Under-compressive shock

A under-compressive Choc (undercompressive shock) is a Shock wave which does not obey the inequalities of Peter Lax on speed characteristic of the waves. They have something of incredible and can be observed during simple experiments.

Definitions

For the compressive shocks, speed characteristic with the back of the shock is larger than the speed of the shock, which is itself larger than characteristic speed in front of the shock. They are the conditions of Lax.

The under-compressive shock waves are shocks which do not obey in the conditions of Lax, by definition. One could thus also say, not-compressive, but it is not use.

They are incredible because the theory (recalled in Shock wave) on the formation of the shock waves seemed of general interest and because one does not include/understand easily how they can be formed and to preserve itself. Why are they preserved? Why the small disturbances are not detached from the shock? Simple experiments show that one can manufacture them easily and that they are preserved.

The first under-compressive shocks were discovered by mathematicians. The experimental discoveries came then, mainly following the meeting between a mathematician, Andrea Bertozzi, and a physician, Anne-Marie Cazabat. They realized that they worked, each one with its way, on the same equation of wave.

Evidence of the existence of the under-compressive shocks

Presentation of the experiments

The studied wave is the surface of a liquid which is spread out. One can force spreading out by gravitation and thermal Effet Marangoni. The gravitation forces the flow of the liquid towards bottom. The thermal Marangoni effect forces the flow on the side where the liquid is coldest. If one puts a drop on a plate whose temperature is not uniform (heated on a side and cooled other) it is spread out the coldest side, as if it wanted to go there.

When one places oneself in a unidimensional geometry (a theoretically infinite and perfectly rectilinear wave in a direction), the conditions of the case studied by Poisson are almost met. The equation of wave is thus very simple at the base. But there are differences, and especially a term associated with the effect of flattening caused by the surface stress. It is this term which opens the possibility of the under-compressive shocks, because it blocks the formation of the small disturbances.

In the presence of the gravitation alone or thermal Marangoni effect only, or when the two effects go in the same direction, the shocks is always compressive. One can make play Marangoni the thermal effect in opposite direction of the gravitation: it is enough to plunge a plate in a hot bath and to cool it at the other end. In this case the liquid goes up on the plate (it should be taken some precautions so that it goes). The shocks which one can then obtain are in general compressive but not all. There are several exceptions which were discovered by Anne-Marie Cazabat and two of its students, Xavier Fanton and myself (TD).

To manufacture a shock, it is enough to let assemble a first film of liquid, then to vary the slope of the plate. A second film of liquid goes up following the first. In certain cases, the walk of liquid thus obtained becomes increasingly soft, the second film remains with the drag and moves away more and more from the first. It is not then a shock wave. In other cases, the walk of liquid goes up without becoming deformed. Often it is unstable, it becomes deformed in fingers of liquid, but as instability develops slowly, one has time to see a propagation without deformation. Measurements by Laser interferometry are very precise. The images of the fingers of liquid are splendid (not those of the shocks, they are only parallel lines). The walk of liquid is never very abrupt, but as it does not change during time, one can see it like a discontinuous transition by an effect of zoom.

Any soft transition can have the air abrupt if scale is changed. This is why these steps of liquid whose slope does not exceed some pourcents can be regarded as shock waves.

The evidence is of several types:

Mathematical validity of the model

It is a question of proving that the found solutions are many solutions of the studied equations and that what one says on their subject is true, as mathematical beings. To refer to the articles of the theorists.

Predictive value of the mathematical model

One knows the result of an experiment before making it. One can then check if the experimental observations are in agreement with the mathematical predictions. One can check all that is measurable. To refer to the thesis of Xavier Fanton and my report of training course. No dissension between the theory and the experiment could be established (taking into account uncertainties of measurement and some terms neglected in the studied equation.)

Predictive value of the experiments

Certain experiments showed the existence of new, unknown under-compressive shocks of the theorists. Their existence, as solutions of the equation, was confirmed then by calculations. An experimenter can make mathematical discoveries. That often occurs, but it is not often noticed. To refer to my report of training course.

Specific evidence of the under-compressive character of the shocks

One can see directly that a shock is compressive or under-compressive by making a small disturbance beside him. If there is a side where the disturbance separates from the shock, it is that it is under-compressive. In the compressive case, the disturbance is swallowed by the shock. To refer to my report of training course.

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