Learning Objectives

The probability that the Bernoulli trials yield four successes followed by a failure is: P(SSSSF)=ppppq =(0.70)4(0.30)=0.0720. Our next tool is extremely powerful and very useful in science. It is the binomial probability distribution. Suppose that we plan to perform/observe n Bernoulli trials. Let X denote the total number of successes in the. Bernoulli’s principle formulated by Daniel Bernoulli states that as the speed of a moving fluid increases (liquid or gas), the pressure within the fluid decreases. Although Bernoulli deduced the law, it was Leonhard Euler who derived Bernoulli’s equation in its usual form in the year 1752.

By the end of this section, you will be able to:

Explain the terms in Bernoulli’s equation.
Explain how Bernoulli’s equation is related to conservation of energy.
Explain how to derive Bernoulli’s principle from Bernoulli’s equation.
Calculate with Bernoulli’s principle.
List some applications of Bernoulli’s principle.

When a fluid flows into a narrower channel, its speed increases. That means its kinetic energy also increases. Where does that change in kinetic energy come from? The increased kinetic energy comes from the net work done on the fluid to push it into the channel and the work done on the fluid by the gravitational force, if the fluid changes vertical position. Recall the work-energy theorem,

[latex]{W}_{text{net}}=frac{1}{2}{mv}^{2}-frac{1}{2}{{mv}_{0}}^{2}[/latex].

There is a pressure difference when the channel narrows. This pressure difference results in a net force on the fluid: recall that pressure times area equals force. The net work done increases the fluid’s kinetic energy. As a result, the pressure will drop in a rapidly-moving fluid, whether or not the fluid is confined to a tube.

There are a number of common examples of pressure dropping in rapidly-moving fluids. Shower curtains have a disagreeable habit of bulging into the shower stall when the shower is on. The high-velocity stream of water and air creates a region of lower pressure inside the shower, and standard atmospheric pressure on the other side. The pressure difference results in a net force inward pushing the curtain in. You may also have noticed that when passing a truck on the highway, your car tends to veer toward it. The reason is the same—the high velocity of the air between the car and the truck creates a region of lower pressure, and the vehicles are pushed together by greater pressure on the outside. (See Figure 1.) This effect was observed as far back as the mid-1800s, when it was found that trains passing in opposite directions tipped precariously toward one another.

Figure 1. An overhead view of a car passing a truck on a highway. Air passing between the vehicles flows in a narrower channel and must increase its speed (v₂ is greater than v1), causing the pressure between them to drop (P_i is less than P_o). Greater pressure on the outside pushes the car and truck together.

Making Connections: Take-Home Investigation with a Sheet of Paper

Hold the short edge of a sheet of paper parallel to your mouth with one hand on each side of your mouth. The page should slant downward over your hands. Blow over the top of the page. Describe what happens and explain the reason for this behavior.

Roulette Bernoulli 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.

Making Connections: Conservation of Energy

Conservation of energy applied to fluid flow produces Bernoulli’s equation. The net work done by the fluid’s pressure results in changes in the fluid’s KE and PE_g per unit volume. If other forms of energy are involved in fluid flow, Bernoulli’s equation can be modified to take these forms into account. Such forms of energy include thermal energy dissipated because of fluid viscosity.

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.

Bernoulli’s Principle—Bernoulli’s Equation at Constant Depth

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.

Example 1. Calculating Pressure: Pressure Drops as a Fluid Speeds Up

In Example 1 from Flow Rate and Its Relation to Velocity, we found that the speed of water in a hose increased from 1.96 m/s to 25.5 m/s going from the hose to the nozzle. Calculate the pressu