Course Title:

  Methods of Applied Math II: Honors

Textbook:

No textbook is mandatory for this course, but students may find the book ”Viscous Flow” by JR Ockendon & H Ockendon (Cambridge) to be a useful resource.  Further supplementary reading (papers etc) will be provided in class.

Prerequisites:

Math 450H (this course is part of the Math 450 - Math 451 sequence). 

Website:

Week

Lecture

Topic

Assignment

1-2

1-2

Basics of Fluid Dynamics: Navier-Stokes equations governing flow of a simple (Newtonian) liquid.  Simple solutions and their interpretation in the context of real flows.

All homeworks   will be assigned in class

2

3

Nondimensionalization: Review of dimensionality and nondimensionalization.  Dimensionless form of the Navier-Stokes equations.  Large and small Reynolds number limits.

3

4

Asymptotic Methods I: Overview of key asymptotic methods.  Expansions in a small parameter.  Applications to stability.

3-4

5-6

Asymptotic Methods II: Thin film modeling for viscous fluids, illustrated by free surface flows and the Hele-Shaw free boundary problem (FBP).

4-5

7-8

Asymptotic Methods III: Linear stability theory as applied to FBPs, in particular the Hele-Shaw problem. When are flows stable and unstable?  Reading assignments: Saffman/Taylor & Paterson papers.

5-6

9-12

LABS: Bubble injection/suction experiments in Hele-Shaw cell with a simple Newtonian fluid (glycerine).  Comparison with the linear stability theory and with Paterson’s results.

7-8

13-14

Complex Fluids: Introduction to Nematic Liquid Crystals (NLCs).  Basic physics and simplified theoretical description.  Effect of electric field.

9

MID-SEMESTER BREAK

10

16-17

LABS: Building a conducting Hele-Shaw cell.

11-12

18-21

LABS: Suction/injection experiments with NLCs.  Effect of electric field on stability.  Comparison with the Newtonian case.

13

22-23

THEORY: Simple math modeling to explain the experiments with NLCs.  Comparison of theory with experiment.

14-15

24-27

Work on final reports & presentations: In teams of theorists & experimentalists.

16

28

final PRESENTATIONS

IMPORTANT DATES

Day Sections

Night Sections

FIRST DAY OF SEMESTER

                                                   January 22, 2013

Common Exam I

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Common Exam II

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LAST DAY TO WITHDRAW

                                                  March 26, 2013

Common Exam III

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LAST DAY OF CLASSES

                                                   May 7, 2013

FINAL EXAM PERIOD

                                                   May 9-15, 2013

Assignment Weighting

Tentative Grading Scale

Hand-in Hw and Quizzes

30 

A

90 -- 100

Online Homework

none

B+

80 -- 90

Midterm Exams

none

B

70 -- 80

Final Report

40

C+

60 -- 70

Final Presentations

30

C

55 - 60

            D

50 - 55