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
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(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
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(12.9) |
From geometry we get Hence, the final form
of Eq. (12.9) becomes
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(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
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or, which finally leads to |
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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.
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