Chapter 9 Fluids. Pressure

Transcription

1 Chapter 9 Fluids States of Matter - Solid, liquid, gas. Fluids (liquids and gases) do not hold their shapes. In many cases we can think of liquids as being incompressible. Liquids do not change their volume (appreciably) when they are heated. Gases do not have a definite volume or shape. ressure fluid particle collides with the surface. The change in momentum is caused by an impulse that acts to the right on the particle. y Newton s third law, the particle pushes to the left on the surface. Definition of pressure F av Surprisingly, pressure is a scalar and not a vector. ressure is measured in N/m which is also called a pascal (a). There are a zillion other units: atmosphere, lbs/square inch, torr, bar, etc. The atmosphere exerts pressure. atm a

2 ascal s rinciple change in pressure at any point in a confined fluid is transmitted everywhere throughout the fluid. ascal s principle is the basis of hydraulics. Hydraulics are the most effective way to transmit a force. Density is the mass per unit volume. It is defined as m V Density is measured in kg/m 3. s we descend down into a fluid the amount of fluid above us increases. That additional fluid pushes down and the pressure increases with depth. t depth d the pressure has increased gd

3 Measuring ressure manometer consists of a U-shaped tube containing some mercury. When both sides are open to the atmosphere, the height in both arms are the same. When one side is connected to the pressure to be measured, the heights are different. The pressures at are the same, gd C Usually, the manometer is open to the atmosphere. It will measure pressures relative to atmospheric. The gauge pressure is the pressure above an atmosphere. gauge abs atm

4 lood pressure is measured using a sphygmomanometer. The pressure in the cuff is higher than the systolic pressure the maximum pressure in the brachial artery that occurs when the heart contracts. The cuff pressure squeezes the artery closed and no blood flows into the forearm. valve on the cuff is then opened to allow air to escape slowly. When the cuff pressure decreases to just below the systolic pressure, a little squirt of blood flows past the constriction in the artery with each heartbeat. The sound of turbulent blood flow past the constriction can be heard through a stethoscope. s air continues to escape from the cuff, the sound of blood flowing through the constriction in the artery continues to be heard. When the pressure in the cuff reaches the diastolic pressure in the artery the minimum pressure that occurs when the heart muscle is relaxed there is no longer a constriction in the artery, so the pulsing sounds cease. uoyant Force When an object is submerged in a fluid, the fluid pushes up on the object. The buoyant force is given by F gd gv rchimedes principle fluid exerts an upward buoyant force on a submerged object equal in magnitude to the weight of the volume of fluid displaced by the object. We still need to use free body diagrams! The force F is the force of the fluid above the block pushing down and he force F is the force of the fluid below the block pushing up. The buoyant force is F F F The specific gravity is defined as the ratio of the density of the material to the density of water. S.G. water

5 If S.G <, the object will float. If S.G. >, the object sinks. rchimedes and the golden crown: The story: We have completed our study of fluids at rest. Now we consider fluids in motion. Fluid Flow fluid moving past a surface can exert a viscous force against the surface. This is similar to the frictional force of an object sliding over a surface. We will start by assuming the viscous force to be small. When flow is steady, the velocity at any point is constant in time. The flow may not be the same everywhere. Steady flow is laminar. The streamlines are clearly defined. s we have done many times this semester, we assume the ideal case first. n ideal fluid is incompressible, undergoes laminar flow, and has no viscosity. The continuity equation Since the fluid is incompressible, the fluid flows faster in the narrow portions of the pipe. The mass flow rate is defined as m v t The volume flow rate is

6 V v t The continuity equation for an incompressible fluid equates the volume flow rates past two different points, v v The continuity equation is a consequence of conservation of mass. ernoulli s Equation Using energy ideas, the pressure of the fluid in a constriction cannot be the same as the pressure before or after the constriction. For horizontal flow the speed is higher where the pressure is lower. This is called the ernoulli effect. For a more general situation where the pipe is not horizontal, we can use energy considerations to derive ernoulli s equation. (The derivation is given on page 335. I will only quote the result.) gy v gy v or gy v Hopefully, this reminds you of constant W nc mgy mv mgy mv ernoulli s equation relates the pressure, flow speed, and height at two points in an ideal fluid.

7 roblem 50. Suppose air, with a density of.9 kg/m 3 is flowing into a Venturi meter. The narrow section of the pipe at point has a diameter that is /3 of the diameter of the larger section of the pipe at point. The U-shaped tube is filled with water and the difference in height between the two sections of pipe is.75 cm. How fast is the air moving at point? Strategy Use the continuity equation to relate the speeds at points and. Then, use ernoulli s equation to find the speed of the air at point. Solution Relate the air speeds at points and. d v v, so v v v 9. v ( d 3) Find the speed of the air at point. Note that y y. v gy v v 40v v v 8 gy (9v v ) The pressure difference is related to the height difference in the manometer gh W Subsituting 40v gh W v W gh 40 3 (000kg/m )(9.8 m/s )( (.9 kg/m ) cm).8 m/s roblem 46. In a tornado or hurricane, a roof may tear away from the house because of a difference in pressure between the air inside and the air outside. Suppose that air is blowing across the top of a 000 ft roof at 50 mph. What is the magnitude of the force on the roof?

8 Strategy Use ernoulli s equation to find the pressure difference at the roof. Solution Let the region above the roof be labeled. ssume the air under the roof is still. v v gy v gy gy gy Now y is almost equal to y and we can assume that the difference in height has negligible effect on the pressure. v v Which side is at the higher pressure, the inside or outside? The magnitude of the force on the roof is F airv 3 50 mi h 609 m m 5 (.0 kg m ) (000 ft ) 5.00 N. h 3600 s mi 3.8 ft which is equal to 56 tons! Viscosity ernoulli s equation ignores viscosity. When real fluids flow, the different layers of fluid drag against each other. pressure difference is needed to maintain the flow. This is similar to needed a constant force to overcome kinetic friction. Fluid layers further away from the wall flow faster than those close to the wall.

9 oiseuille s Law The volume flow rate of viscous fluid through a horizontal cylindrical pipe depends on ressure gradient L Viscosity. The higher the viscosity, the lower the flow rate Radius of the pipe. The French physician oiseuille (pwahzoy) formulated his law after studying blood flow V t / L r 8 4 Viscosity is (Greek letter eta), measured in as. Other units are poise (pwaz) and c. roblem. Water flows through a pipe of radius 8.50 mm. The viscosity of water is.005 c. If the flow speed at the center is 0.00 m/s and the flow is laminar, find the pressure drop due to viscosity along a 3.00 m section of pipe. Strategy Use oiseuille s Law to find the pressure drop. Solution oiseuille s law is V t / L r 8 4 Solving for V t / L r 8 V 8 L 4 t r The volume flow rate is related to the area of the tube and the speed of the flow (see the continuity equation) 4 V t v r v (8.500 m) (0.00 m/s) m /s Viscosity is not in the correct units. = as. The pressure difference is

10 V t 3 8 L (.0050 as)(3 m) (4.540 m /s) 67 a r (8.500 m) Turbulence Turbulence is unsteady fluid flow, not laminar flow. In turbulent flow, swirling vortices appear. The vortices are not stationary and they move with the fluid. The velocity of the fluid flow can change direction and magnitude in an uncontrolled way. Why do golf balls have dimples? From HowStuffWorks.com: The reason why golf balls have dimples is a story of natural selection. Originally, golf balls were smooth; but golfers noticed that older balls that were beat up with nicks, bumps and slices in the cover seemed to fly farther. Golfers, being golfers, naturally gravitate toward anything that gives them an advantage on the golf course, so old, beat-up balls became standard issue. t some point, an aerodynamicist must have looked at this problem and realized that the nicks and cuts were acting as turbulators they induce turbulence in the layer of air next to the ball (the boundary layer ). In some situations, a turbulent boundary layer reduces drag. If you want to get deeper into the aerodynamics, there are two types of flow around an object: laminar and turbulent. Laminar flow has less drag, but it is also prone to a phenomenon called separation. Once separation of a laminar boundary layer occurs, drag rises dramatically because of eddies that form in the gap. Turbulent flow has more drag initially but also better adhesion, and therefore is less prone to separation. Therefore, if the shape of an object is such that separation occurs easily, it is better to turbulate the boundary layer (at the slight cost of increased drag) in order to increase adhesion and reduce eddies (which means a significant reduction in drag). Dimples on golf balls turbulate the boundary layer. The dimples on a golf ball are simply a formal, symmetrical way of creating the same turbulence in the boundary layer that nicks and cuts do. aseball: Magnus effect. Tennis: top spin, slice, side spin; aseball: curveballs; Golf: slice, hook Viscous Drag n object moving through a fluid experiences drag. Clearly the drag depends on the viscosity of the fluid, the speed of the object, and its size. Viscous drag is very complicated, but there is a well understood example. For a sphere of radius r traveling at appropriate speed v (so there is no turbulence), Stoke s Law holds

11 F D 6rv When the viscous drag is equal to a falling object s weight, the object reaches terminal velocity. This is how parachutes work. Surface Tension The surface of a liquid has special properties not associated with the interior of the liquid. The surface acts like a stretched membrane under tension. The surface tension () of a liquid is the force per unit length with which the surface pulls on its edge. Soaps break up the surface tension so that water can reach into small places and clean them. ubbles gas bubble inside a fluid is in equilibrium between the surface tension trying to shrink the bubble and the pressure inside trying to expand it. The pressure inside the bubble must be greater than the fluid pressure outside. It can be shown that the pressure difference,, is in out r s the bubbles rise to the surface from the bottom, they expand. in reduces with the expansion and r increases as well. The difference, in out, becomes smaller.