R. Field 10/29/2013 University of Florida PHY 2053Page 1 Ideal Fluids in Motion Bernoulli’s Equation: The Equation of Continuity: Steady Flow, Incompressible Flow, Non viscous Flow, Irrotational Flow. 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  t we have A 1 v 1  t = A 2 v 2  t. Ideal Fluid: A 1 v 1 = A 2 v 2 (continuity equation) R V = Av = volume flow rate = constant R m =  R V =  Av = mass flow rate = constant Application of W =  KE +  U: F 1 = P 1 A 1 F 2 = P 2 A 2 M M  x 2 = v 2  t  x 1 = v 1  t

R. Field 10/29/2013 University of Florida PHY 2053Page 2 Bernoulli’s Equation: Applications Bernoulli’s Equation: Constant Height (y 1 = y 2 ): If the speed of a fluid element increases as the element travels along a horizontal streamline, the pressure of the fluid must decrease, and conversely. P + ½  v 2 +  gy = constant (conservation of energy for a fluid) v 1 = v 2 A 2 /A 1 P 1 = P 2 = P atm P + ½  v 2 = constant Example (velocity of efflux): We can use Bernoulli’s equation to calculate the speed of efflux, v 2, from a horizontal orifice (and area A 2 ) located a depth h below the water level of a large talk (with area A 1 ). (Torricelli’s Law) (2) (1) (1↔2)

R. Field 10/29/2013 University of Florida PHY 2053Page 3 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 cross- sectional 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? C d (C↔C) (1↔2)

R. Field 10/29/2013 University of Florida PHY 2053Page 4 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 h 1 that a siphon can lift water? y=0 V v v A = Area of container a = area of tube S (S↔A) (B↔A) (A↔C)