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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