1 Chapter 9 Fluid Mechanics 9.1 Properties of Fluids A fluid is a substance that can flow . Liquids and gases are fluids as both substances can flow. A. The density r of a substance of uniform composition is defined as its mass M divided by its volume V. That is, The density of water at 4 o C is 1000 kg/m 3 = 1 g/cm 3 V M = r 2 9.2 Pressure The average pressure P is the perpendicular component of the force F divided by the area A on which the force acts. 𝑃 ≡ 𝐹 ! 𝐴 • The force exerted by a fluid on a submerged object at any point on the object is perpendicular to the surface of the object. 3 • The unit of pressure in the metric system is the Pascal = Pa = 1 N/m 2 • Force is a vector , but pressure is a scalar . No direction is associated with pressure, but the direction of the force associated with the pressure is perpendicular to the surface on wh ich the force acts. A. Variation of Pressure with D epth in a Liquid If a fluid is at rest , t hen all p arts of the fluid are in static equilibrium Consider a liquid i n static equilibrium as shown in the following figure H ow is the pressure P 1 related to the pressure P 2 ? 4 Consider a sample of liquid of cross - sectional area A and height h in static equilibrium . Then, . Thus, use F 2 = P 2 A F 1 = P 1 A so that F 2 – F 1 – mg = 0 becomes P 2 A – P 1 A - r Ahg = 0 or 𝑃 "##$#% &#'#& = 𝑃 ($$#% &#'#& + 𝜌 𝑔 ℎ Clearly, the pressure increases as you go deeper in the liquid if it is in static equilibrium Pressure is constant at the same depth if the liquid is in static equilibrium 0 F y = S 0 mg F F 1 2 = - - ( ) Ah V m r = r = gh P P 1 2 r + = 5 At sea level, the atmospheric pressure P o is equal to 1.01 3 x10 5 Pa = 101 kPa = 1 atm Define Gauge Pressure as the difference between absolute pressure and atmospheric pressure. That is, 𝑮𝒂𝒖𝒈𝒆 𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆 = 𝑷 − 𝑷 𝒐 6 B. Pascal’s Principle (1623 – 1662) A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container. The hydraulic press: Using Pascal’s Principle, one obtains D P 1 = D P 2 2 2 1 1 A F A F = 1 1 2 2 F A A F ÷ ÷ ø ö ç ç è æ = 7 That is, a small force F 1 applied to the left end results in a large force F 2 applied to the right end if A 2 >> A 1 C. Pressure Measurements 1 The open - tube manometer This apparatus is used to measure the pressure in an enclosed fluid. The governing equation is P = P o + r g h , where h represents the vertical separation distance between the levels of the liquid in the two columns of the U - shaped tube 8 2 The Barometer (Torricelli 1608 – 1647) This apparatus is used to measure atmospheric pressure P o The governing equation is: P o = P + r g h P o = r g h For mercury r = 13.6x10 3 kg/m 3 , and atmospheric pressure at sea level is P o = 1.013x10 5 Pa which corresponds to a 9 height h of 76 cm = 0.76 m = 760 mm = 29.92 in ches of mercury. 9.3 Buoyan cy Forces and Archimedes’s Principle Archimedes’s Principle (287 – 212 B.C.) Any object completely or partially submerged in a fluid is buoyed upward by a force whose magnitude is equal to the weight of the fluid displaced by the object. The magnitude of the buoyant force acting on the submerged object exerted by the entire surrounding fluid is F B 10 Derivation of the magnitude of the buoyant force: 𝐹 * = 𝜌 +&(," 𝑉 -(./#%0#" 𝑔 11 9.5 Fluids in Motion Laminar flow : when each particle of the fluid follows a smooth path so that the paths of different particles never cross each other. The Continuity Equation results from conservation of mass in laminar flow. Consider an incompressible fluid (li quid) flowing through a tube as shown in the figure below. If the diameter of the tube changes, then what happens to the speed of the fluid? If a fluid of mass D M 1 enters the tube through A 1 during a time interval D t , then an equal mass of fluid D M 2 must leave the tube through A 2 . Thus, Mass of fluid entering = mass of fluid leaving D M 1 = D M 2 Þ 12 r A 1 ( D x 1 ) = r A 2 ( D x 2 ) b ut , D x 1 = v 1 D t D x 2 = v 2 D t so that , r A 1 (v 1 D t) = r A 2 (v 2 D t) A 1 v 1 = A 2 v 2 o r , 𝐴 𝜐 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Note that Av has units of volume/time . The volume flow rate Q is defined as: 𝑄 ≡ ∆ 𝑉 ∆ 𝑡 = 𝐴 𝜐 13 9.6 Ideal Fluid Dynamics Bernoulli’s Equation (1738) Applies to an ideal liquid with zero friction - like forces results from conservation of energy. Applying the work - energy theorem to the laminar flow described below yields: or Þ Þ 2 2 2 2 1 2 1 1 gh v 2 1 P gh v 2 1 P r + r + = r + r + t tan cons gh v 2 1 P 2 = r + r + 14 Applications of Bernoulli’s principle: 1. Blowing air over sheet of paper in front of your mouth. 2. Canvas top puffs upward in moving convertible cars. 3. Houses may explode during hurric anes or tornados 4. Lift force on airplane wings: Lift force = (pressure difference)*(area of wing) Lift is greater whe n the wing area is large or when the plane moves fast so that the pressure difference across the top and bottom of the wing s is large. 15 5. Magnus effect 9.4 Surface Tension and Capillary Action A. Surface Tension The environment of the molecules on the surface of a liquid (liquid - gas interface) is different from that in liquid bulk. The surface of the liquid behaves like a membrane under tension. This s urface tension arises because the molecules of the liquid exert attractive forces on e ach other. There is zero net force on a molecule beneath the surface in the bulk of the liquid , but a molecule on the surface of the liquid is pulled into the bulk of the liquid (see figure below). Thus, surface tension tends to minimize the liquid’s surfa ce area, just as a stretched membrane does. That is, the surface of a liquid resists being stretched because it tends to contract as if it were tightened into an elastic film. 16 Consider a thin film of water stretched between a U - shaped frame and a slid ing wire. See figure below. 17 A pulling force F pull exerted by someone is necessary to hold the wire in place. If you let go, the wire will snap back to the left as the surface area of the liquid is reduced. The tension force T is independent of the film’s width w , but it is directly proportional to the sliding wire’s contact length L because a longer wire is touched and pulled on by more surface molecules. Define surface tension g as the tension force T provided by the surface per unit contact length L , that is, 𝛾 ≡ 𝑇 𝐿 Note that “surface tension” g is defined as a tensio n force per meter of contact. The units of surface tension g are N/m. The tension force T = g L is tangent to the surface of the liquid at the line of contact. 18 Note from the table of surface tension values that: 1) the surface tension of water is greater than that of other common liquids, and pure water has a stronger surface tension than soapy water. 2) Hot water, however, has less surface tension than cold water because the faster moving molecules of hot water are not bonded as tightly as when the water is cold This also explains why hot soapy water (with less surface tension) is used for washing. To wash clothing thoroughly, water must be forced through the tiny spaces (crevices) between the fibers (see figure below) and surface tension makes it difficult to force water through the small crevices in the fi bers The job is made easier by increasing the temperature of the water and adding soap, both of which decrease the surface tension o f water 19 3) Surface tension also accounts for the spherical shape of liquid drops such as raindrops and oil drops. Raindrops are spherical because their surfaces tend to contract and force each drop into a shape that has the smallest surface area. This shape is a sph ere because the geometrical figure that has the smallest surface area for a given volume is a sphere. B. Bubbles and Droplets Surface tension pulls an isolated gas bubble or liquid droplet into the shape of a sphere. Why don’t the surface tension forces shrink the size down to zero? Because the inward pulling tension increases the pressure inside the spher ical gas bubble until a balance is reached between pressure forces and surface tension forces. The result is that the pressure inside the gas bubble (or liquid droplet) m ust be larger than the exterior pressure to balance the surface tension forces. 20 Consider a free - body diagram of “ half ” a water droplet as shown in the figure below: The pressure inside the water droplet P in is related to the pressure outside P out , the r adius R of the droplet, and the surface tension g by the expression: 𝑃 ,1 = 𝑃 2(3 + 2 𝛾 𝑅 (derive in class...) a) For a liquid droplet in a gas , the pressure inside the liquid is larger than th e pressure in the gas outside the droplet 𝑃 ,1 = 𝑃 2(3 + 2 𝛾 𝑅