Euler’s Equation along a Streamline

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 cross-sectional 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.