Equation of continuity and Bernoulli's theorem

Derivation: equation of continuity

A fluid flows steadily through a tube. At end A the tube is wide (area A₁, speed v₁); at end B it is narrow (area A₂, speed v₂).

1. 1
   In one second, the fluid at A moves a distance v₁. Volume entering per second = A₁v₁.
2. 2
   Mass entering per second = ρA₁v₁, where ρ is the density.
3. 3
   In the same way, the mass leaving at B per second = ρA₂v₂.
4. 4
   No fluid piles up inside or leaks out, so mass in = mass out: ρA₁v₁ = ρA₂v₂.
5. 5★
   So, A₁v₁ = A₂v₂: the speed is greater where the tube is narrower. This is the equation of continuity, a form of conservation of mass.

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Bernoulli's theorem

A flowing fluid has three kinds of energy: pressure energy, kinetic energy and potential energy. Per unit volume, they are P, ½ρv² and ρgh. Bernoulli's theorem: for an incompressible, non-viscous fluid in streamline flow, the sum P + ½ρv² + ρgh stays constant along a streamline. In simple words: where the speed is high, the pressure is low, and where the speed is low, the pressure is high. Your NIOS lesson writes density as d: P + ½dv² + hdg = constant. You need to state and use the theorem; its derivation is not set.

Uses of Bernoulli's theorem

How to draw a Venturi meter

1. 1
   Draw a horizontal pipe with a wide part on the left, labelled A₁, v₁, P₁. It narrows to a throat on the right, labelled A₂, v₂, P₂.
2. 2
   Draw a vertical glass tube rising from the top of the wide part, and another from the throat.
3. 3
   Show the liquid level higher in the tube over the wide part and lower over the throat. Mark the height difference h.
4. 4
   Draw arrows inside the pipe for the direction of flow, closer together in the throat.
5. 5★
   Working: P₁ − P₂ = ½ρ(v₂² − v₁²) and P₁ − P₂ = hρg. With A₁v₁ = A₂v₂, the volume flowing per second, A₁v₁, is proportional to √h.

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Worked example: water in a narrowing pipe

Activity: fractional rate of flow from a burette (an analogy for radioactive decay)

This experiment shows that water draining from a burette behaves like a decaying radioactive sample. You need a 50 mL burette, a thistle funnel, a rubber tube, two stands and a stop clock.

1. 1
   Clamp a 50 mL burette and a thistle funnel on two stands. Join their lower ends with a rubber tube.
2. 2
   Set the mouth of the funnel level with the bottom mark of the burette, so only the water above that mark drives the flow.
3. 3
   Close the stopcock and fill the burette a little above the top mark.
4. 4
   Open the stopcock slightly, so the flow is slow and smooth, not turbulent. Start the stop clock as the level passes 50 mL.
5. 5
   Without touching the stopcock again, record the time at every 5 mL fall, and at every 2 mL fall below 20 mL.
6. 6
   Plot volume V (y-axis) against time t (x-axis). The curve falls fast at first and then more slowly, like a decay curve.
7. 7
   From the graph, read the times for V to fall from 40 mL to 20, 10, 5 and 2.5 mL. Each halving takes about the same time, a 'half-life' of the flow.
8. 8
   Draw tangents at these points to find the rate of flow, ΔV/Δt, and divide it by V. This fractional rate of flow comes out about the same at every stage.
9. 9★
   Why: the rate of flow is proportional to the pressure head, and so to the volume left. In the same way, the decay rate of a radioactive sample is proportional to the number of nuclei left.

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