15/09/14 We have discussed static fluids: Lecture 4 • how pressure changes with depth • what happens when an object is immersed in a fluid (buoyancy) Hydrodynamics • what happens when a fluid is in contact with a something else (surface tension) – fluids in motion Now let’s look at fluids in motion. Consider an ideal fluid: If the pattern of flow lines does not change with • incompressible: ρ constant time, the flow is called steady flow. • no internal friction (non-viscous) We specify the average motion of the fluid at a particular point in space (and time). The path an individual fluid element follows is called a flow line. Steady flow is also called laminar flow. A streamline is parallel to the velocity of fluid elements at every point. When the rate of flow increases, or at abrupt boundary changes, the flow can become irregular Streamlines cannot cross. and chaotic: turbulent flow. A bundle of streamlines which crosses an area A behaves like a pipe: a flow tube. 1 15/09/14 Con5nuity equaon The mass of fluid in a flow tube is constant In steady flow, flowlines and streamlines are (“what goes in must come out”). identical. Fluid enters the tube through area A1 When the flow pattern changes with time, with average speed v and exits streamlines do not coincide with flow lines. 1 through A2 with average speed v2. We will consider only steady-flow situations. Mass flowing in is A1 v1 = A2 v2 dm1 = ρ1 A1 v1 dt, This is just the law of conservation of mass for mass flowing out is fluids. dm2 = ρ2 A2 v2 dt. The speed is highest where the streamlines crowd together. Constant mass ⇒ dm1 = dm2; incompressible ⇒ ρ1 = ρ2. vlow vhigh vlow Hence A1 v1 = A2 v2 Example: Y-junction with pipes of the same diameter Aside: Low speed High speed The product Q = Av is the volume flow rate dV/ dt, the volume which passes by each second. The mass flow rate dm/dt = ρAv. Af = 2 Ai so the flow speed must fall to half: vf = ½ vi 2 15/09/14 To keep the flow speed the same, the pipes after Example: Water flowing from a tap. the branch must have half the cross-sectional area of those before. Same speed High speed Rate of flow Av = constant ⇒ as v increases, A must decrease ⇒ the stream of water narrows Motivation: Look at constricted region. vlow vhigh vlow F F Bernoulli’s Equaon There must be forces accelerating and decelerating the fluid. ⇒ pressure must be highest where the speed is lowest. Conservation of energy leads to the fundamental relation in fluid mechanics: Bernoulli’s equation. volume of fluid Consider a fluid element of a fluid of (uniform) passing c density ρ. volume of fluid in time dt: passing a dV = A2ds2 Let’s work out the change in energy of this in time dt: element. dV = A1ds1 Continuity ⇒ dV = A1ds1 = A2ds2 3 15/09/14 The work-energy theorem says Alternately, ΔW = ΔKE + ΔPE 2 2 p1 + ρ g y1 + ½ ρ v1 = p1 + ρ g y1 + ½ ρ v1 2 2 or, since this is true for any two points along the (p1–p2) dV = ½ ρ dV (v2 – v1 ) + ρ dV g(y2 – y1) flow tube: or Bernoulli’s equation: (p –p ) = ½ ρ (v 2 – v 2) + ρ g(y – y ) 1 2 2 1 2 1 p + ρgy + ½ρv2 = constant Aside: Solving Bernoulli’s equation: p + ρgy + ½ρv2 = constant • Identify points 1 and 2 along a streamline All three terms have dimensions of pressure (or • Define your coordinate system: where y=0 energy density, energy per unit volume). • List your known and unknown variables The pressure p + ρgy is called the static pressure • Solve for your unknowns, possibly using the the pressure ½ρv2 is called the dynamic pressure. continuity equation. Example: A large tank of water has a hole in the Next lecture side, at a depth h. Where does the stream of water hit the ground? Applying Bernoulli’s equation 4 .