Mechanical Properties of Fluids

Bernoulli’s Principle and Equation

Do you know even common garden equipment such as garden hose in certain situations can follow the Bernoulli’s principle? A garden hose can really be an effective piece for tormenting people by simply following Bernoulli’s principle/ Bernoulli’s equation. How? Let us understand.

Water runs out of garden hose but if you use your thumb and block a part of the opening of the hose, it will release the water faster which most definitely can soak anyone near the vicinity, hence, the torment part. However, if we remove the finger, the flow will be back to normal. Bernoulli’s principle forms the basis of this phenomenon.

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Bernoulli’s Equation and Principle

Bernoulli’s principle, also known as Bernoulli’s equation, will apply for fluids in an ideal state. Therefore, pressure and density are inversely proportional to each other. This means that a fluid with slow speed will exert more pressure than a fluid which is moving faster.

In this case, fluid refers to not only liquids but gases as well. This principle forms the basis of many applications. Some very common examples are an aeroplane that tries to stay aloft or even the most common everyday things such as a shower curtain billowing inward.

Browse more Topics under Mechanical Properties Of Fluids

The same phenomenon happens in the case of the river when there is a change in the width of the river. The speed of the water decreases in wider regions whereas the speed of water increases in the narrower regions.

You will think that the pressure within the fluid will increase. However, contrary to the above statement, the pressure within the fluid in the narrower parts will decrease and the pressure inside the fluid will increase in the wider parts of the river.

Swiss scientist Daniel Bernoulli while experimenting with fluid inside the pipes led to the discovery of this concept. He observed in his experiment that the speed of the fluid increases but the internal pressure of the fluid decreases. He named this concept as Bernoulli’s principle.

Definitely,  the concept is difficult to understand and quite complicated. It is possible to think that the pressure of water will increase in tighter spaces. Indeed, the pressure of water increases in the tighter spaces but pressure within the water will not increase.

Thus, the surrounding of the fluid will experience the increase in the pressure. The change in the pressure will also result in the change in speed of the fluid. Now, let us understand this concept clearly.

Deriving Bernoulli’s Equation

Mechanism of fluid flow is a complex process. However, it is possible to get some important properties with respect to streamline flows by using the concept of conservation of energy. Let us take an example of any fluid moving inside a pipe. The pipe has different cross-sectional areas in different parts and is present in different heights. Refer to the diagram below.

Bernoulli’s equation

Now we will consider that an incompressible fluid will flow through this pipe in a steady motion. As per the concept of the equation of continuity, the velocity of the fluid should change. However, to produce acceleration, it is important to produce a force. This is possible by the fluid around it but the pressure must vary in different parts.

Bernoulli’s equation is the general equation that describes the pressure difference in two different points of pipe with respect to velocity changes or change in kinetic energy and height changes or change in potential energy. The relationship was given by Swiss Physicist and Mathematician “Bernoulli” in the year 1738.

Learn more about Surface Tension here in detail.

General Expression of Bernoulli’s Equation

Let us consider two different regions in the above diagram. Let us name the first region as BC and the second region as DE. Now consider the fluid was previously present in between B and D. However, this fluid will move in a minute (infinitesimal) interval of time (∆t).

If the speed of fluid at point B is v1 and at point D is v2. Therefore, if the fluid initially at B moves to C then the distance is v1∆t. However, v1∆t is very small and we can consider it constant across the cross-section in the region BC.

Similarly, during the same interval of time ∆t the fluid which was previously present in the point D is now at E. Thus, the distance covered is v2∆t. Pressures, P1 and P2, will act in the two regions, A1 and A2, thereby binding the two parts. The entire diagram will look something like the figure given below.

Bernoulli's equation

Finding the Work Done

First, we will calculate the work done (W1) on the fluid in the region BC. Work done is

W1 = P1A1 (v1∆t) = P1∆V

Moreover, if we consider the equation of continuity, the same volume of fluid will pass through BC and DE. Therefore, work done by the fluid on the right-hand side of the pipe or DE region is

W2 = P2A2 (v2∆t) = P2∆V

Thus, we can consider the work done on the fluid as – P2∆V. Therefore, the total work done on the fluid is

W1 – W2 = (P1 − P2) ∆V

The total work done helps to convert the gravitational potential energy and kinetic energy of the fluid. Now, consider the fluid density as ρ and the mass passing through the pipe as ∆m in the ∆t interval of time.

Hence, ∆m = ρA1 v1∆t = ρ∆V

Learn more about Viscosity here in detail.

Change in Gravitational Potential and Kinetic Energy

Now, we have to calculate the change in gravitational potential energy ∆U.

Bernoulli's equation

Similarly, the change in ∆K or kinetic energy can be written as

Bernoulli's equation

Calculation of Bernoulli’s Equation

Applying work-energy theorem in the volume of the fluid, the equation will be

Bernoulli's equation

Dividing each term by ∆V, we will obtain the equation

Bernoulli's equation

Rearranging the equation will yield

Bernoulli's equation

The above equation is the Bernoulli’s equation. However, the 1 and 2 of both the sides of the equation denotes two different points along the pipe. Thus, the general equation can be written as

Bernoulli's equation

Thus, we can state that Bernoulli’s equation state that the Pressure (P), potential energy (ρgh) per unit volume and the kinetic energy (ρv2/2)   per unit volume will remain constant.

Important Points to Remember

It is important to note that while deriving this equation we assume there is no loss of energy because of friction if we apply the principle of energy conservation. However, there is actually a loss of energy because of internal friction caused during fluid flow. This, in fact, will result in the loss of some energy.

Limitations of the Applications of Bernoulli’s Equation

One of the restrictions is that some amount of energy will be lost due to internal friction during fluid flow.  This is because fluid has separate layers and each layer of fluid will flow with different velocities. Thus, each layer will exert some amount of frictional force on the other layer thereby losing energy in the process.

The proper term for this property of the fluid is viscosity. Now, what happens to the kinetic energy lost in the process? The kinetic energy of the fluid lost in the process will change into heat energy. Therefore, we can easily conclude that Bernoulli’s principle is applicable to non-viscous fluids (fluids with no viscosity).

Another major limitation of this principle is the requirement of the incompressible fluid. Thus, the equation does not consider the elastic energy of the fluid. However, elastic energy plays a very important role in various applications. It also helps us to understand the concepts related to low viscosity incompressible fluids.

Furthermore, Bernoulli’s principle is not possible in turbulent flows. This is because the pressure and velocity are constantly fluctuating in case of turbulent flow.

What will happen to Bernoulli’s equation if a fluid is at rest or the velocity is zero?

When the velocity is zero, the equation will become

Bernoulli's equation

This equation is the same as the equation of pressure with depth, that is,

P2 − P1 = ρgh.

Solved Examples for You

Question: Which of the following fluid flow do not follow Bernoulli’s equation?

  1. Unsteady
  2. Rotational
  3. Turbulent
  4. All of the above

Solution: Option 4. The equation is applicable only to streamline and steady flows.

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