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 parallelepiped of fluid element as a control mass system in a frame of rectangular Cartesian coordinate axes as shown in Fig. 2. The external forces acting on a fluid element are the body forces and the surface forces.
Let Xx, Xy, Xz 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 Xx, Xy and Xz 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. 2. 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
The mass of a control mass system does not change with time; ρdxdydz is constant with time and can be taken common. Therefore we can write Eqs (1a to 1c) as
Expanding the material accelerations in Eqs (2a) to (2c) in terms of their respective temporal and convective components, we get
The Eqs (3a, 3b, 3c) 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 (3a), (3b), (3c) can be put into a single vector form as
Where 
the velocity vector and the body force vector per unit volume 
are defined as
Euler’s Equation along a Streamline
Derivation: Euler’s equation along a streamline is derived by applying Newton’s second law of motion to a fluid element moving along a streamline. Considering gravity as the only body force component acting vertically downward (Fig. 2), the net external force acting on the fluid element along the directions can be written as
where ∆A is the cross-sectional area of the fluid element. By the application of Newton’s second law of motion in s direction, we get
From geometry we get
Hence, the final form of Eq. (6) becomes
Equation (6) is the Euler’s equation along a streamline.
Let us consider 
along the streamline so that
Again, we can write from Fig. 2
The equation of a streamline is given by
Multiplying Eqs. (3a), (3b) and (3c) by dx, dy and dz respectively and then substituting the above mentioned equalities, we get
Adding these three equations, we can write
This is the more popular form of Euler's equation because the velocity vector in a flow field is always directed along the streamline.
Euler’s Equation in Different Conventional Coordinate System
Euler’s equation in different coordinate systems can be derived either by expanding the acceleration and pressure gradient terms of Eq. (3d), or by the application of Newton’s second law to a fluid element appropriate to the coordinate system.
Euler's Equation in Different Conventional Coordinate Systems
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