Chapter 4 - Fluids This chapter deals ith fluids, hich means a liquid or a as. It covers both fluid statics (fluids at rest) and fluid dynamics (fluids in motion). Pressure in a fluid Pressure is defined as the perpendicular force on a surface per unit surface area. F P [N/m = pascal = Pa] Mass density is mass per unit volume. M [k/m, /cm, ] Water has a density of about /cm (= k/m ). Due to ravity, pressure increases ith depth in a fluid as P P h, here P is the pressure at the top of the fluid and P is the pressure at a depth h. s seen in the diaram to the riht, additional pressure belo a section of a fluid is required to keep this section from sinkin. Since this section of fluid is in equilibrium, P P M But M = = h, so P P h P P h M P h P If the fluid container is open to the atmosphere, then P is atmospheric pressure. t sea level P =. x 5 Pa ( = 4.7 lb/in = atm)
What is the donard force exerted by the atmosphere on top of a m x m desk top. Or, F = P = (. x 5 Pa)( m ) =.6 x 5 N F = (.6 x 5 N)(.5 lb/n) = 45,585 lb = (45,585 lb)( ton/ lb) =.8 tons!! The reason hy this enormous force doesn t crush the desk is because of a nearly equal upard force on the bottom of the table. t hat depth in ater is the pressure atm? P P h P h P P ( h.x k / m 5 Pa )(9.8m / s ).m mercury manometer consists of an inverted tube of mercury as shon to the riht. The top end is closed and the void at the top is essentially a vacuum. The bottom end is open and is in an open container of mercury. What is the heiht of the column of mercury in the tube? The specific ravity of mercury is.6. P P h P Patm ( bottom) P ( top) 5 P x N m h atm. / (.6x k / m )(9.8m / s ) h.76m 76cm P ~ P atm h
Hydraulic press F hydraulic press uses a fluid to manify an applied force. force F applied to the small piston of area increases the pressure in the fluid by P = F /. This pressure increase is transmitted uniformly throuhout the fluid (Pascal s principle). This additional pressure results in a lift on the lare piston. P P F F F F fluid F In a hydraulic press, the diameter of the small piston is.5 cm and the diameter of the lare piston is cm. If the force applied to the small piston is 5 N, hat is the force applied to the lare piston? (.5m) F F ( 5N) 8 N (.5m) Is conservation of enery violated? Not really. The lare piston only moves /6 th as far as the small piston, so the ork done in pushin the to pistons is the same (in the absence of resistance). rchimede s Principle rchimede s principle states that an object submered in a fluid is buoyed up by a force equal to the eiht of the fluid displaced by the object. B B W f m f f obj W The buoyant force, B, is just a consequence of the fact that the pressure belo the object is reater than above it. To understand rchimede s principle, e envision replacin the object ith fluid of the same size and shape. This fluid must be in equilibrium and have the same buoyant force as the object. Thus, its eiht and the buoyant force must be the same.
cubical block of aluminum cm on ede is suspended in ater by a cord. What is the tension in the cord? The density of l is.7 x k/m. Since the block is in equilibrium, T B m T m l l l (.7x 6.7 N B m l k / m m l x ater k / m )(9.8m / s )(.m) B m T n cube floats in a lass of ater. What fraction of its volume is belo ater? The specific ravity of is.97. W m B m belo belo.97 Thus, 9.7% of the volume of the is belo the ater. Fluid Dynamics Equation of continuity The rate at hich fluid mass flos throuh to different parts of the same pipe must be the same. Thus, M M x x v t v t v v 4
If the fluid is incompressible, i.e., its density is nearly the same throuhout the pipe, then e have v v (Eq. of continuity) hose of diameter cm has a nozzle of diameter cm. If the ater flos at m/s in the hose, hat is the ater speed as it oes throuh the nozzle? r v v v () (m / s) 8m / s r () Bernoulli s equation: Bernoulli s equation ives a relationship in a floin fluid beteen the fluid s pressure, flo speed, and elevation. It is based on conservation of enery and holds for an ideal fluid. The ideal fluid ould be () non-viscous, () incompressible, () steady in its flo, and (4) non-turbulent. P v y constant Bernoulli s equation This means that if you pick any to points in a floin fluid, P v y P v y If the fluid is at rest (v = v = ), then Bernoulli s equation is the same as the earlier equation ivin P as a function of depth in a fluid P P ( y y ) Qualitatively, Bernoulli s equation says that the pressure is loer in a reion of a fluid here its speed is reater. n airplane in has curvature and anle of attach such that the air speed above the in is reater than belo. If v(belo) = m/s and v(above) = 5 m/s and the area of the in is m, hat is the lift on the in? The density of air is about. k/m. The difference in elevation belo and above the in is nearly the same, so y ~ y and 5
P P P ( v v ) 666 N / m Lift F P (666N / m )(m ) 6,666 N (.k / m )((5m / s) (m / s) ) liquid in a pressurized container has a hole in the side. What is the speed of the liquid comin out of the hole? P We apply Bernoulli s equation to to points the point here the fluid leaves the hole (point ) and a point at the top of the fluid in the container (point ). P v y P v y y h v y P The pressure at the hole is atmospheric pressure and the pressure, P = P a, and the pressure at the top of the fluid is P =P + P a. lso, e assume that volume of the fluid is such that v. Then, P v y P P y a a Or, usin h = y y, e et P v h P is the aue pressure at the top of the fluid, i.e., the pressure above atmospheric pressure. If the container is open, then P = and v h This is just the speed that the fluid ould obtain if it fell directly a heiht h. Question: Can you calculate ho far the fluid leavin the hole ould travel horizontally before hittin the round? 6
Wind enery Ho much poer can be enerated by a ind turbine? The kinetic enery carried by the ind per unit volume is KE volume v If all this kinetic enery could be converted into poer, then the poer ould be Poer KE volume x volume time ( v ) ( v) v Or, the poer per unit area is Poer v Thus, the poer available varies ith the cube of the indspeed. If the indspeed doubles, then the poer increases by a factor of 8. If the indspeed increases by 5%, then the poer increases by about a factor of. So, indy reions are especially important for ind poer. Not all of the poer iven by the above equation can be extracted from the ind. For one thin, this assumes that the ind ould be brouht completely to rest by the turbine. It also nelects the efficiency of conversion into electricity. The actual poer that can be extracted from the ind no more than about 5% of that iven by the equation. 7