Fluid Mechanics and Energy Transport BIEN 301 Lecture 5 Buoyancy, Archimedes Principle, Rigid Body Motion

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Fluid Mechanics and Energy TransportBIEN
301Lecture 5 Buoyancy, Archimedes Principle,
Rigid Body Motion

• Juan M. Lopez, E.I.T.
• Research Consultant
• LeTourneau University
• Adjunct Lecturer
• Louisiana Tech University

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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.

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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.

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Buoyancy and Stability

• Expressing this as an integral

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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.

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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.

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Buoyancy and Stability

• How can we calculate stability?
• As a portion of a body is submerged, we can
  calculate the degree of stability.

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Buoyancy and Stability

• Metacentric Height
• 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 bodys 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.

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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).

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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.

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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?

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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.

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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.

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Rigid Body Motion

• Special Cases
• Uniform Linear Acceleration

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Rigid Body Motion

• Special Cases
• Rigid Body Rotation

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Rigid Body Motion

• Special Cases Rotation
• For continuous fluids, the parabolic shape is the
  common rigid body rotation result.
• This means that within the fluid, the pressure
  gradient forms a family of curves that depend on
  the position in the fluid, the rate of rotation,
  and the density of the fluid.

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Rigid Body Motion

• Illustrating the Pressure Gradient Field in Rigid
  Body Rotation

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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 Fluideven if its not a perfect
  cylinder.
• More complex container shapes require
  observational derivationstoo difficult to
  analytically predict.

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Student Presentation, Etc.

• Well now have our student presentation. After
  that, well discuss
• New Homework Assignment
• Tutorial Lab Plan
• Exam 1

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Tutorial Lab

• I am reserving the room for Wednesdays, 6 830
  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.

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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 dont
  want to have to remind you. This points reminder
  is itafter this, if you dont follow
  instructions, the problem does not get a grade.

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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)

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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. Ill 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.

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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)

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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.

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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.

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Remember

• Ill 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.

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Questions?