Bernoulli’s Equation

Daniel Bernoulli disclosed the equation used most frequently in engineering hydraulics in 1738. This equation relates the pressure, velocity and height in the steady motion of an ideal fluid. The usual form is v²/2 p/ρ gz = constant, where v is the velocity at a point, p the pressure, ρ the density, g the acceleration of gravity, and z the height above an arbitrary reference level. Students apply the equation without much thought, sometimes inappropriately, and have no clear idea of the conditions under which it is applicable. It appears on every general engineering examination, since it is easy to trap the unwary. Actually, it is not one relation, but three, all apparently of the same form, but applying in different situations. The three forms will be explained in this paper.

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.

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.

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.

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

Change in Gravitational Potential and Kinetic Energy

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

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

Calculation of Bernoulli’s Equation

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

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

Rearranging the equation will yield

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

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

This equation is the same as the equation of pressure with depth, that is,
P2 − P1 = ρgh.