Bernoulli’s equation at a Constant depth

According to hydrodynamics, liquid can flow in two manners: turbulent and laminar. In experimental studies, the laminar flow has its smooth motion and the easy applicability of limitations. When water flows through a narrow pipe, it has a velocity and travels with pressure. To explain the flow of a liquid, we need Bernoulli’s equation at a constant depth. According to his equation, the pressure exerted on the laminar flow of fluid and the velocity of moving water are inversely proportional to meeting the condition of equality.

What is Bernoulli’s equation?

When the liquid flows through a channel with a consistent cross-sectional area, it has a kinetic energy to the velocity. In this condition, the pressure exerted on the flowing water is the same. However, when the channel gets constricted, the cross-sectional area decreases. Due to the decrease in space, the pressure increases—the force changes impact the velocity of the water flow.

Similarly, when the channel suddenly expands, the pressure on the water will fall, further impacting the flow’s velocity. Bernoulli’s equation assumptions draw a relationship between the pressure and velocity of water laminar flow through a narrow channel. Bernoulli has successfully given a mathematical formula for water flow through the track while relating pressure and velocity.

Limitations to Consider for Bernoulli’s Equation

Bernoulli studied the change in the water pressure and velocity when it flowed through a channel having a different height and cross-sectional area. However, he had to put certain restrictions or limitations to present his theory. These are known as the Bernoulli’s equation assumptions, which are as follows:

1. The water flow velocity is uniform, i.e., the change in velocity of different intervals will be the same.
2. The water flow needs to be steady and uniform, which means that there shouldn’t be any turbulence, and the volume of the water must be the same.
3. The water flow needs to be irrotational.
4. The flowing liquid can’t be compressed. When external pressure is applied, the flowing fluid shouldn’t compress.
5. Energy loss will be zero for the water flow. Yes, there can be energy increase or decrease based on the changing velocity, but there shouldn’t be any loss.
6. If the pipe has curves, we must ignore the centrifugal force acting on the liquid flow.
7. No friction between the moving water and the internal walls of the pipes is assumed.
8. Viscosity between the layers of the water is also neglected because it impacts the velocity of the overall liquid flow.

Bernoulli’s equation

Bernoulli’s equation is:

p+1 ⁄ 2pv²+pgh=constant

Here, P is the pressure of the flowing liquid, is the relative fluid density, v is the fluid’s flow velocity, g is the gravitational force, and h the height of the channel. If we apply this equation to two different points, 1 and 2, having different pressure, velocities, and height, then it can be transformed as:

P1   = P2 

 The relative density will remain constant while the g or gravitational force is the same throughout the earth.

Evaluation of Bernoulli’s Equation at a Constant Depth

The above equation is generalised. Therefore, if we consider a condition where the water flows at a constant depth, then h1 and h2 will be the same. Therefore, there won’t be any difference in the depth. So, we can write the Bernoulli’s equation at constant depth as:

P1  = P2 

From this Bernoulli’s equation at constant depth, we get the following evaluations:

1. If P1 > P2, then to clarify the equity symbol, v1 < v2. Hence, if the pressure of the water increases, the velocity will decrease.
2. If P1 < P2, then to clarify the equity symbol, v1 > v2. Hence, if the pressure of the water decreases, the velocity will increase.

What are the applications of Bernoulli’s equations?

Entrainer

An entrainer is the best Bernoulli equation example where the external high pressure and low internal pressure increase the velocity of the fluid. As a result, it moves much faster and drives out a stationary liquid. This mechanism appears in perfume bottles, spray cans, atomisers, etc.

De Laval nozzle

Another Bernoulli equation example is the de Laval nozzle. In this mechanism, hot gas enters the nozzle and flows with constant pressure and velocity. However, at the midsection, the tube is constricted for increasing the pressure and reducing the gas’s velocity. Towards the end of the nozzle, the cross-sectional area is much more significant. As a result, pressure drops to a minimal value which causes the velocity of the flowing gas to skyrocket. Therefore the hot gas coming out of the de Laval nozzle can flow significantly due to the velocity.

Conclusion

Bernoulli’s equation at constant depth proves that when there is no depth difference between two points, the pressure of the flowing fluid will be inversely proportional to its velocity. This means that if the pressure of the liquid increases, then the velocity will decrease and vice versa. Many real-life examples can be defined using this principle. For example, the aeroplane’s wings move through the air due to the collaborative work of drag and lift forces. Even though the equation solves several fluid dynamics problems, there are many limitations. For instance, it is applicable for frictionless fluid with no viscosity.