http://www.ck12.org Chapter 12. Fluid Mechanics
Bernoulli’s Equation:P 1 +^12 ρv 12 +ρgh 1 =P 2 +^12 ρv 22 +ρgh 2 =constantwherePis the pressure of the fluid,ρis the density of the fluid,vis the velocity of the fluid, andhis the height
of the fluid. Though each term in the equation has units of pressure, Bernoulli derived the equation based on the
conservation of energy. The equation is, in fact a statement of the conservation of energy.
A demonstration of this principle can be seen with a device called a venture tube,Figure12.17.
FIGURE 12.17
Notice in theFigure12.17 that where the fluid has a greater velocity, the vertical tube as a lower water level. This
is the result of a lower internal pressure in the fluid at this point. If you have ever placed your finger partly over the
opening of a garden hose, you’ve probably seen water velocity increase. The water sprays out! In the same way, in
Figure12.17, equal volumes of fluid must flow through the large cross sectional area(A 1 )and small cross sectional
area(A 2 )of the tube. Therefore, the flow through theA 2 must have a greater velocity than the flowA 1.
- Pressure is defined asP=FA
- Density is defined as the ratio of mass to volumeρ=Vm
3.Pascal’s Principle: A confined incompressible fluid under pressure will transmit that pressure equally through-
out the system. P 1 =P 2 →AAoutin=FFoutin. The mathematical relationship that follows from Pascal’s principle
shows how the application of a force in one part of the system can be multiplied in another part of the system.
4.Archimedes’ Principle: The buoyant force on a submerged or partially submerged object is equal to the weight
of fluid it displaces.
5.Bernoulli’s Principle: At those points in space where the velocity of a fluid is high, the pressure is low, and at
those points in space where the velocity of a fluid is low, the pressure is high.
The mathematical equation that Bernoulli derived based on this principle is stated below.
Bernoulli’s Equation:P 1 +^12 ρv 12 +ρgh 1 =P 2 +^12 ρv 22 +ρgh 2 =constant