Euler’s Equation: The Equation of Motion of an Ideal Fluid

This section is not a mandatory requirement. One can skip this section (if he/she does not like to spend time on Euler's equation) and go directly to Steady Flow Energy Equation.

Using the Newton's second law of motion the relationship between the velocity and pressure field for a flow of an inviscid fluid can be derived. The resulting equation, in its differential form, is known as Euler’s Equation. The equation is first derived by the scientist Euler.

Derivation:
Let us consider an elementary parallelopiped of fluid element as a control mass system in a frame of rectangular cartesian coordinate axes as shown in Fig. 12.3. The external forces acting on a fluid element are the body forces and the surface forces.

Fig 12.2  A Fluid Element appropriate to a Cartesian Coordinate System
used for the derivation of Euler's Equation

Let X_x, X_y, X_z be the components of body forces acting per unit mass of the fluid element along the coordinate axes x, y and z respectively. The body forces arise due to external force fields like gravity, electromagnetic field, etc., and therefore, the detailed description of X_x, X_y and X_z are provided by the laws of physics describing the force fields. The surface forces for an inviscid fluid will be the pressure forces acting on different surfaces as shown in Fig. 12.3. Therefore, the net forces acting on the fluid element along x, y and z directions can be written as

Since each component of the force can be expressed as the rate of change of momentum in the respective directions, we have

	(12.5a)
	(12.5b)
	(12.5c)

s the mass of a control mass system does not change with time, is constant with time and can be taken common. Therefore we can write Eqs (12.5a to 12.5c) as

	(12.6a)
	(12.6b)
	(12.6c)

Expanding the material accelerations in Eqs (12.6a) to (12.6c) in terms of their respective temporal and convective components, we get

	(12.7a)
	(12.7b)
	(12.7c)

The Eqs (12.7a, 12.7b, 12.7c) are valid for both incompressible and compressible flow. By putting u = v = w = 0, as a special case, one can obtain the equation of hydrostatics .
Equations (12.7a), (12.7b), (12.7c) can be put into a single vector form as

	(12.7d)
	(12.7e)

where the velocity vector and the body force vector per unit volume are defined as