Random stupid thoughts

I don’t know anything about fluid dynamics, bu there are some ideas that I like (they are not original) so I’ll write down here, because my memory is so bad I’ll probably forget them.

What makes water viscous is the intermolecular force (energy potential) which sticks the water molecules together. But it is also this force which makes it incompressible. So viscosity goes hand in hand together with incompressibility. For gases, the intermolecular force is so weak, that gas molecules move rather freely (no viscosity), and at the same time gases can be compressed easily. It means that you can not have both incompressibility and no-viscosity.

It is curious that there are lots of works on incompressible Euler equations (non-viscous incompressible fluids), but those Euler equations must be more geometry than physics. When things are sufficiently regular, you can forget about the viscosity and use the Euler equations as a good approximation of the Navier-Stokes equation. But when things become “turbulent”, the effect of compressibility can no longer be ignored. Shock waves are due mainly to compressibility? There are “soliton” water waves in the ocean which are so big one may think of them as shock waves, but they are on the surface (boundary between water and air) and not down inside deep water ? If one is to show some sort of regularity for water flows, one probably can’t forget the viscosity term because it’s that term which “regularizes” things.

Another stupid idea: the natural space where a dynamical system leaves should be the space where the energy function is well-defined and finite. The L2 space is a wrong space for NSE because of the viscosity term doesn’t live there. It’s a good spacefor the incompressible Euler equation, but then the incompressible Euler equation is a wrong equation! Bifurcations look more natural once one fixes “the right equation” and “the right phase space”.

Dear Prof. Dung,

Thanks for the interesting post!

It’s certainly true that inviscid is for *ideal* gases and fluids, not physically real. There, however, i think people (not me!) who work on incompressible Euler (inEE, for short) might defend by saying that it’s one good way to understand inNSE, right? i.e., to understand how shocks might form from the nonlinearity, convection flux (certainly not from the viscosity). i think this is also the case even for those who work on the (viscous and/or inviscid) Burgers equation?

i seem to not understand what you mean by saying: “But when things become “turbulent”, the effect of compressibility can no longer be ignored.” you meant “viscosity” instead of “compressibility”? shock waves are certainly due to lack of viscosity, regularizing terms as you put it. e.g, inviscid Burgers. it’s great if you could explain more, also on relation between shock and compressibility…

thanks again,

TN.

i’m a complete outsider to fluid dynamics, so don’t expect me to have correct answers to your questions :-)

I guess one can’t separate non-viscosity from compressibility (in the physical world).

When you can compress gas, you can

put lots of energy into a small region, and when the energy concentration is high, it can turn into shocks.

The viscosity (and hence imcompressibility) makes it much more difficult to “store energy”

in small regions of water. It is not only the regularizing effect of viscosity, but also the difficulty

in concentrating energy, which makes water flows (not the surface waves, but deep water) in the oceans

“so regular”.

thanks a lot for the clarification! i’d like the explanation of energy concentration, forming shocks :))

TN.