Math 651: Modeling and Mechanics Fall 2021

Meetings: Randall 4404, Mon/Wed 10-11:20 a.m.

Instructor: Prof.
Silas Alben

Email: alben at
umich.edu

Office: East Hall
4858, tel. 734-647-5518

Office hours: Mon/Tues 3-4 or by appointment.

*Course outline:*

This course will develop mathematical modeling techniques which are useful in many areas of science and engineering, particularly those described by (partial) differential equations.

*Pre-requisities:* An advanced undergrad course in PDE at the level of Math 450 or 454. Some exposure to functions of a complex variable.

*
Assignments:* About 4-5 HW assignments and a short presentation (~10-15 slides) of a mathematical model from your own research area. Presentation could include: Motivating experiments/data, relevant equations, nondimensionalization and scaling of variables, simplifications/approximations of equations and corresponding assumptions made, identification of important parameter ranges/limits, analytical or numerical solutions, comparisons with experiments/data.

The grade will be based on these assignments as well as class attendance and participation.

Dimensional analysis (LS, H)

Scaling (LS, H)

Tensors (S)

Continuum mechanics of solids and fluids (S, LL, F)

Elasticity (S, LL, F)

Potential theory (LS, T)

Organismal locomotion (C)

Fluid-structure interactions (papers)

* indicates freely available through the UM library web site.

*(LS) Lin and Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences

*(S) Segel, Mathematics Applied to Continuum Mechanics

*(LL) Landau and Lifschitz, Elasticity

*(H) Howison, Practical Applied Mathematics

Additional sources:

(AP) Audoly and Pomeau, Elasticity and Geometry

(C) Childress, Mechanics of Swimming and Flying

*(vGM) van Groesen and Molenaar, Continuum Modeling in the Physical Sciences

(M) Middleman, An Introduction to Fluid Dynamics

(T) Tayler, Mathematical Models in Applied Mechanics

(F) Fowler, Mathematical Models in the Applied Sciences

*The
Principle of Similitude* by Lord Rayleigh

Dimensional
Analysis notes by Douglas Hundley

Tadashi Tokieda, Toys in Applied Mathematics

Fluid Dynamics videos