Finally, a starting point?

I must have been attempted the study of Continuum Mechanics and the equations of change a dozen times -as many as the average Spaniard has attempted the study of languages :). However, without the mathematics, the attempt is barren.

I think that a good starting point -at least, it seems to work for me- is the mathematics of the point transformation and the deformation tensor. A couple of solid, key ideas for each of them, and the land smooths out. All of them requires certain familiarity with vector calculus and geometry. There are several shortcuts to show the momentum equations of Navier-Stokes in a reasonable way avoiding the complexities of the pure mathematical thinking -i.e. the transformations indicated above. From my point of view, all of them are flawed and self-defeating. Math can't just be skip. I think Einstein said something like "things must not be made more complex than they are... Nor simpler".

The starting point is to consider a volume of matter, always consisting of the same particles. As time goes by, particles vary, change position and the volume itself gets deformed. However, the position of a given point in the fluid is related to its initial position by the means of a function. This is the key statement of the Fluid Dynamics and its foundations as a continuum.

This approach or philosophy contradicts the kinetic theory of gases -or contradicted it in the mid-19th century-, which focuses on molecular motion. If it is random, as assumed, it cannot exist any analytical mathematical function of the sort to connect one event at t=t and another at origins, t=0. However, if the volume is taken very small but large enough compared to the molecular scale, as to have average properties defined, the approach works. In fact, although molecules move at random, our physical experience shows that fluids also behave as a purposely bulk.

Given the concerns from the kinetic theory, I state that this is the reason for many molecular approaches to the problem done in the early years. And it is nice to show them in this context. Navier arrived to the same results as Stokes', but through molecular considerations (which did not stand the test of time, by the way).

You have to believe me... The whole thing takes on a different, more sensical appearance from this starting point. The subject is difficult but, through this approach, many of its parts are well-settled. It takes time and efforts to learn, assimilate and master them. If you don't apply yourself, you keep making new and challenging what is old knowledge.

Such efforts I poured today... I also enjoyed beer and people.

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