## Saturday, July 10, 2010

### How Euler Derived the Momentum Equations

In this post, I will present Euler's derivation of the momentum balance equations for a fluid flow. Please note that I will use modern notation with a few modifications to the original derivation to clarify certain issues. Nonetheless, the core of Euler's derivation is maintained.

Considering two dimensional flows with axial and transverse velocities denoted by u(x,y,t) and v(x,y,t), respectively, Euler first forms the total differentials of these fields
Next, dividing both equations by dt, we obtain
but
then, upon substitution, we recover
One can infer that Euler is heading towards Newton's law. The previous expressions represent the acceleration of a fluid element. As to the forces, Euler identifies pressure, friction, and gravity. For the time being, he abandons friction and focuses on pressure and gravity.

For gravity, the force is in the transverse direction and is given by
where
is the mass of the fluid element.

For the pressure, he considers a rectangular fluid element as shown below.
Euler then assumes that the pressure at point L is p and deduces the values at the corners of the control volume. The net force acting on each side is calculated by taking the average pressure at the vertices defining that side and multiplying it by the area. This is shown in the figure below

Then, the net forces in the axial and transverse directions are given by
and

Now, by applying Newton's law, we have
or, by projecting in both directions, we have
and
Upon expansion, we get
and
Finally, we get the Euler equations
and

Voila!

[Next: Momentum Eqs. in Cartesian Coordinates]

Cite as: