Viscous Flow Around Metal Spheres Terminal Velocity and Acceleration Profile as a Function of...

Viscous Flow Around Metal Viscous Flow Around Metal SpheresSpheres

Terminal Velocity and Terminal Velocity and Acceleration Profile as a Acceleration Profile as a

Function of RadiusFunction of Radius

IntroductionIntroduction

• Characterize Characterize viscous flow viscous flow around a spherearound a sphere

• Find dependence Find dependence of terminal velocity of terminal velocity on radiuson radius

• Find dependence Find dependence of acceleration of acceleration profile on radiusprofile on radius

IntroductionIntroduction

• Classical problem in fluid dynamicsClassical problem in fluid dynamics

• Special flow regime called Stoke’s Special flow regime called Stoke’s FlowFlow– Viscous forces dominate the flowViscous forces dominate the flow

• Often used to determine fluid Often used to determine fluid viscosityviscosity

Our Experimental SetupOur Experimental Setup

• Graduated cylinder Graduated cylinder filled with glycerinfilled with glycerin

• Different sizes of Different sizes of metal ballsmetal balls

• Digital CamcorderDigital Camcorder

Experimental SetupExperimental Setup

• 6 different sized steel balls6 different sized steel balls


A video data file of the 6.35mm A video data file of the 6.35mm ballsballs

Theoretical ModelsTheoretical Models

• Terminal VelocityTerminal Velocity

Dimensional Analysis of a Metal Ball Falling in a Viscous Fluid

Adding all the forces gives us an equation for the acceleration:

x t( ) F g F b F dF gF g = 4

3r3 o

g F d 1( )

with the initial conditions: x 0( ) x 0( )xx = 0

The goal is to find the terminal velocity

tx t( )limand how it scales with fluid viscosity

and radius of the ball.

x t( ) constant implies that x 0xx and F d4

3r3 o

grr 2( )

The goal here is to find the relationship between F d and x .

= viscosity of fluid [] = M

L T

r = radius of ball [r] = L

F d = drag force [ F d ] = M L

T2

x = velocity [ x ] = L

T

Suppose F d F d x r M L

T2

La

TaLb Mc

Lc Tc

aayields a = b = c = 1

or F d x r 3( )

Plugging (2) into (3) x r 4

3r3 o

grr

x K r2rr where K4

3 o

g

Anticipated terminal velocity v. radius.Anticipated terminal velocity v. radius.

0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

600

700

800

900

Radius Squared (mm2)

Term

nial

Vel

ocity

(

m/s

)Velocity versus square of radius


• Acceleration ProfileAcceleration Profile


••Newton’s Law givesNewton’s Law gives

••Drag Force = Weight – BuoyancyDrag Force = Weight – Buoyancy

••Stokes’ Law => Drag Force = 6Stokes’ Law => Drag Force = 6aUaU

••Velocity = Velocity = (M - 4/ 3(M - 4/ 3 a a33 fluidfluid) g) g /(6 /(6 a) a)

••Velocity = Velocity = 2 a2 a22 g ( g (spheresphere - - fluidfluid)/ 9)/ 9••Velocity = Velocity = (Size, Material, Fluid Properties)(Size, Material, Fluid Properties)

Theoretical ModelsTheoretical ModelsNavierNavier-Stokes Analysis-Stokes AnalysisMomentum & Continuity Momentum & Continuity EqnsEqns

Theoretical ModelsTheoretical ModelsNavier Stokes AnalysisNavier Stokes AnalysisNon-dimensionalizing the Non-dimensionalizing the EqnsEqns

Theoretical ModelsTheoretical ModelsFor Stokes Flow Re<<1For Stokes Flow Re<<1So the Equations simplify toSo the Equations simplify to

Theoretical ModelsTheoretical ModelsNavier Stokes AnalysisNavier Stokes Analysis

Theoretical ModelsTheoretical ModelsAnalytical Soln for the Analytical Soln for the SphereSphere

Theoretical ModelsTheoretical ModelsThe Analytical Expression The Analytical Expression for Drag Force F matches for Drag Force F matches Dimensional AnalysisDimensional Analysis

Stoke’s LawStoke’s Law

Results & AnalysisResults & Analysis

• Used video from camcorder to find Used video from camcorder to find experimental speedsexperimental speeds

• Calculated theoretical speeds using Calculated theoretical speeds using modelmodel

• Compared:Compared:– ExperimentalExperimental– TheoreticalTheoretical– Predicted Scaling Rate from Dimensional Predicted Scaling Rate from Dimensional

Analysis (V ~ r^2)Analysis (V ~ r^2)

Results & AnalysisResults & Analysis• Error sourcesError sources

– Viscosity is a function of temperature!Viscosity is a function of temperature!

Viscosity of Glycerin vs. T

0

0.2

0.4

0.60.8

1

1.2

1.4

1.6

20 25 30 35 40

Temp, C

Vis

cosi

ty,

Pa*

s


• Error Sources (cont.)Error Sources (cont.)– Bubbles effectively reduce viscosity Bubbles effectively reduce viscosity

when they’re in a ball’s pathwhen they’re in a ball’s path– Bubbles effectively increase buoyancy Bubbles effectively increase buoyancy

when they’re piggybacking on a ballwhen they’re piggybacking on a ball– Sidewall effects (disruption of flow lines)Sidewall effects (disruption of flow lines)– Instrument resolution (time and Instrument resolution (time and

distance)distance)


• Velocity Profile AnalysisVelocity Profile Analysis– Terminal velocity reached for smallest Terminal velocity reached for smallest

ball in 0.007 seconds, faster than ball in 0.007 seconds, faster than camera.camera.

– Reached for largest ball in 0.303 second, Reached for largest ball in 0.303 second, but times and distances involved were but times and distances involved were still too fast:still too fast:

ConclusionConclusion

• Experimental terminal velocity matches Experimental terminal velocity matches with dimensional analysis and with dimensional analysis and theoretical modeltheoretical model– significant errors due to temperature and significant errors due to temperature and

other effectsother effects

• Acceleration profile cannot be measured Acceleration profile cannot be measured with current equipmentwith current equipment– resolution is too low relative to phenomena resolution is too low relative to phenomena

to be observedto be observed

Viscous Flow Around Metal Spheres Terminal Velocity and Acceleration Profile as a Function of...

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