13.19:
Bernoulli’s Principle
Bernoulli's equation applies to any incompressible fluid in a laminar, steady flow. A static fluid has zero kinetic energy density. Thus, Bernoulli's equation applied to a static, incompressible fluid reduces to an expression relating the pressures at different heights.
If a fluid moves horizontally, the gravitational energy density is constant at all points. In this case, Bernoulli's equation relates the pressure at any two points on the streamline with the fluid's speed at those points. This simplified equation is called Bernoulli's principle.
A fluid's passage through a region with high speed reduces its pressure. If another fluid can access this region, it will rush in. This process is called entrainment and has many applications.
For instance, Bunsen burners use natural gases to entrain air for combustion, and atomizers use air to entrain perfume droplets.
13.19:
Bernoulli’s Principle
Bernoulli's equation incorporates how fluid pressure changes across a static, incompressible fluid by equating the kinetic energy contribution to zero. It is also helpful in analyzing horizontal flows in which the gravitational energy density is constant throughout. The latter equation is so useful that it is called Bernoulli's principle. According to Bernoulli's principle, the fluid pressure drops if the speed increases and vice versa.
Bernoulli's principle has several applications. It is used to implement a phenomenon called entrainment, wherein one fluid is used to change the pressure of a region accessible to another fluid. This pressure difference then affects the motion of the second fluid. Bernoulli's principle is also helpful in measuring the unknown speed of a fluid.
Consider a manometer, which is a tube with two openings. One opening directly opposes the fluid flow, causing it to come to rest abruptly in front of it. The other is along the fluid flow. The tube contains a liquid of known density. Since the speed of the incoming fluid is different across its ends, the pressure is also different. By applying Bernoulli's principle, it can be shown that the pressure difference is proportional to the square of the speed of the incoming fluid. This pressure difference, in turn, causes the liquid inside the manometer to have different heights on the two ends. Since the height difference is proportional to the pressure difference across the two ends, it is proportional to the square of the incoming fluid's speed. Thus, by measuring the height difference of the liquid inside the manometer, the speed of the incoming fluid is determined.
This text is adapted from Openstax, University Physics Volume 1, Section 14.6: Bernoulli's Equation.