MTH642 Handouts Complete

Lecture notes in fluid mechanics

Laurent Schoeffel, CEA Saclay

These lecture notes have been prepared as a first course in fluid mechanics up to the presentation of the millennium problem listed by the Clay Mathematical Institute. Only a good knowledge of classical Newtonian mechanics is assumed. We start the course with an elementary derivation of the equations of ideal fluid mechanics and end up with a discussion of the millennium problem related to real fluids. With this document, our primary goal is to debunk this beautiful problem as much as possible, without assuming any previous knowledge neither in fluid mechanics of real fluids nor in the mathematical formalism of Sobolev’s inequalities. All these items are introduced progressively through the document with a linear increase in the difficulty. Some rigorous proofs of important partial results concerning the millennium problem are presented. At the end, a very modern aspect of fluid mechanics is covered concerning the subtle issue of its application to high energetic hadronic collisions. §6. Boundary conditions for ideal fluids §7. Introduction to nonlinear differential equations §8. Euler’s equations for incompressible ideal fluids §9. Potential flows for ideal fluids §10. Real fluids and Navier-Stokes equations §11. Boundary conditions for real fluids §12. Reynolds number and related properties §13. The millennium problem of the Clay Institute §14. Bounds and partial proofs §15. Fluid mechanics in relativistic Heavy-Ions collisions Fluid mechanics concerns the study of the motion of fluids (in general liquids and gases) and the forces acting on them. Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to be held true. Modern fluid mechanics, in a well-posed mathematical form, was first formulated in 1755 by Euler for ideal fluids.

Interestingly, it can be shown that the laws of fluid mechanics cover more materials than standard liquid and gases. Indeed, the idea of exploiting the laws of ideal fluid mechanics to describe the expansion of the strongly interacting nuclear matter that is formed in high energetic hadronic collisions was proposed in 1953 by Landau. This theory has been developed extensively in the last 60 years and is still an active field of research. This gives a very simple 3-steps picture of a non-trivial phenomenon observed in hot nuclear matter after the collision of high energetic heavy ions, composed of a large collection of charged particles.

(i) After the collision a nuclear medium, a zone of high density of charges, is formed with high pressure in the middle (center of the collision).

(ii) According to the laws of fluid mechanics, as we shall prove them, this implies that an acceleration field is generated from high pressures to low pressures.