PDE conversion

(Book: 5.7)

Next, we will convert the momentum equation into a PDE as a tool to eventually find detailed solutions of flow fields. The momentum equations for inviscid flow in PDE form are called the Euler equations. The momentum equations for viscous flow in PDE form are called the Navier-Stokes equations.

We start with the derived integral momentum equation,

Write this out in index notation, using the expression for F_i from the previous section:

Use the divergence theorem twice:

So, the conservation PDE form of the momentum equations is:

To get the also useful nonconservative form, we can again differentiate out:

The terms between the first set of parentheses are zero according to the continuity equation derived earlier. So we get the nonconservative Navier-Stokes equations:

We can again make this more physical by recognizing the Lagrangian derivative of the velocity in it: Hence is the gravity force per unit volume and the gradient of the stress tensor is the net surface force per unit volume.

Exercise:

Derive the corresponding Euler equation by restricting the total stress T_ij. Write the equation in vector form. Explain by a simple example that the pressure does not give you the net force per unit volume on the fluid, but that you need the pressure gradient.

You should now be able to do questions 5.4, 5.6