Undercompressive shock wave
Encyclopedia
An undercompressive shock wave is a shock wave
that does not fulfill the Peter Lax
conditions.
s are compressive, that is, they fulfill the Peter Lax
conditions: the characteristic speed
behind the shock is greater than the speed of the shock, which is greater than the characteristic speed in front of the shock. The characteristic speed is the speed of small, traveling perturbations. The Lax conditions seem to be necessary for a shock wave to come to existence; if the top of a wave goes faster than its bottom, then the wave front becomes sharper and sharper and eventually becomes a shock wave (a "discontinuous" wave, a sharp wave front which remains sharp when it travels).
A shock wave is undercompressive if and only if the Lax conditions are not fulfilled. Undercompressive shock waves are astonishing: how can a wave front remain sharp if little perturbations can escape from it? At first sight, it seems that such a wave should not exist. But it exists. It has been observed that a sharp wave front remained sharp in its traveling and that little perturbations behind the front traveled slower than it.
The experiment can be made with traveling liquid steps : a thick film is spreading on a thin one. The liquid steps remain sharp when they travel because the spreading is enhanced by the Marangoni effect
. Making little perturbations with the tip of a hair, one can see whether shock waves are compressive or undercompressive.
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...
that does not fulfill the Peter Lax
Peter Lax
Peter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields...
conditions.
Details
Ordinary shock waveShock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...
s are compressive, that is, they fulfill the Peter Lax
Peter Lax
Peter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields...
conditions: the characteristic speed
Speed
In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
behind the shock is greater than the speed of the shock, which is greater than the characteristic speed in front of the shock. The characteristic speed is the speed of small, traveling perturbations. The Lax conditions seem to be necessary for a shock wave to come to existence; if the top of a wave goes faster than its bottom, then the wave front becomes sharper and sharper and eventually becomes a shock wave (a "discontinuous" wave, a sharp wave front which remains sharp when it travels).
A shock wave is undercompressive if and only if the Lax conditions are not fulfilled. Undercompressive shock waves are astonishing: how can a wave front remain sharp if little perturbations can escape from it? At first sight, it seems that such a wave should not exist. But it exists. It has been observed that a sharp wave front remained sharp in its traveling and that little perturbations behind the front traveled slower than it.
The experiment can be made with traveling liquid steps : a thick film is spreading on a thin one. The liquid steps remain sharp when they travel because the spreading is enhanced by the Marangoni effect
Marangoni effect
The Marangoni effect is the mass transfer along an interface between two fluids due to surface tension gradient...
. Making little perturbations with the tip of a hair, one can see whether shock waves are compressive or undercompressive.
Non-linear waves and the classical theory of shock waves
- J. David Logan. An introduction to nonlinear partial differential equations Wiley-Interscience 1994
- G. B. WhithamGerald B. WhithamGerald Beresford Whitham is an American applied mathematician and the Charles Lee Powell Professor of Applied Mathematics of Applied & Computational Mathematics at the California Institute of Technology. He received his Ph.D. from the University of Manchester in 1953 under the direction of Sir...
. Linear and non-linear waves Wiley-Interscience 1974 - Peter D. Lax. Hyperbolic systems of conservation laws and the mathematical theory of shock waves Society for industrial and applied mathematics Philadelphia, Pennsylvania 1973, Hyperbolic systems of conservation laws II Comm. Pure Appl. Math., 10 :537-566, 1957
The mathematical theory of undercompressive shock waves
- M. Shearer, D.G. Schaeffer, D. Marchesin, P. Paes-Leme. Solution of the Riemann problem for a prototype 2 X 2 system of non-strictly hyperbolic conservation laws Arch. Rat. Mech. Anal. 97 :299-320, 1987
- Andrea L Bertozzi, A. Munch, M. Shearer, Undercompressive Shocks in Thin Film Flow, Physica D, 134(4), 431-464, 1999
- A. Munch. Shock transition in Marangoni and gravitation driven thin film flow 1999
- A. Munch, A. L. Bertozzi, Rarefaction-Undercompressive Fronts in Driven Films, Physics of Fluids (Letters) 11(10), pp. 2812-2814, 1999
Experiments with liquid films
- V. Ludviksson, E. N. Lightfoot. The dynamics of thin liquid films in the presence of surface_tension gradients AIChE Journal 17 :5, 1166-1173, 1971
- Herbert E. Huppert. Flow and instability of a viscous current down a slope Nature Vol. 300, 427-429, 1982
- A.M. Cazabat, F. Heslot, S.M. Troian, P. Carles. Fingering instability of thin spreading films driven by temperature gradients Nature Vol. 346, 824-826 1990
Experimental undercompressive shock waves
- X. Fanton. Etalement et instabilités de films de mouillage en présence de gradients de tension superficielle Thèse, LPMC, Collège de France 1998
- A.L. Bertozzi, A. Münch, X. Fanton, A.M. Cazabat, Contact Line Stability and "Undercompressive Shocks" in Driven Thin Film Flow, Physical Review Letters, Volume 81, Number 23, 7 December 1998, pp. 5169-5172
- T. Dugnolle, Des chocs non-classiques lors de l'étalement forcé d'un liquide, Mémoire de DEA (Paris 6, Physique des Liquides), LPMC, Collège de France 1999