FLUID MECHANICS

                              Bernoulli’s Principle

                                 - Dr. C. Dhayananth Jegan, M.E., Ph.D,

    Bernoulli’s principle formulated by Daniel Bernoulli states that as the speed of a moving fluid increases (liquid or gas), the pressure within the fluid decreases. Although Bernoulli deduced the law, it was Leonhard Euler who derived Bernoulli’s equation in its usual form in the year 1752.

Bernoulli’s principle states that -The total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure and the kinetic energy of the fluid motion, remains constant. Bernoulli’s principle can be derived from the principle of conservation of energy.

Bernoulli’s Principle Formula

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

The formula for Bernoulli’s principle 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 great insight into the balance between pressure, velocity and elevation.

Bernoulli’s Equation Derivation

Consider a pipe with varying diameter and height through which an incompressible fluid is flowing. The relationship between the areas of cross-sections A, the flow speed v, height from the ground y, and pressure p at two different points 1 and 2 is given in the figure below.

Assumptions:

• The density of the incompressible fluid remains constant at both points.
• The energy of the fluid is conserved as there are no viscous forces in the fluid.

Therefore, 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 was due to conservation of gravitational force and change in kinetic energy. The change in kinetic energy of the fluid is given as:

dK=12m2v22−12m1v21=12ρdV(v22−v21)

The change in potential energy is given as:

dU = mgy2 – mgy1 = ρdVg(h2 – h1)

Therefore, the energy equation is given as:

dW = dK + dU

(p1 – p2)dV = 12ρdV(v22−v21) + ρdVg(h2 – h1)

(p1 – p2) = 12ρ(v22−v21) + ρg(h2 – h1)

Rearranging the above equation, we get

p1+12ρv21+ρgy1=p2+12ρv22+ρgy2

This is Bernoulli’s equation.