Bernoulli's Principle

The Bernoulli equation

An incompressible, non-viscous fluid flows steadily through a pipe of varying cross-section that also varies in height. Its speed must change as the area changes, by the equation of continuity. A change in speed needs a net force, and that force comes from a difference in pressure between regions of the pipe. Bernoulli's equation is the general relation that ties these three quantities — pressure, speed and height — together for any two points on a streamline.

Bernoulli's principle (NCERT §9.4): As we move along a streamline, the sum of the pressure $P$, the kinetic energy per unit volume $\tfrac{1}{2}\rho v^2$, and the potential energy per unit volume $\rho g h$ remains a constant.

Written between two points 1 and 2 on the same streamline, and in its general form:

Each term has the dimensions of pressure (energy per unit volume). $P$ is the pressure energy density, $\tfrac{1}{2}\rho v^2$ the kinetic energy density, and $\rho g h$ the gravitational potential energy density. The Swiss physicist Daniel Bernoulli established this relation in 1738.

Derivation from energy conservation

Take the fluid initially between sections B and D in Figure 1. In a small interval $\Delta t$ the fluid at B (speed $v_1$, area $A_1$, pressure $P_1$) moves a distance $v_1\Delta t$ to C; the fluid at D (speed $v_2$, area $A_2$, pressure $P_2$) moves $v_2\Delta t$ to E. By continuity the same volume $\Delta V$ crosses each section. The work done on the fluid at the left face is $W_1 = P_1 A_1 (v_1\Delta t) = P_1\Delta V$, and the work done by the fluid against the pressure at the right face is $W_2 = P_2 A_2 (v_2\Delta t) = P_2\Delta V$. The net work done on the fluid is therefore $(P_1 - P_2)\Delta V$.

This work changes the kinetic and potential energies of the displaced volume. With $\Delta m = \rho\,\Delta V$, the changes are $\Delta K = \tfrac{1}{2}\rho\,\Delta V\,(v_2^2 - v_1^2)$ and $\Delta U = \rho g\,\Delta V\,(h_2 - h_1)$. The work–energy theorem gives:

Dividing every term by $\Delta V$ and rearranging recovers the Bernoulli equation directly:

For a fluid at rest, $v = 0$ everywhere and the equation collapses to $P_1 + \rho g h_1 = P_2 + \rho g h_2$, i.e. $P_1 - P_2 = \rho g (h_2 - h_1)$ — the hydrostatic pressure relation. Bernoulli's equation is thus the dynamic generalisation of the static pressure law you met under pressure in fluids.

Assumptions and limitations

Because the derivation used the work–energy theorem with no dissipative term, the equation holds only for an idealised fluid. NEET routinely tests whether you know the four conditions and where they break.

In short, Bernoulli's equation is exact only for an ideal fluid and a good approximation for low-viscosity, incompressible, smoothly flowing real fluids. For a turbulent rapid in a river, or for blood squeezing through a constricted artery where viscosity dominates, it cannot be applied without correction.

The pressure–velocity inverse relation

Restrict the streamline to a horizontal pipe, so $h_1 = h_2$ and the $\rho g h$ term drops out. Bernoulli's equation reduces to its most-tested form:

The two terms trade against each other. Where the fluid moves faster, the kinetic-energy density $\tfrac{1}{2}\rho v^2$ is larger, so the pressure $P$ must be smaller. This is the single idea behind every Bernoulli application: high speed means low pressure, and low speed means high pressure. Combine it with continuity ($Av = \text{constant}$, so the fluid speeds up where the pipe narrows) and you can predict the pressure everywhere in a flow at a glance.

Torricelli's law of efflux

The word efflux means fluid outflow. Torricelli discovered that the speed at which liquid spurts from an open tank equals the speed of a freely falling body. Consider a tank of liquid of density $\rho$ with a small hole in its side at height $y_1$, the free surface being at height $y_2$, and air above the surface at pressure $P$. Let $A_1$ be the hole area and $A_2$ the tank's cross-section.

By the equation of continuity $v_1 A_1 = v_2 A_2$. If the tank's area is much larger than the hole, $A_2 \gg A_1$, the surface descends so slowly that we take $v_2 \approx 0$. Applying Bernoulli between the surface (point 2) and the hole (point 1), where the pressure at the hole is atmospheric $P_a$:

Writing $h = y_2 - y_1$ and solving for the efflux speed:

Two limiting cases follow. When the tank is sealed and the air pressure is very large, $P \gg P_a$ and $2gh$ is negligible, so $v_1 \approx \sqrt{2(P-P_a)/\rho}$ — the efflux is set by the container pressure, as in rocket propulsion. When the tank is open to the atmosphere, $P = P_a$ and the pressure terms cancel, leaving Torricelli's law:

This is exactly the speed a body acquires falling freely through height $h$. Notice that for an open tank the density $\rho$ has cancelled out entirely — the efflux speed is independent of which liquid is in the tank.

The Venturi-meter

A Venturi-meter measures the rate of flow of a liquid through a pipe. It is a horizontal tube with a deliberate constriction; a manometer connects the wide section (area $A_1$, speed $v_1$, pressure $P_1$) to the narrow throat (area $A_2$, speed $v_2$, pressure $P_2$). Because the pipe is horizontal, the height term vanishes and only pressure and kinetic energy trade.

Continuity gives $A_1 v_1 = A_2 v_2$, so $v_2 > v_1$ at the throat. Bernoulli for the horizontal tube gives:

Eliminating $v_2$ using continuity and reading $P_1 - P_2 = \rho_m g h$ from the manometer (level difference $h$) lets us solve for the inlet speed:

Since every term except $h$ is fixed for a given meter, $v_1 \propto \sqrt{h}$, and the volume flow rate is $Q = A_1 v_1 = K'\sqrt{h}$. The pressure is least where the bore is narrowest — Venturi's principle — which is precisely the pressure–velocity inverse relation in instrument form. NEET 2023 asked the bare fact that the Venturi-meter works on Bernoulli's principle.

Dynamic lift — aerofoil and Magnus effect

Dynamic lift is the force on a body — an aircraft wing, a hydrofoil, a spinning ball — arising from its motion through a fluid. It is the most spectacular consequence of the pressure–velocity relation, and NCERT §9.4.2 treats three cases.

Atomiser, sprayer and blood flow

The same low-pressure-where-fast logic runs through a family of everyday devices and one important physiological caution.