Ideal Fluids in Motion • Ideal Fluid: Steady Flow, Incompressible Flow, Non viscous Flow, Irrotational Flow. • The Equation of Continuity: If a fluid is incompressible, its density is constant throughout. Thus the volume of fluid entering a tube at one end per unit of time must be equal to the volume of fluid leaving the other end per unit time. In the time Dt we have A1v1Dt = A2v2Dt. DV = v2A2Dt DV = v1A1Dt A1v1 = A2v2 (continuity equation) RV = Av = volume flow rate = constant Rm = RV = Av = mass flow rate = constant • Bernoulli’s Equation: Application of W = DKE + DU: W ( P1 A1 )Dx1 ( P2 A2 )Dx2 ( P1 P2 )DV ( P1 P2 ) ( P1 P2 ) M M M 12 M (v22 v12 ) Mg ( y2 y1 ) F1 = P1A1 P1 12 v12 gy1 P2 12 v22 gy2 R. Field 10/29/2013 University of Florida M F2 = P2A2 Dx2 = v2Dt Dx1 = v1Dt PHY 2053 Page 1 Bernoulli’s Equation: Applications • Bernoulli’s Equation: P + ½v2 + gy = constant P1 12 v12 gy1 P2 12 v22 gy2 (conservation of energy for a fluid) • Constant Height (y1 = y2): P1 12 v12 P2 12 v22 v1 >> v2 P + ½v2 = constant P1 << P2 v1 P1 v2 P2 Air Foil If the speed of a fluid element increases as the element travels along a horizontal streamline, the pressure of the fluid must DP P P 1 (v 2 v 2 ) 2 1 1 2 2 decrease, and conversely. • Example (velocity of efflux): Area A1 y-axis We can use Bernoulli’s equation to calculate the speed of efflux, v2, from a horizontal orifice (and area A2) located a depth h below the water level of a large talk (with area A1). P1 v gy1 P2 v gy2 1 2 2 1 P1 = P2 = Patm v2 1 2 v1 = v2A2/A1 2 2 v1 v v 2gh ( A2 / A1 ) v 2gh 2 2 h (1↔2) 2 1 2 2 gh 2 gh (Torricelli’s Law) (1 ( A2 / A1 ) 2 ) A2 A1 R. Field 10/29/2013 University of Florida (1) PHY 2053 (2) 2 2 v2 y=0 Area A2 Page 2 Bernoulli’s Equation: Application • Venturi Meter: A Venturi meter is used to measure the flow of a fluid in a pipe. The meter is constructed between two sections of a pipe, the cross-sectional area A of the entrance and exit of the meter matches the pipe’s cross-sectional area. Between the entrance and exit, the fluid (with density ) flows from the pipe with speed V and then through a narrow “throat” of crosssectional area a with speed v. A manometer (with fluid of density M) connects the wider portion of the meter to the narrow portion. What is V in terms of , M, h, a, and A? P1 V P2 v 1 2 2 1 2 2 C VA va (1↔2) v VA / a A2 2 2 gh V 2 1 ( P1 P2 ) (M ) a PC P1 g (d h) P2 gd M gh (C↔C) 2 P1 P2 ( M ) gh R. Field 10/29/2013 University of Florida d PHY 2053 2 gh M 1 V 2 A 1 a Page 3 Bernoulli’s Equation: Application • Siphon: The figure shows a siphon, which is a device for removing liquid from a container. Tube ABC must initially be filled, but once this is done, liquid will flow until the liquid surface of the container is level with the tube opening A. With what speed does the liquid emerge from the tube at C? What is the greatest possible height h1 that a siphon can lift water? Patm V PA v gd 1 2 2 2 1 2 VA va (S↔A) V va / A a 2 1 Patm PA 12 v 2 1 2 gd a P v gd A 2 A A 2 PA 12 v2 gd Patm 12 v2 g (d h2 ) 1 2 v2 g (d h2 ) A = Area of container a = area of tube S y=0 V (A↔C) v PB 12 v2 gh1 Patm 12 v2 g (d h2 ) (B↔A) v 2g(d h2 ) (P P ) h1 atm B (d h2 ) g R. Field 10/29/2013 University of Florida PHY 2053 (h1 ) max P atm (d h2 ) g v Page 4