# Bernoulli's Theorem and Viscosity

> Bernoulli's theorem and viscosity for JEE Physics: energy per unit volume, Torricelli's law, Stokes law, terminal velocity and Poiseuille flow.

- Canonical URL: https://prepelephant.com/topics/jee/physics/bernoulli-and-viscosity
- Exam / course: JEE · Subject: Physics
- Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team
- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Bernoulli's Theorem and Viscosity", PrepElephant, https://prepelephant.com/topics/jee/physics/bernoulli-and-viscosity

## Direct answer

Bernoulli's theorem is energy conservation written per unit volume for streamline flow of a non-viscous, incompressible fluid: P + (1/2) rho v^2 + rho g h = constant along a streamline, so where the fluid runs faster its pressure must drop. Torricelli's result, efflux speed v = sqrt(2 g h) from a side hole, is Bernoulli applied to a tank. Viscosity enters through Newton's law F = −eta A (dv/dx) with eta in pascal-second, Stokes' drag on a sphere F = 6 pi eta r v, and the terminal velocity v(t) = 2 r^2 (rho − sigma) g/(9 eta); Poiseuille's volumetric flow Q = pi P r^4/(8 eta l) shows the fourth-power tyranny of radius in capillary flow.

## What you must remember

- **Bernoulli equation:** P + (1/2) rho v^2 + rho g h = constant along a streamline; each term is an energy density — pressure energy, kinetic, potential, all in joules per cubic metre.
- **Applications map:** venturimeter measures flow rate from the pressure drop at the throat; pitot tube measures aircraft speed; a spinning ball curves (Magnus effect) because spin changes the relative air speed on the two sides.
- **Torricelli's law:** v = sqrt(2 g h), same as free fall through height h; the range of the jet is x = 2 sqrt(h(H − h)), maximum when the hole is at the middle of the column.
- **Newton's law of viscosity:** F = −eta A dv/dx; eta of water about 10^-3 Pa s at room temperature; liquid viscosity falls with temperature, gas viscosity rises — a viva staple.
- **Stokes law and terminal velocity:** F = 6 pi eta r v; setting drag plus buoyancy against weight gives v(t) = 2 r^2 (rho − sigma) g/(9 eta); doubling the drop's radius quadruples terminal speed.
- **Poiseuille equation:** Q = pi P r^4/(8 eta l); radius is the dominant variable — a slight narrowing of an artery raises the pressure the heart must supply dramatically.
- **Reynolds number:** Re = rho v D/eta; flow turns turbulent beyond roughly 2000, and Re is dimensionless.
- **Pattern note:** Main examines Torricelli and terminal-velocity numericals; Advanced builds multi-level tank problems and viscous-flow reasoning with the energy equation amended for head loss.

## One tank, two holes

A tank filled to height H has a hole at depth h below the surface. Torricelli gives the jet speed sqrt(2gh), and projectile motion from the hole to the floor gives a landing distance that works out to x = 2 sqrt(h(H − h)). Two different holes — one at h, one at H − h — land at the same spot, and the maximum range belongs to the hole punched exactly at mid-depth, h = H/2. These three sentences describe the majority of JEE Main questions on this chapter; the remainder are terminal-velocity substitutions where the only trap is the (rho − sigma) difference: a bubble rising through water has sigma above rho, so the numerator changes sign, not magnitude — the terminal velocity formula returns the same size with reversed direction.

For Poiseuille flow the exam favourite is the artery: when a plaque halves the radius, flow falls sixteen-fold at the same pressure difference, so the body must raise the pressure drop sixteen-fold to keep perfusion constant — the physics of hypertension in one calculation.

## Where students slip

Bernoulli's theorem assumes laminar, incompressible, non-viscous flow, and each assumption has anchored an assertion–reason question; applying it to turbulent flow is invalid. The second error is direction of pressure change: faster flow means lower pressure, and candidates who reverse this cannot explain lift at all. Third, terminal velocity questions forget buoyancy; a sphere falling through a fluid of density sigma has effective weight (4/3) pi r^3 (rho − sigma) g, and omitting sigma inflates v(t) for dense fluids.

## Frequently asked questions

### What does each term of Bernoulli's equation represent?

Pressure energy per unit volume P, kinetic energy per unit volume (1/2)rho v^2 and potential energy per unit volume rho g h, all conserved together along a streamline of an ideal flow.

### Why does a spinning ball curve in flight?

Rotation makes air speed relative to the surface differ on the two sides; by Bernoulli the pressure differs sideways, and the net sideways force (Magnus effect) bends the trajectory.

### How is terminal velocity derived for a sphere in a viscous fluid?

Balance weight minus buoyancy against Stokes drag 6 pi eta r v to get v(t) = 2 r^2 (rho − sigma) g/(9 eta), growing as the square of the radius.

### What happens to flow if the radius of a pipe is halved?

Poiseuille's Q is proportional to r^4, so the flow rate falls sixteen times at the same pressure difference; small constrictions are expensive.

### Why does efflux speed match free-fall speed from the same height?

Bernoulli between the top surface and the orifice at depth h, with both at atmospheric pressure and negligible surface speed, gives v = sqrt(2 g h) exactly.