Kinds of Flow and the Equation of Continuity

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Open a kitchen tap gently, and watch the stream of water. Near the tap, the stream is thick. Lower down, it becomes thin. Hmm. Where did the water go? Is some of it lost on the way? By the end of this lesson, you will explain it in one line. Today: kinds of flow, and the equation of continuity. Questions on this come up often in JEE, so let's begin.

0:30Steady flow and turbulent flow

So, first, how can a fluid flow? The calm kind is called steady flow. You will also hear streamline flow, or laminar flow: three names, one idea, okay? Steady means the velocity at each point does not change with time. Every particle reaching a given point moves just like the one before it. Right? So the fluid particles follow fixed paths. These paths are called streamlines. Think of them as lanes on a road that never change. Clear? Now tell me. Can two streamlines ever cross each other? Never! Correct! Look, if two streamlines crossed, a particle at the crossing point could go along either one. That point would have two different velocities. But in steady flow, each point has one fixed velocity. So streamlines never cross. Got it? But this calm flow does not last at every speed. Above a certain speed, the flow becomes turbulent. You get eddies, small swirling whirlpools, and the motion becomes irregular. Open the tap fully, and the smooth, glassy stream turns rough and messy. With me?

1:44The equation of continuity

Alright, now the main result. Take an ideal fluid, flowing through a tube that is wide at one end and narrow at the other. Incompressible: it cannot be squeezed into less space. Non-viscous: no internal friction, so we can take one speed across the whole section. And the flow is steady. These are the conditions we use. Okay? Now, every second, some volume of water enters the wide end. Tell me, how much volume must leave the narrow end in that same second? Exactly the same volume! Yes! The fluid cannot be squeezed, and in steady flow nothing piles up inside the tube. So, A one v one equals A two v two equals Q, the same at every section. Capital A with subscript one is the area of cross-section at the first section, and small v with subscript one is the speed there. Subscript two means the same at the second section. Right? And this capital Q is the volume flow rate: the volume of fluid passing any section per second. Square metres times metres per second gives cubic metres per second. On screen, it is written m cubed, s to the power minus one: cubic metres per second, because the power minus one means per second. Clear? Suppose at some section, the area becomes half. What happens to the speed there? It doubles! Perfect! Capital A times v stays fixed, so a smaller area means a larger speed. The double arrow you see means so, or implies. Narrow part, faster flow. Got it?

3:24Topper's trick: circular pipes, and our tap

Now, a trick that toppers use. Most pipes in questions are circular. Then the area is pi r squared: pi is the Greek letter, and small r is the radius of the pipe. Okay? Put it into continuity. Pi r one squared v one equals pi r two squared v two, with r one and r two the two radii. Pi cancels. So v is proportional to one over r squared. The symbol that looks like an open loop, between v and one over r squared, means is proportional to. Right? So tell me. If the radius of a pipe is halved, does the speed double? No, no, careful! This is a very common trap. The speed depends on the area, and the area goes as r squared. Half the radius gives one quarter of the area, so the speed becomes four times as large; that is the times four on screen. Always compare areas, never radii. With me? Now, back to our tap. Why does the falling stream get thinner? As the water falls, gravity speeds it up. v squared equals v naught squared plus two g h. Here v naught, with the small zero as subscript, is the speed at the tap, g is the acceleration due to gravity, and h is the height fallen. Okay? And capital A times v must stay constant. So as v grows, the area shrinks, and the stream narrows. No water is lost at all! Puzzle solved. Clear?

4:54Example 1: pipe radius 2 cm to 1 cm, speed in narrow part

Now, Example one. Water flows through a pipe whose radius narrows from two centimetres to one centimetre; c m means centimetres. The speed in the wide part is one metre per second. Find the speed in the narrow part. Pause the video, and try it first. Done? Okay, let's check. The radius goes from two to one; it halves. So the speed doubles: two metres per second, right? Oh no, careful! Remember the trick. Compare the areas: A one over A two equals pi times two squared, over pi times one squared. Pi cancels, and we get four over one, that is four. It is a ratio, so the centimetres cancel too. Got it? So, v two equals v one times A one over A two. One times four: four metres per second. Four, not two! Clear?

5:54Example 2: hose nozzle exit speed and litres per minute

Now a slightly harder one. We have a garden hose of internal area two square centimetres, carrying water at one point five metres per second. At the end is a nozzle of area zero point five square centimetres. Find the exit speed, and the volume that comes out per minute. Pause the video and try it. Hint: speed first, then volume. Done? Let's see. Exit speed first: v two equals v one times A one over A two. Two over zero point five is four, so one point five times four: six metres per second. Again a ratio, so square centimetres are fine here. With me? Now the volume. Q equals A one times v one. So two times one point five, that is three, right? No, no, wait! Units! The speed is in metres per second, so the area must be in square metres. One centimetre is ten to the power minus two metres, so one square centimetre is ten to the power minus four square metres. So A one is two times ten to the power minus four square metres. Okay? Now Q equals two times ten to the power minus four, times one point five. That is three times ten to the power minus four cubic metres per second. Right? Here is a quick check. At the nozzle, A two v two is zero point five times ten to the power minus four, times six. Again three times ten to the power minus four! The same volume per second at both ends. Clear? Last part: per minute. One minute is sixty seconds, so three times ten to the power minus four, times sixty. One point eight times ten to the power minus two cubic metres. With me? But the answer is wanted in litres. Quick question: how many litres make one cubic metre? One thousand litres, yes! Capital L stands for litres. So one point eight times ten to the power minus two, times one thousand: eighteen litres per minute; min to the power minus one means per minute. Six metres per second, and eighteen litres every minute. Got it?

8:14Quick revision: streamlines, A1v1 = A2v2 = Q, compare areas

So, quick revision. Steady flow: the velocity at each point is fixed in time, and streamlines never cross. For an ideal fluid in steady flow, A one v one equals A two v two equals Q, the volume flow rate. For circular pipes, v is proportional to one over r squared: compare areas, not radii, and use S I units when you find the flow rate. Now go and try the practice problems on this page. All the best!