Fluid Mechanics

Short introduction to Navier-Stokes equations

Lecturer: Jakob Hærvig

Slides by Jakob Hærvig (AAU Energy) and Jacob Andersen (AAU Build)

Any experience with the Navier-Stokes Equations?

Richard Feynman on Turbulence

“Turbulence is the most important unsolved problem of classical physics.” - Eames & Flor (2011)

“There is a physical problem that is common to many fields, that is very old, and that has not been solved. It is not the problem of finding new fundamental particles, but something left over from a long time ago—over a hundred years. Nobody in physics has really been able to analyze it mathematically satisfactorily in spite of its importance to the sister sciences. It is the analysis of circulating or turbulent fluids.” - Feynman et al. (1965)

A Millennium Problem

The Clay Mathematics Institute has identified the existence and smoothness of solutions to the Navier-Stokes equations as one of the seven Millennium Prize Problems.

Sketch of the Problem
Prove or give a counter-example that in 3-D, solutions to the incompressible Navier-Stokes equations always exist and are smooth for all time.

Prize
$1,000,000 for a correct solution.

The Navier-Stokes Equations

$x$-direction:

$ \rho $ $ \bigg( $ $ \dfrac{\partial u}{\partial t} $ $ + $ $ u \dfrac{\partial u}{\partial x} $ $ + $ $ v \dfrac{\partial u}{\partial y} $ $ + $ $ w \dfrac{\partial u}{\partial z} $ $ \bigg) = -$ $ \dfrac{\partial p}{\partial x} $ $ + $ $ \mu $ $ \bigg( $ $ \dfrac{\partial^2 u}{\partial x^2} $ $ + $ $ \dfrac{\partial^2 u}{\partial y^2} $ $ + $ $ \dfrac{\partial^2 u}{\partial z^2} $ $ \bigg) + $ $ \rho g_x $

$y$-direction:

$ \rho $ $ \bigg( $ $ \dfrac{\partial v}{\partial t} $ $ + $ $ u \dfrac{\partial v}{\partial x} $ $ + $ $ v \dfrac{\partial v}{\partial y} $ $ + $ $ w \dfrac{\partial v}{\partial z} $ $ \bigg) = -$ $ \dfrac{\partial p}{\partial y} $ $ + $ $ \mu $ $ \bigg( $ $ \dfrac{\partial^2 v}{\partial x^2} $ $ + $ $ \dfrac{\partial^2 v}{\partial y^2} $ $ + $ $ \dfrac{\partial^2 v}{\partial z^2} $ $ \bigg) + $ $ \rho g_y $

$z$-direction:

$ \rho $ $ \bigg( $ $ \dfrac{\partial w}{\partial t} $ $ + $ $ u \dfrac{\partial w}{\partial x} $ $ + $ $ v \dfrac{\partial w}{\partial y} $ $ + $ $ w \dfrac{\partial w}{\partial z} $ $ \bigg) = -$ $ \dfrac{\partial p}{\partial z} $ $ + $ $ \mu $ $ \bigg( $ $ \dfrac{\partial^2 w}{\partial x^2} $ $ + $ $ \dfrac{\partial^2 w}{\partial y^2} $ $ + $ $ \dfrac{\partial^2 w}{\partial z^2} $ $ \bigg) + $ $ \rho g_z $

Continuity equation (incompressible flow):

$ \dfrac{\partial u }{\partial x} + \dfrac{\partial v}{\partial y} + \dfrac{\partial w}{\partial z} = 0 $