Tran- sonic Aerodynamics group of the Max Planck Institute for Fluid Mechanics in Gottingen.¨ Both institutes were founded by Ludwig Prandtl. During his career he has authored more than 400 publications, reports, patents etc. on unsteady and steady transonic flows, multiphase flows with phase transition, supersonic and liquid jets, vortex airfoil interaction, turbulence and experimental methods, especially in flow measurement. He has edited five books on fluid dynamics topics, organized more than 10 international conferences in the field fluid mechanics, and is co-editor of two scientific journals (Ex- periments in Fluids, Progress in Aerospace Sciences). He was previously a member of the congress commit- tees of IUTAM and EUROMECH, founder of the DLR– SCHOOL–LAB and President of the Ludwig–Prandtl– Gesellschaft.

Chapter 8

Beric W. Skews Flow Research Unit Professor Beric Skews is Director of the Flow Research University of the Unit at the University of the Witwatersrand, and has Witwatersrand held visiting appointments in Canada, Japan, and Aus- tralia. He is an active researcher in the field of unsteady PO WITS, 2050 and compressible fluid flow, with particular emphasis Johannesburg on shock waves and flow visualisation. He serves on a South Africa number of international bodies in the field, and is an Tel: +27 11 7177324 editor of the journal Shock Waves. [email protected] Shock Wave Science and Technology Reference Library, Volume 1 Multiphase Flows I

Chapter 9

David Smeulders GeoTechnology David Smeulders holds an MSc in Aeronautics from Department Delft University, Netherlands and a PhD in Physics Delft University of from Eindhoven University, Netherlands. He has been working in the ﬁeld of acoustics and porous media Technology since 1988, currently as an associate professor of Petro- PO Box 5028 physics at the Delft University, Netherlands. 2600GA Delft, Netherlands [email protected] Shock Wave Science and Technology Reference Library, Volume 1 Multiphase Flows I

Chapter 10

Victor Golub Institute for High Energy Dr. Victor Golub is a head of the Physical Gasdynam- Densities ics Department at the Institute for High Energy Densi- Russian Academy of ties of the Russian Academy of Sciences. He is working in shock and detonation waves investigations in gas Sciences and granular media, ﬂow vizualisation and blast wave [email protected] attenuation. He has been corresponding member of ER- COFTAC (European Community on Flow Turbulence and Combustion) since 1993, member of the Scientiﬁc Council on Combustion and Explosion of the Presidium of the Russian Academy of Sciences since 1999, member of the International Advisory Committee of Inter- national Symposium on Shock Waves since 2002, and member of FLUCOME since 2004.

Olga Mirova Institute for High Energy Mrs. Olga A. Mirova, is a researcher of Physical Gas- Densities dynamics Department of the Institute for High Energy Russian Academy of Densities, Russian Academy of Science. She attended the Moscow Institute of Physics and Technology, where Sciences she obtained her Master of Science in 1996. She is [email protected] working in shock wave investigation in gas and granular media and blast wave attenuation.