Chapter 12. Fluid Mechanics. A. The density ρ of a substance of uniform composition is defined as its mass M divided by its volume V.


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1 Chapter 12 Fluid Mechanics 12.1 Density A. The density ρ of a substance of uniform composition is defined as its mass M divided by its volume V. That is,! = M V The density of water at 4 o C is 1000 kg/m 3 = 1 g/cm 3 B. The specific gravity of a substance is defined as the ratio of the density of that substance to the density of water at 4 o C. The density of gold is 19.3g/cm 3. Hence the specific gravity of gold is Pressure in a Fluid The average pressure P is the perpendicular component of the force F divided by the area A on which the force acts. 1
2 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. The unit of pressure in the metric system is the Pascal = Pa = 1 N/m 2. Force is a vector and 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 of interest. Variation of Pressure with depth Consider a fluid at rest. Then all portions of the fluid are in static equilibrium. In the following figure, how is the pressure P 1 related to the pressure P 2? Consider a sample of liquid of crosssectional area A and height h. Then since the fluid is in static equilibrium,! F y = 0. Thus, F! F! mg = use F 2 = P 2 A F 1 = P 1 A m =! V =! Ah ( ) 2
3 so that F 2 F 1 mg = 0 becomes P 2 A P 1 A  ρahg = 0 P deeper level = P upper level + "gh Clearly, the pressure increases as you go deeper in the fluid. At sea level, the atmospheric pressure is P o = 1.013x10 5 Pa = 1 atm. Pressure is constant at the same depth. Pascal s Principle ( ) If an external pressure is applied to an enclosed fluid, the pressure at every point within the fluid increases by that amount. Pascal s principle underlies the operation of a hydraulic press. ΔP in = ΔP out F in = F out A in A out " F out = A % out $ ' F in # & A in 3
4 That is, a small force F in applied to the left end results in a large force F out applied to the right end if A out >> A in. Gauge pressure: The excess pressure above atmospheric pressure is called gauge pressure, and the total pressure is called absolute pressure. Pressure Measurements A. The opentube manometer This apparatus is used to measure the pressure in an enclosed fluid. The governing equation is P = P o + ρgh. 4
5 B. The Barometer (Torricelli ) This apparatus is used to measure atmospheric pressure. The governing equation is: P o = P + ρgh P o = ρgh. For mercury ρ = 13.6x10 3 kg/m 3, and atmospheric pressure at sea level is P o = 1.013x10 5 Pa which corresponds to a height of 76 cm = 0.76 m = 760 mm = inches of mercury. 5
6 12.3 Buoyancy Archimedes s Principle ( 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. Buoyant force = F B F B = " fluid V submerged g volume only F B " F 2 # F 1 F B = P 2 A " P 1 A F B = ( P 2 " P 1 )A but P 2 " P 1 = # g $h so that F B = " g A#h The magnitude of the buoyant force F B is equal to the weight of the fluid displaced by the submerged object. 6
7 12.4 Fluid Flow 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. dv = A 1 v 1 = A 2 v 2 Mass Flow Rate! mass of fluid per unit time passing through any crosssection. The law of conservation of mass in fluid dynamics states that mass flow rate through A 1 = mass flow rate through A 2 7
8 dm 1 = dm 2 dm 1 = " 1 (dv 1 ) dm 2 = " 2 (dv 2 ) " 1 (dv 1 ) = " 2(dV 2 ) but dv 1 = A 1 ds 1 dv 2 = A 2 ds 2 so that the above becomes " 1 A 1 ds 1 = " 2A 2 ds 2 using ds 1 = v 1 ds 2 = v 2 one obtains ρ 1 A 1 v 1 = ρ 2 A 2 v 2 8
9 If the fluid is incompressible, then ρ 1 = ρ 2 and dv = A 1 v 1 = A 2 v 2 or Av = constant Note that Av has units of volume/time = volume flow rate Bernoulli s Principle (1738) Bernoulli s principle results from conservation of energy. Applying the workenergy theorem to the laminar flow of the entire shaded fluid described below at a particular instant of time: 9
10 E 1 + "W other = E 2 where E 1 is the initial mechanical energy of the fluid element, E 2 is the final mechanical energy of the fluid element, and "W other is the work done by the nonconservative forces (or the forces other than the conservative forces). Note that E 1 = 1 2 dm 1 v dm 1 g y 1 = 1 2 "dv v "dv g y 1 "W other = F 1 ds 1 # F 2 ds 2 = P 1 A 1 ds 1 " P 2 A 2 ds 2 = P 1 dv " P 2 dv E 2 = 1 2 dm v dm 2 g y 2 = 1 2 "dv v "dv g y 2 Plugging all these into the workenergy theorem equation yields E 1 + "W other = E 2 10
11 1 2 "dv v "dv g y 1 + P 1 dv # P 2 dv = 1 2 "dv v "dv g y 2 note that the volume elements cancel out, and rearranging terms yields P "v "gy 1 = P "v "gy 2 or P "v2 + "gy = cons tant Applications of Bernoulli s principle: 1. Blowing over sheet of paper in front of your mouth. 2. Canvas top puffs upward in moving convertible cars. 3. Houses may explode during thunderstorms. 4. Lift force on airplane wings: Lift force = (pressure difference)*(area of wing) Lift is greater when the wing area is large or when the plane moves fast so that the pressure difference across the top and bottom of the wing is large. The Magnus force is indicated by the red arrow in the figure below. 11
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