Fluid Mechanics (Book): Difference between revisions

Continuum mechanics and fluid mechanics in particular are unusual as branches of mechanics. It has the most apparent relevance to geometry yet its development in this aspect is not as popular, and theory initially lagged behind the basic physical features - there was no account for drag. This was corrected towards the end of the 19th century, as physics developed overall and computational techniques coming directly from theory began (and still do) dominate. Most recently, Vladimir Arnol'd helped to show topological aspects of fluid mechanics: ideal fluid flows are geodesics in the infinite dimensional diffeomorphism group (such Lie groups also appear in QFT and CFT). The difficulties in describing turbulence persist, and it remains a testament to the difficulty of describing dynamics at multiple interacting scales. The Reynolds number expresses the scale invariance of specific phenomena, and is the entry point to the study of turbulent flows. In the simplest case, this number is calculated as the ratio uL/𝜈 where u is the velocity of the flow, L is the length of an immersed object, 𝜈 is the viscosity. Zero viscosity corresponds of the limit of the Navier Stokes equations to the Euler equations as studied in Arnol'd's book. Realistic fluids have some viscosity however small, and this is accounted for in the layer of fluid flow near the immersed object where the gradient of the velocity is large (friction takes over near the object). This theory, known as Boundary Layer Theory, was developed by Prandtl. A typical example would be air, which is very inviscid and corresponds to high Reynolds numbers especially in higher speed flows associated with aeronautics.