phy2053 lecture 20 - university of florida · phy2053, lecture 4, motion in a plane illustrations...
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PHY2053 Lecture 20 Ch. 9.7 - 9.11: Fluid Flow,
Bernoulli’s Equation
PHY2053, Lecture 4, Motion in a Plane
Overview
• last lecture - fluid statics (fluid not moving)• this lecture - fluid dynamics - allow the fluid to move• main difference - moving fluid can exert force parallel
to the surface (container) • viscous force - opposes fluid flow (analogy - friction)• first let’s consider an ideal fluid (incompressible)• steady flow - velocity at any given point is constant• laminary flow - fluid flows in layers; every small piece
follows the trajectory of the piece in front of it
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PHY2053, Lecture 4, Motion in a Plane
Continuity Equation• incompressible fluid has constant volume (by definition)
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• fluid flow through a pipe with variable cross-section
∆V
∆V
A1 A2
∆x1
∆x2
PHY2053, Lecture 4, Motion in a Plane
Bernoulli’s Equation• can we connect the velocity of fluid flow, pressure,
and height / depth ?
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P2∆x1
∆x2
∆m
∆m
PHY2053, Lecture 4, Motion in a Plane
Bernoulli’s Equation, Part 2
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Demos: Bernoulli Tube
PHY2053, Lecture 4, Motion in a Plane
Illustrations of Bernoulli’s Equation
• lift force generated by airplane wing profile
• simple mechanical vacuum pumps
• calculating speed of fluid flow due to static pressure
• trajectory of a spinning object
• “unexpected” behavior of objects placed into jets
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PHY2053, Lecture 4, Motion in a Plane
H-ITT: Water TowerWater towers are usually installed to help stabilize water pressure in a city’s water system. Assume that the height of the local water tower is 30 m, and that the inside diameter of a standard pipe is 0.5 cm. At full pressure, how long would it take to fill a 4L (1 gal) container with city water?
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A) roughly 8 secondsB) roughly 12 secondsC) roughly 16 secondsD) roughly 20 secondsE) roughly 24 seconds
Demos:Floating Ball
Bernoulli FunnelBernoulli Cart
PHY2053, Lecture 4, Motion in a Plane
Viscous Flow• another type of laminary flow,
interactions (friction) between fluid molecules becomes important
• due to “friction” between layers, layers closer to the stationary surface (container) move slower
• Poiseulle’s Law for viscous flow computes volume flow rate:
∆V
∆t=
π
8∆P/L
ηr4
• ∆P - pressure drop; L - pipe length, r - pipe radius• η - fluid viscosity, measured in Pa sec 10
PHY2053, Lecture 4, Motion in a Plane
Example: Two pipes
• A water tank holds water at a constant gauge pressure of 400 kPa. A pipe of radius 0.5 cm and length 5 m is connected to a riser of radius 0.25 cm and length 1 m. What is the volume flow rate at the end of the riser? The viscosity of water is 10-3 Pa sec.
• What would one expect the laminary flow to be just from Bernoulli’s equation?
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PHY2053, Lecture 4, Motion in a Plane
Drag Forces• drag force opposes object movement in a fluid• two types of drag force: turbulent and viscous drag• turbulent drag produces a counter-force ~ v2
• viscous drag produces a counter-force ~ v
• viscous drag is more appropriate for air resistance• turbulent drag more appropriate for water resistance
or resistance of air at high speeds • Stokes’ Law: viscous drag on a sphere
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FD = 6πηrvη - fluid viscosityr - sphere radiusv - sphere velocity
PHY2053, Lecture 4, Motion in a Plane
Example: Terminal Velocity
• drag force increases with velocity• Earth’s gravitational acceleration
increases velocity• eventually, the drag force matches
the gravitational pull• equilibrium established - velocity
no longer increases• equilibrium velocity is called the
terminal velocity
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mg
mg
mg
v=0
v < vterminal
v = vterminal
Fdrag
Fdrag
Demo:Terminal Velocity
γ =F
Lboundary
PHY2053, Lecture 4, Motion in a Plane
Surface Tension• force due to asymmetry at surface• molecules “like” to be in the state
of minimum potential energy• achieved when surrounded with
molecules of the same kind
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• asymmetry @surface - no molecules outside surface• a fluid will try to minimize the surface area, thus
causing force at the boundaries to pull inwards• surface tension: force per unit surface boundary length
Surface Tension
PHY2053, Lecture 4, Motion in a Plane
Demos
• Bernoulli Tube• Floating Ball• Bernoulli Funnel• Bernoulli Cart• Terminal Velocity• Surface Tension
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