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)

Where *F _{fluid}* 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:

Thus Newton's Second law equates to:

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:

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

Which leads us to:

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