Physics Part 1MECHANICS

Hydrodynamics

W. Pezzaglia

Updated: 2014Jan26

Hydrodynamics

A. Continuity Equation

B. Bernoulli’s Law

C. Viscosity

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A. Continuity Equation

1. Mass flow is conserved

2. Flow Rate

3. Continuity Equation

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1. Definition of Fluid Flow

(a) Volume Flux: Measured in units of:• Imperial: gallons per min• SI: cubic meters per second

(b) Relate to “flow speed” v• A=cross section area of pipe• v=speed of flow

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Avt

V

2. Mass Flux

(a) Mass Flux: Measured in units of:• SI: kg/sec

(b) Relate to “volume flux” =density of fluid• A=cross section area of pipe• v=speed of flow

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vAt

V

t

m

3. Continuity Equation

• Based upon conservation of mass, when a fluid (liquid or gas) is forced through a smaller pipe, the speed must increase.

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222111 AvAv

3b. Incompressible Fluids

• Unlike gasses, liquids are difficult to compress. Hence the density is constant. The continuity equation reduces to the velocity inversely proportional to the cross section area:

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2211

21

AvAv

B. Bernoulli Equation

1. Torricelli's law (1643)

2. Bernoulli effect

3. Bernoulli equation (1738)

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1. Torricelli's law (1643)

In brief, the velocity of a fluid exiting at the bottom of a tank of depth “h” is independent of the fluid’s density (i.e. the fluid analogy of Galileo’s law that all bodies fall at same rate independent of mass)

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ghv 2

2. Bernoulli Effect

1738, Daniel Bernoulli notes that pressure decreases when a fluid’s velocity increases.

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21222

1 vvP

Eulerian Flow (1757)

Note on your Lab:

If no friction (no viscosity), the volumetric flow rate out of a pipe of radius “r” with pressure difference P will hence be proportional to the square of the radius:

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2

221

2 rPQ

AvQ

vP

3. Bernoulli Equation

1738, at any point in the fluid, the sum of the pressure, kinetic energy density and potential energy density is a constant

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constant221 ghvP

From this can derive Pascal’s law of depth, Torricelli’s law and Bernoulli effect

3b. Venturi Tube

Can be used to measure flow rate v1 of a liquid (or gas) from the observed pressure difference (inferred from “h”) when cross section area is decreased.

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2

112

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21

222

121

A

Avv

ghPP

vvPP

4. Rayleigh’s drag force

For high velocity, drag force (on object with surface area “A”) increases quadratically with velocity due to inelastic collisions of object with molecules of fluid (density ).

The coefficient of drag “Cd” depends upon geometry.

Hence “power” goes likethe cube of velocity (i.e.windmills most efficient athigh velocities).

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221 AvCF Dd

C. Viscosity

1. Definition

2. Drag Force

3. Reynolds number

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1. Fluid Shear

• Inviscid fluid, no viscosity fluid all flows at same rate

• “Eulerian flow”

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1. Fluid Shear

• With viscosity, there is a flow gradient.

• Near the wall the velocity is zero• Velocity increases linearly as

move away from wall.

• Velocity gradient:ratio of velocity to distance “h” from wall

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h

v

1c. Definition of Viscosity

• Viscosity defined by ratio of force per wall area to the velocity gradient.

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h

vAF

Viscosity Units

• SI Unit: Poiseulle=Pl=Pasec

• Cgs: poise “P”=gm/(cms)=0.1 Pl

• Most common: centipoise: 100 cP=P

• Imperial: reyn=lbsec/inch2=6.894 Pl

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Viscosity Table

• Water: 1 cP

• Corn oil: 50 cP

• SAE 10W 3500 cP

• SAE 20W 4500 cP

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2. Drag Force

• Stoke's Law (1851) for sphere (radius “r”) moving at slow speed “v” in a fluid of viscosity “”:

• If falling, terminal velocity reached (density of object )

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vrFd 6

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2 2grv

2. Viscometer

• Measure viscosity of fluid from terminal velocity (“settling velocity”) of object. When drag plus buoyant force equal gravity we have:

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2

092 gr

v

3. Poiseuille’s Law (1838)

• Pressure required to make fluid flow with (average) velocity “v” in pipe of length “L”, radius “r”

• Expressed in terms of volumetric flow “Q”, we find an r4 dependence!

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2

8

r

vLP

L

rPQ8

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4. Reynold’s Number

• Dimensionless ratio (invented by Stokes 1851, popularized by Reynolds 1883) of the kinetic effects to the frictional effects.

R<1 viscosity dominates, Stoke’s law valid

R=1000 Rayleigh’s drag force dominates

R>2000 unstable

R>3000 turbulence

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vL

R

Notes

• Demo PHET Bernoulli: http://phet.colorado.edu/en/simulation/fluid-pressure-and-flow

• Demo PHET for Gas Law: http://phet.colorado.edu/en/simulation/gas-properties

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