Streamline flow and continuity
Steady flow and streamlines
So far our fluids have been at rest. Now let them move. The study of fluids in motion is called fluid dynamics.
Open a tap just a little. The water flows out smoothly, as clear as a glass rod. Light an agarbatti in still air: its smoke rises in a smooth, thin thread.
Watch one point in a gently flowing stream. Every drop of water that passes this point moves with the same velocity as the drops before it. The flow pattern does not change with time.
This is called steady flow. The velocity can be different at different points, faster here and slower there, but at any one point it stays the same.
Now follow a single drop of water. The path it traces is called a streamline. In steady flow, every drop that passes through a point follows the same streamline.
At any point on a streamline, the tangent gives the direction of the fluid’s velocity there. Watch the green velocity arrows lie along the curve.
Two streamlines can never cross. If they did, a drop at the crossing would have two velocities at once, which is impossible.
A bundle of neighbouring streamlines forms a tube of flow. Fluid never crosses the walls of this tube; it just flows along inside it, like water in an invisible pipe.
Open the tap fully, or watch the agarbatti smoke higher up. The smooth flow breaks into swirls and eddies, and the velocity at a point keeps changing. This irregular flow is called turbulent flow.
The equation of continuity
Put your thumb over the end of a garden hose. The water shoots out much faster and further. Why does a narrower opening make the water faster?
Take a tube of flow in a liquid. Mark three cross-sections: P, R and Q, with areas A P, A R and A Q, where the speeds are v P, v R and v Q.
In a short time Δt, the liquid at P moves forward by v P Δt. So the volume that crosses P is a short cylinder: A P times v P times Δt.
Liquids are almost incompressible, and no liquid leaks out through the sides of the tube. So exactly the same volume must cross R, and Q, in the same time.
That gives A P v P equals A R v R equals A Q v Q. In short, A v is constant along the tube. This is the equation of continuity.
A v is the volume of liquid flowing per second, called the volume flux or flow rate. Where the tube is narrow, the speed must be larger. Watch the streamlines crowd together where the flow is fast.
Worked example: a hose of inner radius 1 centimetre carries water at 2 metres per second. The flow rate is π times 0.01 squared times 2, about 6.3 times 10⁻⁴ cubic metres per second: 0.63 litres every second.
At that rate, a 15 litre bucket fills in about 24 seconds. Now fit a nozzle of radius 0.5 centimetres. The area is four times smaller, so the water leaves four times faster, at 8 metres per second.
The same idea explains why a river flows fastest where it is narrow, and why a stream of water from a tap gets thinner as it falls: it speeds up, so its cross-section must shrink.
Try it: water through a narrow pipe
- Each dot is a drop of water following a streamline. Watch it speed up in the throat.
- Make the throat narrower: the streamlines crowd together and v₂ goes up.
- A₁v₁ and A₂v₂ always come out equal: that is the flow rate Q.
- Press Space to pause the flow.
A narrowing stream
Water flowing gently from a tap forms a smooth stream that gets narrower as it falls. Why?
- Surface tension squeezes the stream more and more
- The water speeds up as it falls, and Av stays constant, so A must decrease
- Air pressure is larger lower down and squeezes the stream
- Some of the water evaporates on the way down
Answer
B. Gravity makes the water speed up as it falls. By the equation of continuity, Av = constant: the same volume crosses every section each second. So where v is larger, the area A must be smaller.
A narrow section
Water flows through a pipe of diameter 4 cm at 1.5 m/s. The pipe then narrows to a diameter of 2 cm. What is the speed of the water in the narrow part?
- 0.375 m/s
- 3 m/s
- 6 m/s
- 12 m/s
Answer
C. A₁v₁ = A₂v₂, and the area goes as the diameter squared. Halving the diameter makes the area 4 times smaller, so v₂ = 4 × 1.5 = 6 m/s. 3 m/s comes from using the diameters instead of the areas.
Summary
- Equation of continuity:
A₁v₁ = A₂v₂ - Flow rate (volume flux):
Q = Av = constant (m³/s)
- In steady flow, the velocity at each point does not change with time (it can differ from point to point).
- A streamline is the path of a fluid particle in steady flow. Its tangent gives the velocity direction. Streamlines never cross.
- For an incompressible liquid, Av is constant: the liquid is fastest where the pipe is narrowest, and the streamlines are closest together there.
- At high speeds, smooth flow breaks up into turbulent flow with eddies.