Fluid Flow-Bernoullis Equation Derivation and Fluid Mechanics - Physics

What is Bernoulli’s Principle?

Bernoulli's principle describes how the speed of a fluid relates to its pressure. Bernoulli's principle asserts that places of higher fluid speed will have less pressure than points of slower fluid speed in a horizontal flow of fluid.

Bernoulli’s equation formula gives the relation between pressure, gravitational potential energy, and kinetic energy of a fluid in a container.

Bernoulli’s Equation is given as:

p + 12 ρ v² + ρgh = constant

Where,

p is the pressure exerted by the fluid
v is the velocity of the fluid
ρ is the density of the fluid
h is the height of the container

Bernoulli’s equation gives insight into the balance between pressure, velocity and elevation.

Derivation of Bernoulli’s Equation

Let us now derive Bernoulli's Principle equation, and demonstrate more clearly what it means.

Let us assume that there is a pipe with varying diameter and height. Let us also assume that an incompressible fluid is flowing through it. The relationship between the area of cross-section A, height from the ground y, the flow speed v, and pressure p at two different points 1 and 2 is given in the figure below -

Bernoulli

Let us make a few more assumptions -
1. The density of the fluid will remain constant at both points 1 and 2.
2. The energy of the fluid will be conserved since there is no viscous force in the fluid.

Now, the work done on the fluid is given as -

dW = F₁dx₁ – F₂dx₂

dW = p₁A₁dx₁ – p₂A₂dx₂

dW = p₁dV – p₂dV = (p₁ – p₂)dV

We know that the work done on the fluid is because of conservation of gravitational force and change in kinetic energy. Change in kinetic energy of the fluid is given as -

dK = ½ m₂v₂² − ½ m₁v₁² = ½ ρdV (v₂²−v₁²)

The change in potential energy is given by -

dU = mgy₂ – mgy₁ = ρdVg (y₂ – y₁)

Therefore, the energy equation is given as -

dW = dK + dU

(p₁ – p₂) dV = 12 ρdV (v₂²−v₁²) + ρdVg (y₂ – y₁)

(p₁ – p²) = 12ρ (v₂²−v₁²) + ρg (y₂ – y₁)

Rearranging the above equation, we get Bernoulli’s Equation -

p₁ + ½ ρ v₁² + ρg y₁ = p₂ + ½ ρ v₂² + ρg y₂

Fluid Mechanics

Conservation of energy is applied to the fluid flow to produce Bernoulli’s equation. The net work done is the result of a change in fluid’s kinetic energy and gravitational potential energy.

Bernoulli's principle states that higher pressure regions have lower fluid speed and lower pressure regions have higher fluid speed at locations along a horizontal streamline. Bernoulli's principle can be thought of as fluid flowing from a high-pressure zone to a low-pressure region would accelerate due to the net force acting in the same direction.

It may seem surprising that locations where the fluid is moving quickly, will have lower pressure. A fast-moving fluid striking you must surely exert more pressure on your body than a slow-moving fluid, right? Yes, you are correct. But we're now discussing two distinct pressures.

The internal fluid pressure that would be exerted in all directions during the flow, including on the pipe's sides, is what Bernoulli's principle refers to. This is not the same as the pressure fluid will apply to you if you come in the path and stop it from moving. It's worth noting that Bernoulli's principle does not rule out the possibility of considerable pressures in a fast-moving fluid. It simply states that the pressure in a slower part of the same flowing system must be higher than the pressure in the quicker part.