Chapter 4 : Conservation Equations and Analysis of Finite Control Volume
Lecture 12 :

Euler’s Equation along a Streamline

Fig 12.3  Force Balance on a Moving Element Along a Streamline

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


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. (12.9) becomes


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




This is the more popular form of Euler's equation because the velocity vector in a flow field is always directed along the streamline.