17 From F=ma to Navier-Stokes

In the last section, we developed a framework to quantify how flux divergence creates changes over time in the concentration of a substance.  We will now apply that framework to the concentration of momentum. This will lead us to the Navier-Stokes equation, the fluid mechanics equivalent of F=ma that you studied in first quarter physics.  We usually just call it the momentum equation.

First, recall the definition:  momentum = mass times velocity.

We will consider the momentum per unit mass of the fluid – its concentration.  Notice that the momentum per unit mass is the same thing as the velocity of the fluid.  Thus, when we are thinking about a time rate of change of momentum concentration, what we really mean is simply the acceleration.   Convergences and divergences of momentum fluxes within the fluid will lead to acceleration and are thus equivalent to forces.  We will have advective and diffusive fluxes of momentum as well as external and internal forces which act like a source terms for momentum.

The Navier-Stokes equation:

    \[{ \frac{D \underline{\textbf{u}}}{Dt}}={ \frac{\partial \underline{\textbf{u}}}{\partial t}}+\left( \underline{\textbf{u}} \cdot \nabla \right) \underline{\textbf{u}} = K \nabla^2 \underline{\textbf{u}} + \sum \frac{Forces}{mass} \]

Compare this momentum equation to the advection-diffusion equation for substances that we defined in the previous section. This is the Lagrangian form, where the advective terms appear on the left side.

This is a vector equation for the three-dimensional velocity and so it can also be broken out into three separate equations for the acceleration along each axis.