Fluid Mechanics and Energy Transport
BIEN 301
Lecture 5
Buoyancy, Archimedes Principle, Rigid Body Motion
Juan M. Lopez, E.I.T.
Research Consultant
LeTourneau University
Adjunct Lecturer
Louisiana Tech University
Note on Accuracy

I hold YOU accountable, so you hold ME
accountable.
 The deal: if you catch 3 or more math or theory
errors in any given lecture slide set, the entire
class gets 1 point on the next homework
assignment.
 I am going to apply this to the last two
assignments.
12/14/2006
BIEN 301 – Winter 2006-2007
Buoyancy and Stability

Archimedes (White 2.8)


A body immersed in a fluid experiences a vertical
buoyant force equal to the weight of the fluid it
displaces.
A floating body displaces its own weight in the fluid in
which it floats.
FB
 FTopSurface  FBottomSurface
 ( Weight of fluid above top surface) (Weight of the fluid above the bottom surface)
 Weight of fluid equivalent to body
12/14/2006
BIEN 301 – Winter 2006-2007
Buoyancy and Stability

FB
Expressing this as an integral:

Body
 p2  p1 dAH
   z2  z1 dAH
  body volume
12/14/2006
BIEN 301 – Winter 2006-2007
Buoyancy and Stability

What about when multiple fluids are involved?
• We can simply extend this to a more generalized
form that deals with the individual fluids and the
portion of the body inside of that fluid.
FB LF  g displaced volumes
Where
FB LF  Buoyant force due to layered fluids (LF).
12/14/2006
BIEN 301 – Winter 2006-2007
Buoyancy and Stability

What if my body is not aligned in the fluid?
• Except in special cases, an unbalanced body will seek and
obtain a stable, static position.
12/14/2006
BIEN 301 – Winter 2006-2007
Buoyancy and Stability

How can we calculate stability?
• As a portion of a body is submerged, we can calculate the
degree of stability.
12/14/2006
BIEN 301 – Winter 2006-2007
Buoyancy and Stability
x
tan 

 MB

Metacentric Height

IO
vsubmerged

 MG  GB

Which gives us :
MG 
IO
vsubmerged
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 GB


The metacentric height (MB) is
calculated from the submerged
center of buoyancy projected onto
the original vertical axis.
It can be expressed as a function of
the moment of inertia and volume as
well.
The sign of the metacentric height
relative to the body’s center of gravity
and the metacentric height (MG) as
well as the height between the center
of gravity and the center of buoyancy
(GB) tells you much about the
stability.
If MG is positive, the body is stable
for SMALL displacements.
If GB is negative, the body is always
stable.
BIEN 301 – Winter 2006-2007
Buoyancy and Stability

Example (White Example 2.10)

A barge has a uniform rectangular cross section of
width 2L and vertical draft height H, as in the figure
below. Determine (a) the metacentric heigth for a
small tilt angle and (b) the range of ratio L/H for
which the barge is statically stable if G is exactly at
the waterline (as shown).
12/14/2006
BIEN 301 – Winter 2006-2007
Buoyancy and Stability

Example (White Example 2.10)

The moment of inertia
gives us a measure of
the effort required to
rotate the item about a
particular axis. For this
rectangular cross
section, I = b(2L)3/12.
MG

IO
vsubmerged
L3
12  H

2 LbH 2
L2 H


3H 2
8b
From earlier, stability exists when MG is positive.
Therefore, stability exists when :
L2 H

3H
2
or
2 L  2.45 H
12/14/2006
 GB
BIEN 301 – Winter 2006-2007
Rigid Body Motion

Definition (White 2.9)


All particles are in combined translation and rotation,
and there is no relative motion between the particles.
Recalling our last lecture, what terms drop out?

a

p
12/14/2006
2


 p   g    V


   g   a
 
  g  a 
BIEN 301 – Winter 2006-2007
Rigid Body Motion

Example



A block of fluid free-falling in air (Patm = 101 kPa).
Assume negligible drag, no container.
Find the pressure at the bottom of the fluid element.

 
p
  g  a 


but a is actually g (free fall), so :

 
p
  g  g   0
Therefore
Pbottom  101kPa
12/14/2006
BIEN 301 – Winter 2006-2007
Rigid Body Motion



Therefore, the pressure gradient is aligned with the
vector (g-a).
This means the lines of constant pressure are aligned
perpendicular to the gradient.
So we need some general forms for expressing body
motion and acceleration.

V
  
 V0    r
and


  d 
dV0 
a
 r 
r
dt
dt

12/14/2006

BIEN 301 – Winter 2006-2007
Rigid Body Motion

Special Cases
 Uniform Linear Acceleration
 a 
tan 1  x 
 g  az 
So, why no viscous terms?
Why no density te rms?
For the problem shown :


 a 
 tan 1  x 
 g  az 
7. 0 m / s 2

1 
 tan 

2  0.0m / s 2
9
.
81
m
/
s


 35.5 deg
12/14/2006
BIEN 301 – Winter 2006-2007
Rigid Body Motion

Special Cases
 Rigid Body Rotation
12/14/2006
BIEN 301 – Winter 2006-2007
Rigid Body Motion

Special Cases – Rotation


1 2 2
 p0  z  r 
2
For continuous fluids, the
p
parabolic shape is the
common rigid body rotation
where
result.
This means that within the
p0  p r 2  2
z


fluid, the pressure gradient

2g
forms a family of curves that
depend on the position in the
2

a

br
fluid, the rate of rotation, and
the density of the fluid.
Why can we assume this reduces to two constants
and a function of the radial position?
12/14/2006
BIEN 301 – Winter 2006-2007
Rigid Body Motion

Illustrating the Pressure Gradient Field in Rigid
Body Rotation:
12/14/2006
BIEN 301 – Winter 2006-2007
Pressure

Special Cases – Rotation


Note: the parabola continues
to be applicable, even if the
fluid configuration does not
appear to lend itself to the
analysis.
Why?
• Continuous Fluid…even if
it’s not a perfect cylinder.

More complex container
shapes require observational
derivations…too difficult to
analytically predict.
12/14/2006
BIEN 301 – Winter 2006-2007
Student Presentation, Etc.
 We’ll
now have our student presentation.
After that, we’ll discuss:



New Homework Assignment
Tutorial Lab Plan
Exam 1
12/14/2006
BIEN 301 – Winter 2006-2007
Tutorial Lab
I am reserving the room for Wednesdays, 6 –
8:30 pm. We should only go to 8 pm most days.
 The classroom is as of yet undefined, but it will
either be 305 (from last night), or 327 (this
room). I will get this confirmation today.
 I will be hosting another session tonight for 2
reasons: people who may not have been able to
make it to the first, people who may be wanting
some more help before Exam 1.

12/14/2006
BIEN 301 – Winter 2006-2007
Homework




Homework 5 has been posted on blackboard.
Most of HW2 and HW3 have been graded. I should have those
available tomorrow (Friday). You may pick up your homework in the
Biomedical Engineering office in BH, from my mailbox. (Check with
Arlene)
If you turn in your homework by Sunday Noon, I will do everything I
can to make it available by Monday evening. Sunday you can turn it
in at my office.
On the items graded thus far: no severe concerns for the homework
health of the class other than 1 specific thing: FOLLOWING
INSTRUCTIONS. I don’t want to have to remind you. This points
reminder is it…after this, if you don’t follow instructions, the problem
does not get a grade.
12/14/2006
BIEN 301 – Winter 2006-2007
Exam 1

Exam 1 - Reminder


Reviews have been posted
In-class exam materials allowed:
•
•
•
•
•
1 Calculator
Writing Implements
Chapter Reviews from course documents, NO NOTES.
Lecture slides, 6 to a page, NO NOTES.
Fluid Mechanics, fifth edition, by White (class textbook), NO
NOTES.
• If I find handwritten notes in these sections, your materials will be
removed. Any additional cheating will result in failing the test, and
maybe the course.

Take-Home Option (closed now)
12/14/2006
BIEN 301 – Winter 2006-2007
Exam 1
 Format:




1st part, Closed Book, Notes, Calculator (basically,
you may use your pencil, and the paper I hand you)
2nd part, you may use the reference materials listed.
I’ll have scratch paper available if you need it.
Cellphones strictly forbidden. Turn them off and put
them away.
Total time expected: 1st part, 20 minutes; 2nd part, 70
minutes. Total time expected: 1.5 hrs.
12/14/2006
BIEN 301 – Winter 2006-2007
Exam 1
 Topics



on exam:
Principal topics as listed in the title pages of
the various lectures.
Principal topics covered by the homework
assignments
Principal topics covered by the in-class
presentations (1, 2, and 3)
12/14/2006
BIEN 301 – Winter 2006-2007
Exam 1
 Part

1-
Will be general knowledge of fluids:
• Definitions, Descriptions, Short Answers, Multiple
Choice, True False. The basic mathematics and
operators from our introduction to fluid mechanics.
• If any calculations are present, they will be simple.
• 1/3 of extra credit will be available on this section.
12/14/2006
BIEN 301 – Winter 2006-2007
Exam 1

Part 2
Will cover specific problems:
• 8 Questions divided into 4 sections (2 questions apiece).
• Pick 1 question from each, and solve it.
• The problem solving procedures expected in your homework
WILL BE EXPECTED HERE.
• SHOW YOUR WORK, it lends to better partial credit.
• 2/3 of Extra Credit available in this section. Confidence
Checks will be available (5 points per question), You are free
to ignore the confidence check.
12/14/2006
BIEN 301 – Winter 2006-2007
Remember
 I’ll
be available tonight for another tutoring
session (special case because of exam).
 Show of hands on whether I need to show
up.
 Review your homework, the study guides,
and the student presentations. This should
make you well prepared for the types of
questions you will have to encounter.
12/14/2006
BIEN 301 – Winter 2006-2007
Questions?
12/14/2006
BIEN 301 – Winter 2006-2007