Bernoulli’s Equation For Static Fluids

Bernoulli’s principle, developed by Daniel Bernoulli, is one of the most important theorems in Fluid Mechanics. It may be seen as a fluid mechanical equivalent of the conservation of energy concept. Bernoulli’s principle states that the speed of a flowing fluid is inversely proportional to the pressure and potential energy of the fluid.

As a result, Bernoulli’s principle states that a decrease in fluid pressure in the surrounding area results in the flow velocity rising. Bernoulli’s equation may be tweaked to account for head gains and losses. Bernoulli’s equation helps address most fluid flow issues.

What is Bernoulli’s equation?

As a generalised and mathematical version of Bernoulli’s principle, Bernoulli’s equation also considers variations in gravitational potential energy. Before we go into how to derive this equation, let’s look at Bernoulli’s equation.

When applied to any two locations (1 and 2) in a steady, streamlined flowing fluid with density, Bernoulli’s equation connects the fluid’s pressure, speed, and height (potential energy). Bernoulli’s equation is commonly stated as follows:

P1 + 1/2 ρv12 + ρgh1 = P2 + 1/2 ρv2 2+ ρgh2

The variables P1, v1, and h1 relate to the fluid’s force, speed, and elevation at point 1, while P2, v2, and h2 correspond to the fluid’s force, speed, and elevation at point 2. 

Bernoulli’s Equation for Static Fluids

Let’s start with a very basic scenario where we apply Bernoulli’s equation on static fluids. That is, v1 = v2 = 0. Bernoulli’s equation in that case is

P1 + ρgh1 = P2 + ρgh2.

We may reduce the equation even further by using h2 = 0. (we may always pick a height to be zero and treat all other heights in comparison to that height). In that case, we get

P2 = P1 + ρgh1

Pressure rises with depth in static fluids, according to this equation. The depth of the fluid rises by h1 as we go from point 1 to point 2; hence P2 is bigger than P1 by an amount of ρgh1. To simplify things, let’s assume P1 is zero at the fluid’s surface, in which case P = ρgh is the correct equation to use.

(Recall that P = ρgh and ΔPEg = mgh)

Bernoulli’s Equation Derivation

In the derivations of Bernoulli’s equation when incompressible fluids approach a tightly restricted segment, they must accelerate to maintain a constant volume flow rate. The same principle explains why a hose with a small nozzle allows water to flow faster.

A restriction must be causing the water to accelerate, which implies that it is also acquiring energy. So, where does all this additional kinetic energy come from? Working on something is the only method to provide it with kinetic energy. The work-energy principle encapsulates it.

W = ΔKE = 1/2mvf2 − 1/2mvi2

Where W stands for work, ΔKE stands for change in kinetic energy, vf stands for final velocity, and vi stands for beginning velocity.

In this case, it is reasonable to assume that something external to the fluid is causing the speeding up of a section of the liquid. What is the driving factor for the fluid’s work?

The response is that there are many dissipative forces in most real-world systems that may be producing negative work, but, for simplicity, we’ll suppose that these viscous forces are nearly non-existent and that the flow is continuous and completely laminar (streamline).

The fluid moves in parallel layers, without intersecting pathways, in a laminar or streamlined fashion. There is no whirling or formation of vortices in the laminar flow fluid. As a result, we’ll assume there is no energy loss owing to dissipative forces.

In this situation, what non-dissipative forces may work on our fluid, causing it to accelerate as a result? The surrounding fluid’s pressure will create a force that may function and speed up a piece of the fluid.

Conclusion

Bernoulli’s equation is a variant of the general energy equation and is arguably the most extensively utilised technique for resolving fluid flow issues. It’s a simple approach to connecting a fluid’s elevation, velocity, and pressure heads. Bernoulli’s equation may be tweaked to account for pump effort and head losses.