Homework

Difficulty: *

Summary: Your first assignment is to develop your own syllabus and timeline for the course. Every student must come up with a timeline they wish to commit to to getting homework complete.

If you are a principle investigator for a team you will have to submit your own individual plan, your team's roster, and the paper plan for the term. Your paper plan must include 3 milestones to accomplish through out the term.

Difficulty: ***

Summary: You will learn to deal with boundary value problems using numeric solvers, and the issues that come about from non-linearity. Knowledge of how these solvers work make the assignment trivial, but discovery thru frustration will instill some hard lessons you will never forget. The point is you will think more carefully before hopping right over to the computer in the middle of an analysis.

This assignment can be completed in EXCEL, Mathematica, MATLAB, or Python. Part of the lesson is really to dive into online documentation for computer tools that can solve nonlinear differential equations and get used to the process of varying inputs.

Hopefully you come away learning that as long as you have some of math in front of you, a computer can be an investigation tool. I also hope you will learn to appreciate well designed plots as they will aid in troubling shooting issues.

Difficulty: ****

Summary: See how non-dimensionalization actually plays out to take your physical problem statement, remove the dimensions, and allow you to play around in pure number land. You should get the hand of pre-conditioning your problems before just jumping in and trying to write down the solution. Selecting a scale is an art that can only be learned from doing problems, failing, doing it again, then jumping for joy when it works perfectly.

There are also some algebraic exercises to help you get adjusted to using '~' instead of '='.

Difficulty: ***

Summary: Solve the velocity field for some classical in-compressible laminar flow problems. You will start from the Navier Stokes each time and see all the steps work out yourself. These questions are very similar to qualifying exam problems.

I have quite a dinosaur approach and think that actually writing out this "inserting scales into differential equations" process creates some short circuits in your brain that help you skip this step later. However one can program Mathematica to do even the worst of substitutions for you at lightning speed! So the entire assignment could be completed in a Mathematica Notebook for the new age student.

I also would like to comment that we will not at all address this Buckingham Pi stuff you may have heard of. I don't see it as useful mathematics and believe any non-dimensional group has some physical argument hiding behind it, and that is what's important for reasoning fluids.

Difficulty: *****

Summary: Now you will solve for velocity fields with the more advanced methods of series and similarity. You will have a go at formulating problems in terms of stream functions and hopefully see how the they are interlinked to these similarity solutions we've talked about. Not much more to summarize accept this is the just the more difficult version of the last assignment.

Difficulty: *****

Summary: Implement the numerical solution of the a nonlinear partial differential equation. You are asked to actually create a quick finite difference solution for this problem. While today this isn't really necessary it is still very instructive to learn how these solvers are actually providing an answer. The boundary conditions are also nonlinear in this model which makes pen and paper analysis even more out of reach. In fact, these nonlinear boundary condition even give the tools you have been using a hard time. The model equation is for liquid flowing in an interior corner absorbing gas across the meniscus surface. Coding Website

This will be a toss up as for some students that have experience implementing algorithms wont find this difficult, but others that have never written a line of code will struggle. Reach out on the Slack Channel for help, share code, and help each other out. I do not recommend doing this assignment in Mathematica for first time coders. Mathematica is what is a called an functional language and has quite the learning curve when making your own programs.

Difficulty: ***

Summary: Preform stability analysis for some flow problems yourself. You will get an appreciation of making locus plots numerically to investigate stability and find a balance of analytical and numerical methods for these types of questions. Here's a fascinating video of how they made locus plots experimentally back in the day.