Fluids: Bernoulli's equation

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This text is meant to accompany class discussions. It is not everything there is to know about fluids, hydrostatics, and fluid dynamics. It is meant as a prep for class.

Click for the questions that go with this reading

Bernoulli's Equation

This video can be seen on YouTube at https://youtu.be/ozoC_v_jwcA

Bernoulli's Equation is an application of the conservation of energy law. With fluids this is saying that the kinetic energy per unit volume plus the gravitational potential energy per unit volume, plus the gauge pressure is conserved.

Bernoulli's Equation is not perfect. It has its limitations.

The fluid must have a constant density.
The fluid flow is laminar, steady.
There is no friction.

Some implications of Bernoulli's Equation.

When there is no change in elevation, the pressure OF THE FLUID in the faster moving section of a streamline is lower than the slower moving region.

When a fluid is exposed to the atmosphere, its absolute pressure is 1 atm. It does not matter is the fluid is moving or not.

Without additional pressure a fluid that starts from rest, cannot reach a height higher that the starting height.

Bernoulli's equation is a relationship between energies. It's the kinetic energy per unit volume, the gravitational potential energy per unit volume, the absolute pressure, and the total energy per unit volume. The energy starts to looks similar to the energy chart for mechanical energy. Below is what it looks like now.

These can be used to understand how energy flows along a STEAMLINE. But it only applies along a stream line. However, this understanding will help with other style problems too.

Example 1 (energy charts)

Question1
Solution

A squirt gun has a pump on it that pressurizes the gun. When gun is pressurized and the trigger is pulled, the pressure pushes the water out. The water is squirted vertically to its maximum height. What to the energy charts look like at the identified locations?

Example 1.5

Question
Solution

Example 2 (energy charts)

Question2
Solution

Below is a water slide. Water comes out at the top with a small velocity. Draw the energies associated with the locations shown below.

Unlike the next tow example problems, The water flows along a streamline.

Solution Strategy 1

When fluids are involved, see if there is a streamline of fluid flow. (A streamline is a smooth line connecting two locations.) Examine two locations along the streamline with Bernoulli's principle in mind. Look at the fluid's gauge pressure, velocity, and relative heights at both locations. The energy/volume flow bar graphs may help you to think about these energies. Compare these with Bernoulli's principle. This is done in the next example.

Example 3

Question
Solution

In a post apocalyptic world, after all of the zombies have starved themselves out of existence, some clever survivors have take a large industrial pipe and turned it into a huge water container. The top of the very larger pipe is open to the atmosphere and there is a very, very, small opening a the bottom of the pipe that can be opened and closed. The opening at the bottom is opened, how fast will the water flow out if the pipe is filled to the top.

Solution Strategy 2

Sometimes you cannot connect two locations by a steam line. But you can compare two locations. Especially two locations where there are different pressures. This is demonstrated in the 3 examples problems below.

Example 4

Question
Solution on Paper
Video Solution

The pipe in the figure to the right is transporting oil (density 800 kg/m³). The velocity at point 1 is 5m/s, but at point 2 it is 10m/s. Calculate the difference in height in the two open thin tubes.

Example 5

Question
Solution
Your Paper

In a factory, a venturi tube is used to show the flow rate of a lubricant through some pipes. The diagram above illustrates the various heights the fluid rises to in the tubes. How fast is the fluid traveling under neath the pipe located at the right?

This problem is examining the change in the fluid's velocity between two locations in a pipe. The continuity equation is not an option because the areas, or diameters, under the vertical pipes are not known. Bernoulli’s equation can relate the fluid’s velocity and the gauge pressure. The gauge pressure is the difference in the fluid’s heights in the pipes.
Each vertical pipe is open the atmosphere. The difference in the fluids’ heights in the two pipes shows the gauge pressure between these locations.

Example 6

Question
Solution on Paper
Video Solution

Perfume in a bottle has a density of 940 kg/m³ and its level is h=0.020m below the nozzle as shown in the figure. Calculate the minimum speed of the air, so the liquid will reach the nozzle. [For the density of air use ρ_air =1.29 kg/m³ ]

This video can be found on YouTube at https://youtu.be/tWY7_PhAmOE

Using Bernoulli's Equation to find a force

To use pressure to find the net force, you need to find the change in pressure from Bernoulli's equation.

Here's a typical problem setup. The air, (or fluid,) is traveling at two speeds on opposite sides of a barrier -like a sail, tarp, or hard wall.

Another assumption that is often made is that the velocity of air behind a perpendicular surface that is moving through the air is zero. This might be a hand that is held outside of a moving car.

See example 7, below.

Example 7

Question
Solution on Paper
Video Solution

You are driving down the road at 70 mph and stick your hand out the window. Your hand is perpendicular to the wind. The area of your hand is 400 cm². How much force is exerted on your hand?

This video can be found on YouTube at https://youtu.be/RPgiPTESUkw