Bernoulli Principle

Key Ideas of Bernoulli Principle (in simple Indian English)

1. Fluid Element: Think of a very small packet of water or air moving along a path called a streamline.

2. Energy Conservation: For that packet, total mechanical energy (per unit mass) stays constant if the flow is steady, non-viscous (no stickiness), incompressible (density $\rho$ constant) and along the same streamline.

3. Three Forms of Energy per unit mass

   • Pressure energy: $\frac{P}{\rho}$
   • Kinetic energy: $\frac{v^2}{2}$
   • Potential energy: $gz$ (because of height $z$)

4. Bernoulli Equation

   $$\frac{P}{\rho} + \frac{v^2}{2} + gz = \text{constant along a streamline}$$

5. Why every step?

   • Apply work–energy theorem to a tiny fluid element.
   • Pressure difference does work, gravity does work, and kinetic energy changes.
   • Add them → total stays same.

6. Typical Uses

   • Speed of fluid from a tank (Torricelli),
   • Lift on airplane wing,
   • Velocity measurement with pitot tube,
   • Why roof can fly in storm.

Imagine water as a group of kids running in a narrow street.

• When the street becomes narrow, kids have to run faster so that everyone can pass.
• When they run faster, they do not have enough energy left to push on the side walls, so the side-push (pressure) becomes low.
• Where the street is wide, kids slow down and start pushing the walls more, so the pressure goes high.
  Bernoulli Principle says exactly this for any liquid or gas that flows smoothly:

Fast flow = Low pressure, Slow flow = High pressure

It is like balancing three things that stay constant along one streamline:

1. Height energy (because of gravity)
2. Pressure energy (side push)
3. Kinetic energy (speed energy)

If one goes up, at least one other must come down so that the total stays the same.

Deriving Bernoulli Equation (Step-by-Step)

1. Assume incompressible, non-viscous, steady flow.

2. Choose two nearby cross-sections A and B on same streamline. Fluid element moves from A to B in time $\Delta t$.

3. Work done by pressure forces

   • At A: Force $P_A A_A$, displacement $v_A \Delta t$, work $P_A A_A v_A \Delta t = P_A \Delta V$ (where $\Delta V$ is volume).
   • At B (opposite direction): work $-P_B \Delta V$.
   • Net work = $(P_A - P_B)\Delta V$.

4. Work done by gravity
   $W_g = -mg(z_B - z_A) = -\rho \Delta V g (z_B - z_A)$

5. Change in kinetic energy
   $\Delta KE = \frac{1}{2} \rho \Delta V (v_B^2 - v_A^2)$

6. Work–Energy theorem
   $\text{(pressure work)} + \text{(gravity work)} = \Delta KE$

   $(P_A - P_B)\Delta V - \rho g (z_B - z_A)\Delta V = \frac{1}{2}\rho (v_B^2 - v_A^2)\Delta V$

7. Divide by $\rho \Delta V$ to get

   $\frac{P_A}{\rho} + gz_A + \frac{v_A^2}{2} = \frac{P_B}{\rho} + gz_B + \frac{v_B^2}{2}$

8. Thus along any point on same streamline

   $\frac{P}{\rho} + gz + \frac{v^2}{2} = \text{constant}$

This is the Bernoulli Equation.