EULER EQUATION OF MOTION ALONG THE STREAMLINE
- Consider a streamline of steady flow of an ideal fluid along a stream tube with flow taking place in s-direction. Take a cylindrical element of cross-sectional area dA and length ds figure showing a free body diagram for the cylindrical element.
- Since, the forces due to gravity and pressure are considered in Euler’s equation of motion, therefore, the forces acting on the cylindrical dement are:
(i) Pressure force pdA in the direction of flow.
(ii) Pressure force (p + dp)dA opposite to the direction of flow.
(iii) Weight of the element ρgdAdS - Applying Newton’s second law of motion,
KINETIC ENERGY CORRECTION FACTOR
- In Bernoulli’s equation, the velocity head is computed on the basis of the assumption that velocity is uniform
over the entire cross section of the stream tube. But in the case of flow of real fluids, the velocity distribution across any cross sectional area of the flow passage is not uniform - So, the actual kinetic energy possessed by the fluid is different from that computed by using the mean velocity.
- To obtain actual kinetic energy, kinetic correction factor (α) is multiplied to kinetic energy obtained through mean velocity.
- Consider fluid (ρ) is flowing in a circular pipe of area A. Take an elementary strip of area dA at which velocity of flow is ‘v’ at a cross-section.
Kinetic energy passing through the elementary area dA is
- For laminar flow in circular pipe, α = 2 and for parallel plate, α = 1.543.
- Value of α for laminar flow is greater than for the turbulent flow because in laminar flow velocity gradient across the section is greater than the turbulent flow.
- Accordingly the modified Bernoulli’s equation between any two section is
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