Concept explainers
To Derive:The Bernoulli Equation in more general way
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Explanation of Solution
Introduction:
Bernoulli’s equation describes the quantitative relationship between the pressure and velocity in fluids. This is derived by Daniel Bernoulli, a Swiss Physicist and Mathematician in the year 1738.
FIGURE: 1
Consider the fluid flowing through a tube which varies in elevation as well as in cross-sectional area, as shown in the figure 1.The work-energy theorem can be applied to a parcel of fluid between the points 1 and 2. Let
The expression for work-energy theorem can be written as:
Where,
Substituting for
Where,
Substituting for
The fluid behind the parcel (that is fluid in parcel’s left in the diagram) pushes on the parcel with a force of magnitude
Since area
Where,
The fluid in the front of the parcel (that is fluid in parcel’s left in the diagram) pushes on the parcel with a force of magnitude
Since area
Where,
This work done by the fluid in the front of the parcel is negative because the applied force and displacement are in the direction opposite to the flow of fluid.
Thus, the total work done is,
Substituting for
On rearranging,
This is the Bernoulli’s equation for steady, non-viscous flow of an incompressible fluid.
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