Undercompressive shock wave

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Ordinary shock waves 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 travelling little 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, than 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 a wave front can 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 travelling and that little perturbations behind the front travelled slower than it.

The experiment can be made with travelling liquid steps : a thick film is spreading on a thin one. The liquid steps remain sharp when they travell 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.

For precisions about the experimental setup, see : 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

Bibliography :

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. Whitham. 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