Explanation – Bernoulli’s equation is derived from the principle of conservation of mechanical energy. It states: the total mechanical energy of an object remains constant as the object moves, provided that the net work done by external, nonconservative forces is zero. The mechanical energy and the net work done are actually the same thing. Nonconservative just means depending on path taken. So a nonconservative force would be friction, since the work done by friction depends on its path, for example in a zigzag or straight path. External force refers to any force outside the object of interest. Fluid pressure is caused by collisional forces, which are non-conservative. Collisional forces refers to the forces acting perpendicular to the surface. Fluid can be accelerated due to an unbalance in pressures, thus, work is being performed by these collision, non-conservative forces, and this work is equal to the difference in total mechanical energy of the fluid.

Hydrodynamics And Hydraulics

Bernoulli’s equation is based on two main observations. First, if a fluid is traveling in a horizontal pipe and meets a section of tighter cross sectional area, the fluid pressure lessens. Newton’s second law affects this. When coming from the larger section to the smaller section, the fluid accelerates constantly with the conservation of mass (based on equation of continuity). According to second law, the accelerating fluid must be acting under an unbalanced force. An unbalanced force exists if the pressure at the larger end exceeds the pressure at the smaller end. The difference in their pressure is the net force. Bernoulli’s equation simply calculates that difference. The 2nd observation is based on elevation. When a fluid is moved to a higher elevation, the pressure at the lower elevation is greater compared to the higher elevation. Bernoulli’s equation can prove that, only if the cross section is the same all around.

Application:

These ideas benefit us every day.

The influence pressure and fluid flow has had on us is enormous. One example shows how household plumbing makes use of Bernoulli’s equation. The U-shaped region under our sinks is known as the “trap” because water collects there, and serves as a blockage against sewer gas escaping into the house. When water from a machine flows through the sewer pipe, the fast velocity causes the pressure at that point to drop. The pressure at the trap is higher due to a higher elevation. Because of this unbalance in pressure, the water is forced out of trap and into sewer line, leading to an opening in sewer gas. A properly designed plumbing system has a vent to the outside of the house. The vent makes sure the pressure in the sewer pipe and trap is constant, even when water from a machine is flowing through the sewer pipe. Thus, the vent disallows the trap from emptying, which will not allow a 2nd path for sewer gas.

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Question – What other applications use Bernoulli’s equation?

Answer – Other applications dealing with Bernoulli’s equation are the dynamic lift on airplane wings, the curve ball in baseball and the speed of water through a large tank.

Ajit Srinivas