From Newton's Second Law to the Equations of Change for a fluid

Posted in math on Tuesday, June 10 2014

Most often when you see the derivation for the general form of the equations of change for fluid dynamics it is done by considering some differential element. What follows is a way of arriving at the same result by considering Newton's second law for an arbitrary volume of fluid moving through space (problem 3D.1 in Transport Phenomena)

equation of change 1

Where Ffluid is the force exerted on the fluid by pressure and viscous forces.

We can find expressions for the momentum and the force of gravity acting on the fluid volume by considering a small volume element then integrating over the entire volume. For the forces acting on the fluid element from the outside, we consider a small surface element and integrate over the surface:

equation of change 2

Thus Newton's Second law equates to:

equation of change 3

We can simplify the left hand side of the second law expression by using the Liebniz formula for differentiating a volume integral, and the Gauss-Ostrogradskii divergence theorem:

equation of change 4

We can simplify the right hand side of the second law expression by application of the Gauss-Ostrogradskii divergence theorem:

equation of change 5

Which leads us to:

equation of change 6

Since the choice of volume was completely arbitrary we can drop the integration, leading to... $$ \frac{\partial}{\partial t} \left( \rho \mathbf{v} \right) = - \left[ \mathbf{\nabla \cdot} \rho \mathbf{v v} \right] - \nabla p - \left[ \mathbf{\nabla \cdot \tau } \right] + \rho \mathbf{g} $$