Fluid Mechanics Chapter 13

2 Fluid Anything that can flow A liquid or a gas Physics Chapter 13

3 Density Mass per unit volume Where  (rho) is the density, m is mass, and V is volume A homogeneous material has the same density throughout The SI unit of density is the kg/m 3. Physics Chapter 13

4 Specific gravity Should be called relative density, but we are stuck with the traditional term The ratio of its density to the density of water. Physics Chapter 13

5 Pressure Pressure is force per unit area, expressed in Pascals (Pa). 1 Pa = 1 N/m 2 Physics Chapter 13

6 Pressure changes Pressure increases with depth –Atmospheric pressure is greater at sea level than on top of a mountain –Water pressure is greater in deeper water When y 2 is greater, p 2 is less. Physics Chapter 13

7 Open containers The pressure at the surface is atmospheric pressure, or p 0. If we are at a depth, h, below the surface, Physics Chapter 13

8 Pascal’s Law If we increase the pressure at the surface, the pressure at any depth increases by the same amount. The pressure is transmitted throughout the fluid – if it has a uniform density – this is a fairly safe assumption for most liquids and for gases over small distances. Physics Chapter 13

9 Pascal’s Law Used in hydraulics to use a small force over a small area to exert a large force over a large area. –See page 304 Physics Chapter 13

10 Gauge pressure If the pressure in your tire equals atmospheric pressure, the tire is flat. When your pressure gauge reads 32 psi, that means the pressure in the tire is 32 psi above the atmospheric pressure. Physics Chapter 13

11 Absolute pressure Total pressure Atmospheric pressure at sea level is kPa or 14.7 psi. Also called 1 atm Physics Chapter 13

12 Example A residential hot water heating system has an expansion tank in the attic, 12 m above the boiler. If the tank is open to the atmosphere, what is the gauge pressure in the boiler? What is the absolute pressure? p g = 118 kPa, p a =219 kPa Physics Chapter 13

13 Open-tube manometer Measures the pressure in a container. Physics Chapter 13

14 Barometers Long glass tubes full of mercury used to measure atmospheric pressure. h Physics Chapter 13

15 Buoyancy When an object is less dense than water, it floats. Physics Chapter 13

16 Archimedes's principle When an object is completely or partially immersed in a fluid, the fluid exerts an upward force on the object equal to the weight of the fluid displaced by the object. We call this upward force the buoyant force. Physics Chapter 13

17 Weight of water displaced Physics Chapter 13

18 Example A cork has a density of 200 kg/m 3. Find the fraction of the volume of the cork that is submerged when the cork floats in water. Physics Chapter 13

19 Example continued Physics Chapter 13

20 Example An ore sample weighs N in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 9.00 N. Find the total volume and the density of the sample. Physics Chapter 13

21 Example continued Physics Chapter 13

22 Example continued Physics Chapter 13

23 Example continued Physics Chapter 13

24 Fluid Flow An ideal fluid is incompressible and has no internal friction. We will only deal with laminar flow, which has a steady-state pattern. We will not deal with turbulent flow, which is chaotic. Physics Chapter 13

25 Continuity equation The mass of a moving fluid doesn’t change as it flows. Physics Chapter 13

26 Volume flow rate The rate at which volume crosses a section of the tube: Physics Chapter 13

27 Example Blood flows from an artery of radius 0.3 cm, where it’s speed is 10 cm/s into a region where the radius has been reduced to 0.2 cm. What is the speed of the blood in the narrower region? 22.5 cm/s Physics Chapter 13

28 Bernoulli’s Equation Relates pressure, flow speed, and height for flow of ideal fluids. Derived in book by applying the work energy theorem to a flowing fluid Physics Chapter 13

29 Bernoulli’s Equation Make sure that your units are consistent. –Always use pascals, kg/m 3, and m/s Always use either all absolute pressures or all gauge pressures. Physics Chapter 13

30 Example A large tank of water has a small hole a distance h below the water surface. Find the speed of the water as it flows from the tank. V b =sqrt(2gh) Physics Chapter 13

31 On your own Water enters a house through a pipe with an inside diameter of 2.0 cm at an absolute pressure of 4.0 x 10 5 Pa. A 1.0 cm diameter pipe leads to the second-floor bathroom 5.0 m above. When the flow speed at the inlet pipe is 1.5 m/s, find the pressure and volume flow rate in the bathroom. 3.3 x 10 5 Pa4.7 x m 3 /s Physics Chapter 13