Euler’s Equation along a Streamline
Fig 12.3 Force Balance on a Moving Element 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. 12.3), the net external force acting on the
fluid element along the directions can be written as

(12.8) 
where ∆A is the crosssectional area of the ﬂuid element. By the application of Newton’s second law of motion in s direction, we get

(12.9) 
From geometry we get Hence, the final form
of Eq. (12.9) becomes



(12.10) 
Equation (12.10) is
the Euler’s equation
along a streamline. Let us consider
along the streamline so that
Again, we can write from Fig.
12.3
The equation of a streamline is given by


or, which finally leads to 

Multiplying Eqs (12.7a), (12.7b)
and (12.7c) by dx, dy and dz respectively and then substituting
the above mentioned equalities, we get
Adding these three equations, we can write
=
=
Hence,
This is the more popular form of Euler's equation because the velocity vector in a flow field is always directed along the streamline.
