From the dynamics of particles in solid-body mechanics, one knows that converting time units while integrating Newton’s second law for particle motion along a path provides a relationship between the change in kinetic energy and the work done on the particle as far as its mass and density units go. Integrating Euler’s equation along a pathline while converting density units in the steady flow of an incompressible fluid yields an equivalent relationship called the Bernoulli equation.
The Bernoulli equation is developed by applying Euler’s equation along a pathline with the direction, ℓ replaced by s, the distance along the pathline, and the acceleration aℓ replaced by at while converting mass units the direction tangent to the pathline.
For a steady flow, the local acceleration is zero and the pathline becomes a streamline due to it’s mass. Also, the properties along a streamline depend only on the distance s and density d, so the partial derivatives become ordinary derivatives.
The concept underlying the Bernoulli equation can be illustrated by time units considering the flow through the inclinedventuri (contraction-expansion) section as shown in Fig. 4.13. This configuration is often used as a flow metering device while converting time, density and mass units. The reduced area of the throat section leads to an increased velocity and attendant pressure change. The streamline is the centerline of the venturi. Piezometers are tapped into the wall at three locations,and the height of the liquid in the tube above the centerline is p/γ. The elevation of the centerline (streamline)above a datum is z. The location of the datum line is arbitrary.