Fluid Dynamics. Equation of continuity Bernoulli s Equation Bernoulli s Application Viscosity Poiseuilles law Stokes law Reynolds Number


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1 Fluid Dynamics Equation of continuity Bernoulli s Equation Bernoulli s Application Viscosity Poiseuilles law Stokes law Reynolds Number
2 Fluids in Motion steady or laminar flow, if each particle of the fluid follows a smooth path, such that the paths of different particles never cross each. turbulent flow is irregular flow characterized by small whirlpoollike regions viscosity is used in the description of fluid flow to characterize the degree of internal friction in the fluid.
3 ideal fluid flow 1. The fluid is nonviscous.(i.e.) internal friction is neglected.. The flow is steady. In steady (laminar) flow, the velocity of the fluid at each point remains constant. 3. The fluid is incompressible. The density of an incompressible fluid is constant. 4. The flow is irrotational. the fluid has no angular momentum about any point.
4 Fluids in Motion The steady flow the velocity of the fluid particles at any point is constant as time passes. The steady flow is often called streamline flow. At any point along a streamline, the velocity vector of the fluid particle at that point is tangent to the streamline The mass flow rate: is the mass of fluid per second that flows through a tube.
5 equation of continuity for fluids v 1 v x 1 A 1 x A An incompressible fluid, like water, flowing through a pipe will slow down if the pipe gets wider. If the green volume and the purple volume both travel through the pipe in the same amount of time, green mass (m 1 )= purple mass (m ) A 1 x 1 = A x Av = constant m m 1 A x 1 A 1 x A v 1 A 1 v t t The bigger the area, the slower the fluid speed.
6 Examples: Each second, 5 55 m 3 of water flows over the 670mwide cliff of the Horseshoe Falls portion of Niagara Falls. The water is approximately m deep as it reaches the cliff. What is its speed at that instant? v = 4 m/s
7 Bernoulli Equation: v The relationship between fluid speed, pressure, and elevation P + ½ v + gy = constant P 1 v 1 P y y 1 P = pressure v = fluid speed = fluid density (a constant) y = height As a nonviscous, incompressible fluid flows through a pipe that changes in both area and height, the pressure and fluid speed change, but the above expression remains constant everywhere in the pipe.
8 Bernoulli v Equation Proof F P x A P 1 v 1 F 1 y y 1 A 1 x 1 Let green volume = purple volume = V. The volumes travel through the pipe in the same time. Let s look at the work done on all the fluid from A 1 to A by the pressure in the pipe at each end as the fluid at the bottom moves a distance x 1 : W = F 1 x 1  F x = P 1 A 1 x 1  P A x = P 1 V  P V continued on next slide
9 Bernoulli Equation v Proof (cont.) F P x A P 1 v 1 F 1 y y 1 A 1 x 1 So the net work done by the fluid pressure is W = (P 1  P )V. This work goes into changing the potential and kinetic energy of the fluid: (P 1  P ) V = U + K = m g y  m g y 1 + ½ m v  ½ m v 1 where m is the mass of the moving volume of fluid. Dividing by the volume, we get: P 1  P = g y  g y 1 + ½ v  ½ v 1 P 1 + ½ v 1 + g y 1 = P + ½ v + g y continued
10 Bernoulli Equation Proof (cont.) The last equation shows that P + ½ v + gy is the same before and after traveling from the left end of the pipe to the right end. Since these two places are completely arbitrary, our derivation shows that P + ½ v + gy is a constant throughout the pipe, and the Bernoulli equation is proven! This equation is useful in many applications, from aviation to medicine.
11 Bernoulli s Principle Bernoulli s principle says that the faster a fluid is moving the less pressure it exerts. This is true for a nonviscous fluid flowing at a constant height. It follows directly from the Bernoulli equation: P + ½ v + gy = constant. If y is a constant, then P + ½ v = constant. This shows that if pressure increases, then v decreases, and versa vise.
12 Airplanes Bugs Bunny & Yosemite Sam are taking a little plane ride. What does Bernoulli s principle have to do with this situation? answer: Air is not incompressible, but the Bernoulli principle can explain, in part, why an airplane flies. The upper surface of the wing has a smaller radius of curvature than the bottom surface. Air on top must travel farther, so it moves faster, and the pressure there is lower, creating lift. Also, because of the wing s upward tilt, air is pushed downward. So, the air pushes back on the wing in the direction of F. F
13 Bernoulli Example 1 In an unfortunate mishap, the Tidy Bowl man gets flushed. With the info given below, let s figure out the pressure difference he and his boat experience as he travels across the pipe. Since the wider pipe has 4 times the area, the water speed there is 4 times slower (recall A v = constant). So, v = m/s, which means P > P 1. From Bernoulli s equation at a constant height, we get: P 1 + ½ v 1 = P + ½ v P = P  P 1 = ½ v 1  ½ v = ½ (v 1  v ) = ½ (1000 kg / m 3 ) (64 m /s  4 m /s ) = kg/(ms ) = kg m/(s m ) = N/(m ) = Pa P 1 8 m/s v P A 4 A
14 Bernoulli Example air flow h w a t e r Three vertical pipes open up inside the top pipe, in which air is flowing. Because air flows faster in the thin section of the top pipe, the pressure is lower there, and the water level beneath it rises more than in the other two. The difference in pressure between the thick section of the top pipe and the thin section is given by: P = gh.
15 The Venturi Tube The horizontal constricted pipe illustrated in this Figure 1 known as a Venturi tube, can be used to measure the flow speed of an incompressible fluid. Determine the flow speed at point if the pressure difference P 1 & P is known Pressure P 1 is greater than pressure P because v 1 < v. This device can be used to measure the speed of fluid flow. (b) A Venturi tube, located at the top of the photograph. The higher level of fluid in the middle column shows that the pressure at the top of the column, which is in the constricted region of the Venturi tube, is lower.
16 Torricelli s Law After eating some of Popeye s spinach Olive Oyl club a ball clear across the course and into a water tower. How far from the base of the tower does the water land? answer: This is like water moving downward through a very large pipe and then moving sideways through a very small pipe. We ll find v h using Bernoulli s equation and then do projectile motion. Both at the hole and the top the water is exposed to the air, so the pressure there is normal air pressure. Bernoulli says: v t 8 m v h 15 m P air + ½ v t + g(8) = P air + ½ v h + g(0)
17 Torricelli (cont.) P air + ½ v t + g(8) = P air + ½ v h + g(0) v t ½ v t + 8 g = ½ v h Since the area at the top is so much larger than the area of the hole, the water is shooting out much, much faster the level is dropping at the top. This means v t is negligible, and our equation becomes: 8 m 15 m v h 8 g = ½ v h v h = g (8) = 1.5 m/s. In general, the speed of a fluid leaking from a hole is given by: v = g h This is known as Torricelli s principle. continued
18 Torricelli (cont.) The water molecules shooting out of the hole are projectiles being shot horizontally at 1.5 m / s from 15 m up. y = v 0 t + ½ a t 15 = t t = 1.75 s 8 m 15 m v h The range, then, is: (1.5 m/s) (1.75 s) = 1.9 m Note: As the water level decreases, the speed decreases at the hole, and so does the range.
19 Bernoulli: Wind Example The Big Bad Pig is about to blow down the house of the Three Little Wolves. The little wolves live in a little flatroofed house. The wolf home has very sturdy walls, so the Big Bad Pig decides to incorporate a little physics into his attack. Instead of blowing directly on the walls, he blows over the roof. He blows hard enough that the air above the roof is moving fast enough to create a large pressure difference. Inside the air is at normal atmospheric pressure. Outside it is much lower. The pressure difference can blow the roof right off the Three Little Wolves house. Strong, naturally occurring winds can damage structures in the same way.
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