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Download Lecture 14c - TTU Physics
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Sect. 14.5: Fluid Dynamics Section 14.5: Fluid Dynamics • We’ve done fluid statics. Now, Fluid Dynamics (fluid flow), which is much more interesting! COURSE THEME: NEWTON’S LAWS OF MOTION! NOW • Sects. 14.5 - 14.7: Methods to analyze the dynamics of fluids in motion. • First, we need to discuss FLUID LANGUAGE. We’ve introduced a lot of this language while talking about fluid statics. But, there is some other terminology we need to discuss before we discuss Newton’s Laws (Especially Newton’s 2nd Law!) in Fluid Language! Types of Fluid Flow • Two main types of fluid flow: 1. Laminar Flow (or Streamline Flow) – – – – Steady flow Each particle of the fluid follows a smooth path The paths of the different particles never cross each other Every fluid particle arriving at a given point has the same velocity – The path taken by the particles is called a streamline Paths of the particles look qualitatively like this! We’ll assume this type of flow Types of Fluid Flow • Two main types of fluid flow: 2. Turbulent Flow – Irregular flow which has small whirlpool-like regions – It’s turbulent flow when the particles go above some critical speed Streamlines can cross each other Paths of the particles can Look qualitatively like this! We’ll not discuss this type. Viscosity • Viscosity is a measure of the amount of internal friction in the fluid. • This internal friction or VISCOUS FORCE, comes from the resistance that two adjacent layers of fluid have to moving relative to each other. • Viscosity causes part of the kinetic energy of a fluid to be converted to internal energy. Ideal Fluid Flow • We make four simplifying assumptions in our treatment of fluid flow to make the analysis easier: 1. The fluid is nonviscous Internal friction is neglected 2. The flow is steady The velocity of each point remains constant 3. The fluid is incompressible The density remains constant 4. The flow is irrotational The fluid has no angular momentum about any point Streamlines • The path a particle takes in steady flow is a streamline • The velocity of each particle is tangent to a streamline • A set of streamlines is called a TUBE OF FLOW Equation of Continuity • Consider a fluid moving through a pipe of nonuniform diameter. The particles move along the streamlines in steady flow. • The mass m1 in the small portion of pipe of length Δx1, crossing area A1 in some time Δt, must be exactly the same as the mass m2 in length Δx2, crossing area A2 in the same time Δt. • Why? Because no fluid particles “leak” out of the pipe! The fluid has Conservation of Mass! m2 = mass of fluid in this volume m1 = mass of fluid in this volume Conservation of Mass: m1 = m2 (1) For point 1 & point 2, the definition of density ρ in terms of mass m & volume V gives: m = ρV. For points1 & 2, use V = Ax (1) gives r1A1v1 = r2A2v2 (2) • Fluid is incompressible so, r = constant (2) gives: A1v1 = A2v2 ρAv “mass flow rate” Units: mass per time interval or kg/s (3) – (3) is called the EQUATION OF CONTINUITY FOR FLUIDS – The product of the area and the fluid speed at all points along a pipe is constant for an incompressible fluid Av “volume flow rate” Units: volume per time interval or m3/s • Mass flow rate (mass of fluid passing a point per second) is constant: ρ1A1v1 = ρ2A2v2 Equation of Continuity PHYSICS: Conservation of Mass!! • For an incompressible fluid (ρ1 = ρ2 = ρ) Then Or: A1v1 = A2v2 Av = constant Where cross sectional area A is large, velocity v is small, where A is small, v is large. • Volume flow rate: (V/t) = A(x/t) = Av Implications of Equation of Continuity A1v1 = A2v2 The fluid speed v is low where the pipe is wide (large A) The fluid speed v is high where the pipe is constricted (small A) • The product, Av, is called the volume flow rate or flux. Av = constant says that the volume that enters one end of the pipe in a given time equals the volume leaving the other end in the same time (If no leaks are present!) • PHYSICS: Conservation of Mass!! A1v1 = A2v2 Or Av = constant • Small pipe cross section larger v • Large pipe cross section smaller v Example: Estimate Blood Flow rcap = 4 10-4 cm, raorta = 1.2 cm v1 = 40 cm/s, v2 = 5 10-4 cm/s Number of capillaries N = ? A2 = N(rcap)2, A1 = (raorta)2 A1v1 = A2v2 N = (v1/v2)[(raorta)2/(rcap)2] N 7 109 Example: Heating Duct Speed in duct: v1 = 3 m/s Room volume: V2 = 300 m3 Fills room every t =15 min = 900 s A1 = ? A1v1 = Volume flow rate = (V/t) = V2/t A1 = 0.11 m2