98 Bernoulli’s Equation

The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant:

[latex]P+\frac{1}{2}\rho v^{2}+\rho gh=\text{constant}\\[/latex],

where P is the absolute pressure, ρ is the fluid density, v is the velocity of the fluid, h is the height above some reference point, and g is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes

[latex]{P}_{1}+\frac{1}{2}{{\rho v}_{1}}^{2}+\rho {gh}_{1}={P}_{2}+\frac{1}{2}{{\rho v}_{2}}^{2}+\rho {gh}_{2}\\[/latex].

Bernoulli’s equation is a form of the conservation of energy principle. Note that the second and third terms are the kinetic and potential energy with m replaced by ρ. In fact, each term in the equation has units of energy per unit volume. We can prove this for the second term by substituting ρ = m/V into it and gathering terms:

[latex]\frac{1}{2}{\rho v}^{2}=\frac{\frac{1}{2}{\text{mv}}^{2}}{V}=\frac{\text{KE}}{V}\\[/latex]

So [latex]\frac{1}{2}{\rho v}^{2}\\[/latex] is the kinetic energy per unit volume. Making the same substitution into the third term in the equation, we find

[latex]\rho {gh}=\frac{mgh}{V}=\frac{{\text{PE}}_{\text{g}}}{V}\\[/latex],

so ρgh is the gravitational potential energy per unit volume. Note that pressure P has units of energy per unit volume, too. Since P = F/A, its units are N/m². If we multiply these by m/m, we obtain N ⋅ m/m³ = J/m³, or energy per unit volume. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, we will look at a number of specific situations that simplify and illustrate its use and meaning.

Let us first consider the very simple situation where the fluid is static—that is, v₁ = v₂ = 0. Bernoulli’s equation in that case is

P₁ + ρgh₁ = P₂ + ρgh₂.

We can further simplify the equation by taking h₂ = 0 (we can always choose some height to be zero, just as we often have done for other situations involving the gravitational force, and take all other heights to be relative to this). In that case, we get

[latex]{P}_{2}={P}_{1}+\rho {gh}_{1}\\[/latex]

This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by h₁, and consequently, P₂ is greater than P₁ by an amount ρgh₁. In the very simplest case, P₁ is zero at the top of the fluid, and we get the familiar relationship P = ρgh. (Recall that P = ρgh and [latex]\Delta{\text{PE}}_{\text{g}}= mgh\\[/latex]) Bernoulli’s equation includes the fact that the pressure due to the weight of a fluid is ρgh. Although we introduce Bernoulli’s equation for fluid flow, it includes much of what we studied for static fluids in the preceding chapter.

Another important situation is one in which the fluid moves but its depth is constant—that is, h₁ = h₂. Under that condition, Bernoulli’s equation becomes

[latex]{P}_{1}+\frac{1}{2}{{\rho v}_{1}}^{2}={P}_{2}+\frac{1}{2}{{\rho v}_{2}}^{2}\\[/latex].

Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli’s principle. It is Bernoulli’s equation for fluids at constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path.) As we have just discussed, pressure drops as speed increases in a moving fluid. We can see this from Bernoulli’s principle. For example, if v₂ is greater than v₁ in the equation, then P₂ must be less than P₁ for the equality to hold.

There are a number of devices and situations in which fluid flows at a constant height and, thus, can be analyzed with Bernoulli’s principle.

Entrainment

People have long put the Bernoulli principle to work by using reduced pressure in high-velocity fluids to move things about. With a higher pressure on the outside, the high-velocity fluid forces other fluids into the stream. This process is called entrainment. Entrainment devices have been in use since ancient times, particularly as pumps to raise water small heights, as in draining swamps, fields, or other low-lying areas. Some other devices that use the concept of entrainment are shown in Figure 2.

Velocity measurement

Figure 3 shows two devices that measure fluid velocity based on Bernoulli’s principle. The manometer in Figure 3(a) is connected to two tubes that are small enough not to appreciably disturb the flow. The tube facing the oncoming fluid creates a dead spot having zero velocity (v₁=0) in front of it, while fluid passing the other tube has velocity v₂. This means that Bernoulli’s principle as stated in [latex]{P}_{1}+\frac{1}{2}{{\rho v}_{1}}^{2}={P}_{2}+\frac{1}{2}{{\rho v}_{2}}^{2}\\[/latex] becomes [latex]{P}_{1}={P}_{2}+\frac{1}{2}{{\rho v}_{2}}^{2}\\[/latex].

Thus pressure P₂ over the second opening is reduced by [latex]\frac{1}{2}{{\rho v}_{2}}^{2}\\[/latex], and so the fluid in the manometer rises by h on the side connected to the second opening, where

[latex]h\propto \frac{1}{2}{{\rho v}_{2}}^{2}\\[/latex]

(Recall that the symbol ∝ means “proportional to.”) Solving for v₂, we see that

[latex]{v}_{2}\propto \sqrt{h}\\[/latex].

Figure 4(b) shows a version of this device that is in common use for measuring various fluid velocities; such devices are frequently used as air speed indicators in aircraft.

Glossary

Bernoulli’s equation:

the equation resulting from applying conservation of energy to an incompressible frictionless fluid: P + 1/2pv² + pgh = constant , through the fluid

Bernoulli’s principle:

Bernoulli’s equation applied at constant depth: P₁ + 1/2pv₁² = P₂ + 1/2pv₂²