Transcript Viscous Flow Around Metal Spheres

Viscous Flow Around Metal
Spheres
Terminal Velocity and Acceleration
Profile as a Function of Radius
Introduction
• Characterize viscous
•
•
flow around a sphere
Find dependence of
terminal velocity on
radius
Find dependence of
acceleration profile on
radius
Introduction
• Classical problem in fluid dynamics
• Special flow regime called Stoke’s Flow
– Viscous forces dominate the flow
• Often used to determine fluid viscosity
Our Experimental Setup
• Graduated cylinder
•
•
filled with glycerin
Different sizes of
metal balls
Digital Camcorder
Experimental Setup
• 6 different sized steel balls
Experimental Setup
Experimental Setup
A video data file of the 6.35mm balls
Theoretical Models
• Terminal Velocity
D imensional Analy sis of a Metal Ball Falling in a Vis c ous Fluid
Adding all t he f orc es giv es us an equat ion f or t he ac celerat ion:
x (t)
Fg
Fb
Fd
=
4  3 
r 
3
wit h the initial c ondit ions :
Fd
( 1)
x( 0 ) = 0
x ( 0)
The goal is to f ind the t erminal v elocit y
and how it s cales wit h f luid v isc osit y
and radius of the ball.
x ( t ) c onst ant implies t hatx
x
 o g
l im x ( t )
t 
0 and
Fd
4  3 
r 
3
 o g
( 2)
The goal here is t o f ind the relationship between
F d and x .
M
 = v is cosity of f luid[] =
L T
r = radius of ball
F d = drag f orc e
[ r] =
L
M L
[ Fd ] =
T
x
= v eloc ity
2
L
[x ] =
T
Suppos e F d
M L
F d x  r 
T
a
c
L  b M
L
a
c c
T
L T
2
y ields a = b = c = 1
or F d
Plugging (2) int o (3)
 x r
x
4  3 
r 
3
Kr
2
 x r
 o g
where K
4 

3
g
o 
 
( 3)
Anticipated terminal velocity v. radius.
Velocity versus square of radius
900
800
Termnial Velocity (m/s)
700
600
500
400
300
200
100
0
0
5
10
15
20
25
30
2
Radius Squared (mm )
35
40
45
Theoretical Models
• Acceleration Profile
Theoretical Models
Theoretical Models
Theoretical Models
Theoretical Models
Theoretical Models
• Newton’s Law gives
• Drag Force = Weight – Buoyancy
• Stokes’ Law => Drag Force = 6
6 aU
aU
• Velocity = (M - 4/3  a3 fluid) g /(6   a)
• Velocity = 2 a2 g (
(sphere - fluid)/ 9
9
• Velocity = (Size, Material, Fluid Properties)
Theoretical Models
Navier-Stokes Analysis
Momentum & Continuity Eqns
Theoretical Models
Navier Stokes Analysis
Non-dimensionalizing the Eqns
Theoretical Models
For Stokes Flow Re<<1
So the Equations simplify to
Theoretical Models
Navier Stokes Analysis
Theoretical Models
Analytical Soln for the Sphere
Theoretical Models
The Analytical Expression for
Drag Force F matches
Dimensional Analysis
Stoke’s Law
Results & Analysis
• Used video from camcorder to find
experimental speeds
• Calculated theoretical speeds using model
• Compared:
– Experimental
– Theoretical
– Predicted Scaling Rate from Dimensional
Analysis (V ~ r^2)
Results & Analysis
Results & Analysis
• Error sources
– Viscosity is a function of temperature!
Viscosity, Pa*s
Viscosity of Glycerin vs. T
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
20
25
30
Temp, C
35
40
Results & Analysis
• Error Sources (cont.)
– Bubbles effectively reduce viscosity when
they’re in a ball’s path
– Bubbles effectively increase buoyancy when
they’re piggybacking on a ball
– Sidewall effects (disruption of flow lines)
– Instrument resolution (time and distance)
Results & Analysis
• Velocity Profile Analysis
– Terminal velocity reached for smallest ball in
0.007 seconds, faster than camera.
– Reached for largest ball in 0.303 second, but
times and distances involved were still too
fast:
Results & Analysis
Results & Analysis
Conclusion
• Experimental terminal velocity matches with
dimensional analysis and theoretical model
– significant errors due to temperature and other
effects
• Acceleration profile cannot be measured with
current equipment
– resolution is too low relative to phenomena to be
observed