Winter 2002

MEAM 626

Singular-Perturbation Methods in Fluid Mechanics

 

Instructor:    W. Schultz (schultz@umich.edu)

Office:        2027 Auto Lab (936-0351)

References:    Bender and Orszag (1978), Advanced Mathematical Methods for Scientists & Engineers, McGraw-Hill.

               Kervorkian & Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, NY.

         Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford.

Prerequisite:  One graduate fluids course or consent of instructor

Class Hours:   11:30-1:00 MW in 104 EWRE

 

Class is small and therefore the material covered will be adapted to meet the needs of the students.  Some emphasis will be placed on using asymptotics to aid computations and the use of symbolic algebra (MAPLE or equivalent). A small project and homework will be required.

 

Tentative Outline

 

       Introduction to asymptotic methods

             Convergent and asymptotics sequences

             Gauge Functions

             Properties of asymptotic sequences

             Regular and singular perturbations

             Uses in Fluid Mechanics

             Symbolic Computer Algebra

       Unsteady Problems

             Poincare-Linstedt Method

             Multi-time methods

       Method of Matched Asymptotic Methods

             Oseen Flow

             Prandtl Boundary Layers

       Derivation of Lower-Dimensional Theories

             Lubrication Theory (Boundary Layer Theory)

             Fin Equation

             Slender Body Theory

       Asymptotic Evaluation of Integrals

             Integration by Parts

             Saddle Point Methods

       WKBJ Method

             Geometric Theory

             Evaluation of Orr-Sommerfeld Equation

 

 

                  ME626 Reserves, UMMU

 

Abramowitz & Stegun, Handbook of Mathematical Functions, Dover.

Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw Hill

Hinch, E.J. Perturbation Methods, Cambridge Texts in Applied Math.

Kervorkian & Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, NY.

Lamb, Hydrodynamics, Dover Publications.

Landau & Lifshitz, Fluid Mechanics, Pergamon Press.

Milne-Thomson, Theoretical Hydrodynamics, Macmillan.

Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press.

Warsi, Fluid Dynamics: Theoretical and Computational Approaches, CRC.

Yih, Fluid Mechanics, West River Press, Ann Arbor Michigan.

 

 

Maple commands (as described in LaTex output) for MMAE

 

 

      Hwk #1 (Due Feb 15)

 

  1. Find the roots of u3 – u2 + e = 0

 

  1. Write a computer program (subroutine) that uses expansions for small and large argument that will find erf(x) to 5 digits for 0 < x < ∞. How would the program change for complex argument?

 

  1. Solve y'' = ey'+e/(1+x),   y(0) = y’(0) = 0 using e=1
  2. The flow between nearly concentric, slowly rotating cylinders can be described by the biharmonic equation. Let the inner cylinder radius and rotation rate equal one, the outer cylinder radius = R and the offset between the cylinder centers = e. Discuss the differences between this problem and that when R is only slightly greater than one.

 

Hwk #2 (Due late March)

 

  1. Solve ey” + 2y’ + ey = 0, y(0)=y(1)=0  using MMAE

 

  1. Starting from where we ended in class with multiple scale methods, show which solutions of the Duffing equation are stable for b > 0 and b < 0.

 

  1. Use the P-L method for the quadratic nonlinear problem:

x” + p2x - ebx2 = 0

 

  1. Find the limitations for the fin equation on slope and Bi = hl/k, L= A/p from e2T2 << T0 It is possible to have Bi≠0 and a sloping fin (Y=ax+b) simultaneously, but for simplicity you can do them one at a time.

 

  1. Write up to 5 pages on a research problem.  Mark all parts that are new and have been done to satisfy this requirement.