Ideal Fluids in Motion Ideal Fluid Steady Flow

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 r 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 A 1 v 1 Dt = A 2 v 2 Dt. A 1 v 1 = A 2 v 2 (continuity equation) RV = Av = volume flow rate = constant Rm = r. RV = r. Av = mass flow rate = constant • Bernoulli’s Equation: Application of W = DKE + DU: M F 1 = P 1 A 1 M F 2 = P 2 A 2 Dx 2 = v 2 Dt Dx 1 = v 1 Dt R. Field 10/29/2013 University of Florida PHY 2053 1

Bernoulli’s Equation: Applications • Bernoulli’s Equation: P + ½rv 2 + rgy = constant (conservation of energy for a fluid) • Constant Height (y 1 = y 2): P + ½rv 2 = constant 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. • 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). (1) (1↔ 2) P 1 = P 2 = Patm (2) v 1 = v 2 A 2/A 1 (Torricelli’s Law) R. Field 10/29/2013 University of Florida PHY 2053 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 r) 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 r. M) connects the wider portion of the meter to the narrow portion. What is V in terms of r, r. M, h, a, and A? d C (1↔ 2) (C↔C) R. Field 10/29/2013 University of Florida PHY 2053 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 h 1 that a siphon can lift water? (S↔A) A = Area of container a = area of tube S y=0 V (A↔C) v (B↔A) v R. Field 10/29/2013 University of Florida PHY 2053 4