Q.In a horizontal pipe of non-uniform cross section, water flows with a velocity of 1 m s$^{-1}$ at a point where the diameter of the pipe is 20 cm. Find the diameter of the pipe (in cm) at another point where the velocity of water is 1.5 m s$^{-1}$.

The Intuition: Why a River Speeds Up in a Narrow Stretch

Imagine you're standing on a bridge watching a river. The river is wide and slow-moving upstream. Then it passes through a narrow gorge — suddenly the water races through, fast and turbulent. Yet the same amount of water must pass every point every second. Water can't pile up or vanish.

That's the core idea: what flows in must flow out. If the pipe (or river) gets narrower, the fluid must move faster to get the same volume through in the same time. If it widens, the fluid slows down.

This is not a guess — it's a direct consequence of mass being conserved. Fluid cannot be created or destroyed inside a pipe (assuming no leaks). So the mass that enters a section per second must equal the mass that leaves it per second.

The Precise Statement

For a fluid flowing steadily through a pipe of varying cross-section, the product of cross-sectional area and flow speed is constant at every point along the pipe.

Where:

• $A$ = cross-sectional area of the pipe (in $\text{m}^2$)
• $v$ = flow speed of the fluid (in $\text{m/s}$)

The product $A v$ is called the volume flow rate (or discharge), often denoted $Q$. Its SI unit is $\text{m}^3/\text{s}$.

Why It Works: The Derivation in One Minute

Consider a pipe with cross-sectional area $A$. In a small time $\Delta t$, a fluid particle moves a distance $v \Delta t$. The volume of fluid that crosses the section in that time is:

So the volume flow rate is:

Now take two different cross-sections (1 and 2) along the same pipe. If the fluid is incompressible (density constant) and no fluid is added or removed between them, the volume entering section 1 per second must equal the volume leaving section 2 per second:

That's it. The equation is a direct statement of conservation of mass for an incompressible fluid.

What It Tells You (and What It Doesn't)

It tells you: If you know the area and speed at one point, you can find the speed at any other point. A garden hose with a nozzle: wide at the tap ($A_1$ large, $v_1$ small), narrow at the nozzle ($A_2$ small, $v_2$ large). That's why water shoots out fast when you cover part of the opening with your thumb.

It does NOT tell you: Why the fluid speeds up or slows down. That's the job of Bernoulli's equation, which relates speed to pressure. The continuity equation is purely geometric — it's about how much fluid must move, not about the forces that make it move.

A Common Mistake to Avoid

Quick Example …