Bernoulli’s Principle

What is Bernoulli’s Principle?

Bernoulli's principle is also known as Bernoulli’s equation. It can be applied for fluids in an ideal state. We already know that pressure and density are inversely proportional to each other, which means, a fluid with slow speed will exert more pressure than fluid, which is moving faster. In this case, fluid refers to not only liquids but gases as well. Bernoulli's principle forms the basis of many applications in our day-to-day lives.  

Some examples are - an airplane that tries to stay aloft, shower a certain billowing inward; this phenomenon happens in the case of rivers as well 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.

Most of you will think that the pressure within the fluid in the narrower parts 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 in the wider parts of the river will increase. Daniel Bernoulli, a Swiss Scientist discovered this concept while experimenting with fluid inside the pipes. He observed in his experiment that the speed of the fluid increases, but its internal pressure decreases. He referred to this concept as Bernoulli’s principle.

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 tighter spaces, but pressure within the water will not increase. Thus, the surrounding of the fluid will experience an increase in pressure. The change in the pressure will also result in a change in the speed of the fluid.

Bernoulli’s Principle Formula

Bernoulli's principle is a seemingly counterintuitive statement about how the speed of a fluid relates to its pressure. Many people across the globe believe that Bernoulli's principle isn't correct; however, this might be due to a misunderstanding about what Bernoulli's principle says. Bernoulli's principle states that within a horizontal flow of fluid, points of higher fluid speed will always have less pressure than the points of slower fluid speed.

Therefore, inside a horizontal water pipe that changes diameter, the regions where the water is moving fast will experience less pressure than the regions where the water is moving slowly. Bernoulli's equation is usually written or expressed as follows: P1+1/2 ρv12+ ρgh1=P2+1/2 ρv22+ ρgh2 

Where ρ = density, g = gravitational acceleration, and v = velocity. 

In the equation mentioned above, the variables P1, v1, h1 denote the pressure, speed, and height of the fluid at point 1 respectively, whereas the variables P2, v2, h2 denote the pressure, speed, and height of the fluid at point 2 respectively.

Bernoulli’s Equation Derivation

For maintaining a constant volume flow rate, incompressible fluids have to increase their speed when they reach a narrow constricted section. The same accounts for why a narrow nozzle on a hose causes water to speed up. Now, something might be bothering you about this phenomenon. If the water is speeding up at a constriction, it must be gaining kinetic energy as well. So, from where is this extra kinetic energy coming? 

The only possible way of giving kinetic energy to something is by doing work on it. It is expressed by the work-energy principle.

W=ΔKE=1/2mv2f −1/2mv2i 

Where, W = work, ΔKE = change in kinetic energy, vf = final velocity, and vi = initial velocity. 

So, if a portion of the fluid is speeding up, something external to that portion of fluid must be doing work.

What is the force that causes work to be done on the fluid? The answer is, in most of the real-world systems, there are lots of dissipative forces that could be doing negative work; however, we're going to assume for the sake of simplicity that these viscous forces are almost negligible, and there is a continuous and perfectly laminar (streamline) flow. 

Laminar or streamline flow implies that the fluid flows in parallel layers, that too, without crossing paths. In laminar streamline flow, there is no swirling or vortices in the fluid. Therefore, we'll assume that we have no loss in energy due to dissipative forces. In such a case, what non-dissipative forces could be doing work on our fluid that causes it to speed up? The pressure from the surrounding fluid will be causing a force that can do work and speed up a portion of the fluid as well.

Principle of Continuity

The principle of continuity tells us that what flows into a defined volume in a defined time, minus what flows out of that volume at that time, must accumulate in that volume. If the accumulation is negative, then the material in that volume is being reduced. Bernoulli's principle is a result of the law of conservation of mass. It fully describes the behavior of fluids in motion, along with a second equation - based on the second Newton’s laws of motion, and a third equation - based on the conservation of energy.

The Relation Between Conservation of Energy and Bernoulli’s Equation

The Bernoulli Equation can be expressed or defined as a statement of the conservation of energy principle appropriate for flowing fluids. The qualitative behavior that is usually labeled with the term "Bernoulli effect" is the decrease in the fluid pressure in regions where the flow velocity increases. The lowering of pressure in a constriction of a flow path may seem counterintuitive, but it seems less when you consider the pressure to be energy density. The kinetic energy must increase at the expense of pressure energy in the high-velocity flow through the constriction.

Bernoulli’s Equation at Constant Depth

The situation in which the fluid moves, but its depth is constant- that is h1=h2. Under that condition, Bernoulli’s equation becomes P1+1/2pv21 = P2+1/2pv22.

The situations in which fluid flows at a constant depth are so crucial that this equation is often called Bernoulli’s principle. It is Bernoulli’s equation for fluids at a constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path).

Bernoulli’s Equation for Static Fluids

Let us consider the equation given below in which the fluid is static - that is, v\[_{1}\] = v\[_{2}\]=0. 

Bernoulli’s equation, in that case, is: P1+pgh1=P2+ ρgh2

We 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 P2=P1+ ρgh1

This equation conveys that in static fluids the pressure increases with the increase in 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 pgh1. In the simplest possible case, P1 is 0p at the top of the fluid, and we come to a familiar relationship as mentioned below:

P = ρgh

PEg= mgh

This equation includes the fact that the pressure due to the weight of the fluid is ρ*g*h.