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14-1 Note 14 Fluid Dynamics

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14-1 Note 14 Fluid Dynamics Note 14 Fluid Dynamics Sections Covered in the Text: Chapter 15, except 15.6 To complete our study of fluids we now examine Flow Rate fluids in motion. For the most part the study of fluids When a fluid occupies a pipe of cross sectional area A in motion was put into an organized state by scientists and flows with average speed v, the rate of flow Q is working a generation after Pascal and Torricelli. We given by shall see here that much of the physics of fluids is Q = Av . …[14-1] encapsulated in the statement of Bernoulli’s principle. The study of fluid flow was driven by the demands The units of Q are m3.s–1. These are units of volume.s–1, of the industrial revolution. It was vital for progress so flow rate is the same as volume rate. and profit that the movement of water, steam and oil The paths of€ particles in a fluid moving with laminar through pipes, and the movement of aerofoils through flow are called streamlines. Streamlines never cross the air, were understood mathematically. Today, the one another (Figure 14-1). physics of fluids in motion is of special interest in the various branches of the environmental and life sciences. Fluids in Motion There are two types of fluid motion called laminar flow and turbulent flow. A fluid will execute laminar flow when it is moving at low velocity. The particles of the fluid follow smooth paths that do not cross, and the rate of fluid flow remains constant in time. This is the easiest type of flow to describe mathematically. A fluid will execute turbulent flow when it is moving above a certain critical velocity. Strings of vortices Figure 14-1. Particles in a fluid moving with laminar flow form in the fluid, resulting in highly irregular motion. follow streamlines that do not cross. A white water rapid is a good example. A system moving irregularly is difficult to describe mathemat- ically, so we shall not be concerned with turbulent The Equation of Continuity flow here. A fluid moving in laminar flow in a flow tube (that may be a pipe) can be shown to satisfy a simple rela- An Ideal Fluid tionship. Consider an ideal fluid flowing through a Since a fluid is in general a complicated medium to tube of variable cross-section (Figure 14-2). In an describe mathematically, even in laminar flow, we elapsed time ∆t, a volume A1v1∆t of fluid crosses area shall assume for simplicity that the fluid is ideal. By A1, and a volume A 2v2∆t crosses area A2. Since the ideal we mean fluid is incompressible and the streamlines do not cross the volume of fluid crossing A1 must equal the 1 The fluid is moving in laminar flow; viscous forces volume of fluid crossing A2, so A1v1∆t = A2v2∆t, from between adjacent layers are negligible. which it follows that A 1v1 = A2v2 or Q 1 = Q2. This 2 The flow is steady, that is, the flow rate does not means that the flow rate change with time. 3 The fluid has a uniform density and is thus incom- Q = Av = const . …[14-2] pressible. 4 The flow is irrotational, that is, the angular momen- An important consequence of eq[14-2] is that if the tum about any point is zero; in common parlance cross sectional area of the flow tube is reduced at the fluid does not “swirl”. some point,€ then the flow speed increases. Eq[14-2] is 14-1 Note 14 known as the equation of continuity.1 Example Problem 14-1 Speed of Blood Flow in Capillaries The radius of the aorta is about 1.0 cm and the blood flowing through it has a speed of about 30.0 cm.s–1. Calculate the average speed of the blood in the capillaries using the fact that although each capillary has a diameter of about 8.0 x 10–4 cm, there are literally billions of them so that their total cross section is about 2000 cm2. Solution: From the equation of continuity the speed of blood in the capillaries is −1 2 2 v1A1 0.30(m.s ) × 3.14 × (0.010) (m ) v2 = = −1 2 A2 2.0 ×10 (m ) = 5.0 x 10–4 m.s–1 € or about 0.5 mm.s–1. This is a very low speed. Bernoulli’s Equation Daniel Bernoulli (1700-1782), a Swiss mathematician and scientist, lived a generation after Pascal and Torricelli. The equation he derived is a more general statement of the laws and principles of fluids we have examined thus far. Bernoulli allowed for the flow tube to undergo a possible change in height (Figure 14-3). Consider Figure 14-2. Illustration of the equation of continuity. The points 1 and 2. Let point 1 be at a height y1 and let v1, flow tube of a fluid is shown at two positions 1 and 2. At no A1 and p1 be the speed of the fluid, cross sectional area time does fluid enter or leave the flow tube. of the tube and pressure of the fluid at that point. Similarly let v2, A2 and p 2 be the same variables at point 2. The actual system is the volume of fluid in the The equation of continuity can be applied to explain flow tube. the various rates of blood flow in the body. Blood In an elapsed time ∆t the amount of fluid crossing A1 flows from the heart into the aorta from which it is ∆V1 = A1v1∆t and the amount of fluid crossing A2 is passes into the major arteries; these branch into the ∆V2 = A2v2∆t. But from the equation of continuity, A1v1 small arteries (arterioles), which in turn branch into = A2v2. So the volume of fluid crossing either area is myriads of tiny capillaries. The blood then returns to the same; let us simply write it as the heart via the veins. Blood flow is fast in the aorta, but quite slow in capillaries. When you cut a finger ΔV = AvΔt . (capillary) the blood oozes, or flows very slowly. We can show this by means of a numerical example. Fluid is moved in the flow tube as the result of the work done on the fluid by the surrounding fluid (the environment).€ The net work W done on the fluid in 1 the elapsed time ∆t is The equation of continuity can be thought of as a statement of the conservation of fluid. As the fluid flows through the pipe the volume of fluid remains constant; it neither increases nor decreases. 14-2 Note 14 ρ 2 p + v + ρgy = const . …[14-6] 2 Many of the “principles” and “laws” we have seen can be shown to be special cases of Bernoulli’s equa- tion.€ We shall consider a number of them. Many homes and buildings in the colder climates are heated by the circulation of hot water in pipes. Even if they are not explicitly aware of it, architects must ensure that their designs conform to Bernoulli’s principle in order to avoid system failure. Let us consider an example. Figure 14-3. Illustration of Bernoulli’s equation. Example Problem 14-2 Application of Bernoulli’s Principle V Δ Water circulates throughout a house in a hot water W = (F2 − F1)Δx = (F2 − F1) = (p2 − p1)ΔV . A heating system. If the water is pumped at a speed of 0.50 m.s–1 through a pipe of diameter 4.0 cm in the This work goes into achieving two things: basement under a pressure of 3.0 atm, what will be the flow speed and pressure in a pipe of diameter 2.6 € 1 changing the kinetic energy of the fluid between cm on the second floor 5.0 m above? the two points by the amount: Solution: ρΔV 2 2 Let the basement be level 1 and the second floor be ΔK = v − v . …[14-3] 2 ( 2 1 ) level 2. We can obtain the flow speed by applying the equation of continuity, eq[14-2]: 2 changing the gravitational potential energy of the 2 fluid between the two points by the amount mg∆h v1A1 −1 π(0.020m) v2 = = 0.50(m.s ) 2 or € A2 π(0.013m) ΔU = ρΔVg(y2 − y1). …[14-4] = 1.2 m.s–1. Since W = (p2 − p1)ΔV = ΔK + ΔU , € To find the pressure p2 we use Bernoulli’s equation, we have,€ by substituting eqs[14-3] and [14-4]: eq[14-5]: 1 2 2 p2 = p1 + ρg(y1 − y2 ) + ρ(v1 − v2 ). € (p1 − p2 )ΔV 2 5 –3 ρΔV Substituting p1 = 3.0 x 10 Pa, ρwater = 1000 kg.m , g = 2 2 9.8 m.s –2, y = 0, y = 5.0 m, v = 0.50 m.s–1, v = 1.2 m.s–1 = ( v2 − v1 ) + ρΔVg(y2 − y1) . 1 2 1 2 2 € we get € 5 p2 = 2.5 x 10 Pa. Dividing through by ∆V we obtain the general form of Bernoulli’s equation: Thus p2 < p1. Obviously, the pipe on the second floor € of the house must be able to withstand less pressure ρ 2 ρ 2 than the pipe in the basement. p + v + ρgy = p + v + ρgy . 1 2 1 1 2 2 2 2 …[14-5] Eqs[14-5] and [14-6] have the look of conservation of energy expressions including work, because that, in We can put this equation into the simpler form: effect, is what they are.
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