Bernoulli's Equation and Applications

For steady, incompressible, non-viscous flow along a streamline, the sum of pressure energy, kinetic energy per unit volume and gravitational potential energy per unit volume is constant. This is Bernoulli's equation, and it underlies Venturi meters, Pitot tubes, atomisers and Torricelli's law.

Concept

The equation of continuity for an incompressible fluid is

$A_{1} v_{1} = A_{2} v_{2} \quad \text{(volume flow rate)}.$

Bernoulli's theorem then states that along a streamline,

$P + \tfrac{1}{2}\rho v^{2} + \rho g h = \text{constant}.$

A narrower section of a horizontal pipe carries faster fluid (continuity) at lower pressure (Bernoulli). This is the principle behind aerofoils, atomisers and the Venturi meter.

Formula Derivation

Apply the work-energy theorem to a fluid element of mass $\Delta m$ moving from point 1 (cross-section $A_{1}$, speed $v_{1}$, height $h_{1}$, pressure $P_{1}$) to point 2:

$W_{\text{net}} = \Delta KE + \Delta PE.$

The work done by pressure forces is $(P_{1} - P_{2})\,\Delta V$, while the changes in KE and PE per unit volume are $\tfrac{1}{2}\rho(v_{2}^{2} - v_{1}^{2})$ and $\rho g(h_{2} - h_{1})$. Rearranging,

$P_{1} + \tfrac{1}{2}\rho v_{1}^{2} + \rho g h_{1} = P_{2} + \tfrac{1}{2}\rho v_{2}^{2} + \rho g h_{2}.$

For a tank with a small hole at depth $h$ below the free surface, setting $P_{1} = P_{2} = P_{0}$ and $v_{1} \approx 0$ gives Torricelli's law:

$v = \sqrt{2 g h}.$

NEET-style Worked Example

Water flows through a horizontal pipe whose cross-section narrows from $20\ \text{cm}^{2}$ to $5\ \text{cm}^{2}$. The speed at the wider section is $1\ \text{m/s}$ and the pressure is $2 \times 10^{5}\ \text{Pa}$. Find the pressure at the constriction.

Continuity: $v_{2} = v_{1} A_{1}/A_{2} = 1 \times (20/5) = 4\ \text{m/s}$.

Bernoulli (horizontal pipe):

$P_{2} = P_{1} + \tfrac{1}{2}\rho(v_{1}^{2} - v_{2}^{2}) = 2 \times 10^{5} + \tfrac{1}{2}(10^{3})(1 - 16) = 2 \times 10^{5} - 7500 \approx 1.925 \times 10^{5}\ \text{Pa}.$

Common Confusions

• Bernoulli applies to steady, incompressible, non-viscous, irrotational flow along a streamline.
• The same equation does not hold across streamlines unless flow is irrotational.
• Speed at a constriction increases (continuity), so pressure must decrease (Bernoulli) — common in NEET MCQs.
• Torricelli's law uses the height of free surface above the orifice, regardless of the tank shape.

Key Takeaways

• Continuity: $A v =$ constant for incompressible flow.
• Bernoulli: $P + \tfrac{1}{2}\rho v^{2} + \rho g h =$ constant.
• Torricelli: $v = \sqrt{2gh}$.
• Faster flow corresponds to lower pressure in a horizontal pipe.
• Applications: Venturi meter, Pitot tube, atomiser, aerofoil lift, Magnus effect.