Math Detours · Grade 7 · Connections · 12

Why narrow pipes matter so much

Halve the width of a pipe and, for smooth flow at the same pressure, only one sixteenth as much gets through. Water seeps slowly through fine sand for a related reason.

Name ____________________Date ____________

Tonight's idea

Try drinking a thick milkshake through a thin straw. It is hard work. A wide straw is far easier, much more than you would guess. The reason is that fluid touching the wall of a pipe hardly moves. It sticks. The fastest fluid is in the middle, far from the wall. In a narrow pipe, nothing is far from the wall.

For smooth, steady flow of one fluid through a straight, stiff pipe of fixed length, the volume per minute goes with the pressure difference between the ends, and with radius \(\times\) radius \(\times\) radius \(\times\) radius.

This is called Poiseuille's law. Double the push, double the flow. Double the radius, 16 times the flow. Sand and ground coffee act like a tangle of very narrow pipes. For slow seepage, the flow again goes with the pressure difference. That is Darcy's law. Both are models. Fast flow turns rough and swirly, and both rules fail. Arteries stretch, and blood is not a simple fluid, so for blood this is only a rough guide.

Watch one: four tubes of oil

Cooking oil is pushed smoothly through a tube of radius 2 mm, and 160 mL comes out each minute. (The figures on this sheet are illustrative.) With the same oil, push and length, what do tubes of radius 1, 3 and 4 mm carry? Compare each radius with 2 mm, then multiply four copies of that number together.

Radius 1 mm is \(\tfrac12\) of 2 mm, and \(\tfrac12 \times \tfrac12 \times \tfrac12 \times \tfrac12 = \tfrac1{16}\). The flow is \(160 \div 16 = 10\) mL per minute.

Radius 4 mm is twice 2 mm: \(2 \times 2 \times 2 \times 2 = 16\). The flow is \(16 \times 160 = 2560\) mL per minute.

Radius 3 mm is \(1.5\) times 2 mm. \(1.5 \times 1.5 = 2.25\) and \(2.25 \times 2.25 = 5.0625\). The flow is \(5.0625 \times 160 = 810\) mL per minute.

Why four copies? The opening's area goes with radius \(\times\) radius. So does the average speed in smooth flow, as the middle is farther from the sticky wall. Flow is area times average speed.

radius 1 mm 10 mL/min radius 2 mm 160 mL/min radius 3 mm 810 mL/min radius 4 mm 2560 mL/min

Your turn

  1. A tube is narrowed to \(\tfrac34\) of its radius. The fluid, the push and the length stay the same, and the flow stays smooth. (a) What fraction of the old flow is left? Give it as a fraction, then as a decimal to two places. (b) The tube carried 256 mL per minute. What does it carry now? (c) Another tube loses just one tenth of its radius, so \(0.9\) of it is left. What fraction of its flow is left? About how much is lost?
  2. Model an artery as a stiff pipe with smooth flow. (a) A stretch of artery narrows to half its radius. How many times bigger must the pressure difference across that stretch be to keep the same flow? (b) Answer again for a stretch narrowed to \(\tfrac34\) of its radius, to two decimal places. (c) The radius halves and the pressure difference doubles. What fraction of the old flow gets through?
  3. All these pipes have the same length, fluid and push. (a) One pipe of radius 2 against two pipes of radius 1, side by side. Which carries more, and how many times as much? (b) How many pipes of radius 1 match one pipe of radius 2? (c) Area goes with radius \(\times\) radius. Show that four pipes of radius 1 have the same total opening as one of radius 2. What fraction of its flow do they carry?
  4. A coffee filter, with made-up numbers. Model the ground coffee as slow seepage: the drip rate goes with the depth of water above it, because deeper water pushes harder. At 4 cm deep it drips 30 drops per minute. (a) Find the drip rate at 8 cm and at 1 cm. (b) As the water drains, does the drip speed up or slow down? (c) A finer grind. Model its gaps as pipes half as wide, with the same total opening. Use 3(c) to find the drip rate at 4 cm. (d) What depth makes the fine grind drip 30 drops per minute?

Stretch: one big vessel or many small ones?

Compare one vessel of radius 10 with a bundle of capillaries of radius 1, in the same unit. Model them all as stiff pipes of equal length with smooth flow and the same pressure difference. (a) How many capillaries carry what the big vessel carries? (b) How many times bigger is the total opening of those capillaries than that of the big vessel? (c) The same flow passes through that bigger opening. How does the average speed in the capillaries compare with that in the big vessel?

Answers

  1. (a) \(\tfrac34 \times \tfrac34 \times \tfrac34 \times \tfrac34 = \tfrac{81}{256}\), which is \(0.32\) to two places (more exactly \(0.316\)). Less than a third of the flow is left. (b) \(256 \times \tfrac{81}{256} = 81\) mL per minute. (c) \(0.9 \times 0.9 = 0.81\) and \(0.81 \times 0.81 = 0.6561\), so about \(0.66\) is left. About a third of the flow is lost from a tenth of the radius.
  2. (a) Half the radius leaves \(\tfrac1{16}\) of the flow. Flow goes with the pressure difference, so that must be 16 times bigger. (b) \(\tfrac{81}{256}\) of the flow is left, so the pressure difference must be \(\tfrac{256}{81}\) times bigger, which is \(3.16\) times. (c) \(2 \times \tfrac1{16} = \tfrac18\) of the old flow. Remember this is a model. A real artery stretches, blood is not a simple fluid, and the body has other ways to respond.
  3. (a) A radius 2 pipe carries \(2 \times 2 \times 2 \times 2 = 16\) units for every 1 unit a radius 1 pipe carries. Two small pipes carry 2 units, so the one wide pipe carries more: \(16 \div 2 = 8\) times as much. (b) 16 pipes. (c) Four small openings: \(4 \times (1 \times 1) = 4\). One big opening: \(2 \times 2 = 4\). The same. But the four carry 4 units against 16, which is \(\tfrac14\) of the flow. The same opening, split into narrower pipes, carries much less.
  4. (a) Twice the depth gives twice the rate: 60 drops per minute. A quarter of the depth gives a quarter of the rate: \(30 \div 4 = 7.5\) drops per minute. (b) It slows down, because the depth falls and the rate goes with the depth. (c) Pipes half as wide with the same total opening carry \(\tfrac14\) of the flow, as in 3(c): \(30 \div 4 = 7.5\) drops per minute. (d) 4 times the push is needed, so \(4 \times 4 = 16\) cm. Real coffee is messier than this model. The grounds swell and shift, and the gaps are not neat pipes.

Stretch. (a) The big vessel carries \(10 \times 10 \times 10 \times 10 = 10\,000\) units for every 1 unit a capillary carries, so it takes 10,000 capillaries. (b) Their total opening is \(10\,000 \times (1 \times 1) = 10\,000\) against \(10 \times 10 = 100\), which is 100 times bigger. (c) Flow is area times average speed. The same flow through 100 times the area means the average speed is \(\tfrac1{100}\) as much. In this model, fluid creeps through the capillaries. In your body, too, blood moves far more slowly through capillaries than through large arteries.

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