Analytical Methods in Fluid Dynamics
MA 595 - Spring 2026
Department of Mathematical Sciences
Worcester Polytechnic Institute
Prof. B. S. Tilley
MA 595 - Spring 2026
Department of Mathematical Sciences
Worcester Polytechnic Institute
Prof. B. S. Tilley
Instructor: Prof. B.S. Tilley
Department of Mathematical Sciences
Worcester Polytechnic Institute
Tilley Home Page
Office: SH 419
e-mail: tilley@wpi.edu
Phone: (508) 831-6664
Office Hours: T: 3:00-3:50pm ;
F: 3:00-3:50pm
or by appointment
Additional References:
An Introduction to Fluid Dynamics, G.K. Batchelor, Cambridge University Press
Boundary Layer Theory, H.T. Schlichting, (any edition) Spring-Verlag.
Fluid Mechanics, 2nd Edition, L.D. Landau and E.M. Lifshtiz, Butterworth-Heinemann
Hydrodynamic Stability, P.G. Drazin and W.H. Reid, Cambridge University Press (electronic version available)
Linear and Nonlinear Waves, G.B. Whitham, Wiley Interscience
Waves in Fluids, J. Lighthill, Cambridge Mathematical Library
Perturbation Methods in Fluid Mechanics, M. Van Dyke, Parabolic Press
Scientific Papers, G.I. Taylor (ed. By G.K. Batchelor,) Cambridge University Press
Class Expectations: As a graduate-level mathematics course, collaborative learning and active engagement are expected. Collaborative learning meas that students collaborate together to learn the material in the course. Active engagement by students means that students accept the responsibility for their own learning of the material and do not perceive the instructor (professor) as a source of all knowledge.
In order to meet these expectations, the classroom environment must be professional and supportive. Students are expected to treat each other with mutual respect, provide constructive feedback to other students, and to realize that as humans we all need guidance at times.
Homework (40%): There will be multiple homework problem sets given over the semester. The assignments should be done in red ink, scanned in grey-scale (not color) into a PDF file, and uploaded to the Canvas site. You are strongly encouraged to work on the homework problems together, but the written solutions should be done individually. Copying homework, either from a human, a text, or a bot, is cheating, and will be dealt with according to the academic integrity policies of WPI.
Projects (60%): Two projects (one before Spring Break, one after) one which focuses on a paper by G.I. Taylor and the second on its current impact in the field. Each project is evaluated based on student presentations at the end of each project. A convenient rubric for project grading at WPI can be found at this website.
Academic Policies
Grading Policy: Students have two business days after grades are posted to contact the instructor about potential errors in grading any assignment (homework or exams) after receiving their graded work via Canvas. Beyond this time, the grade on that assignment is final.
Generative AI Policy: Students should feel free to find online resources that describe the topics covered here that may present the material in a different style or structure that may be helpful. These resources are designed with the intention to aid students in learning the material. Generative AI tools like Gemini, Chat-GPT, or Copilot, are more general use tools to help the user find potentially relevant information quickly and in a format that appears conversational. Do: experiment with it and use if for brainstorming, and do cite the source/engine when communicating the output, Do not: give an assignment prompt as input and paste the output as your submission. Students who submit AI-generated content as their own work will receive no credit for that work.
Accommodations: Students with approved academic accommodations should plan to submit their accommodation letters through the Office of Accessibility Services Student Portal. Should you have any questions about how accommodations can be implemented in this particular course, please contact me as soon as possible. Students who are not currently registered with the Office of Accessibility Services (OAS) but who would like to find out more information about requesting accommodations, documentation guidelines, and what the accommodated interactive process entails should plan to contact OAS either by email: AccessibilityServices@wpi.edu, by phone (508) 831-4908, or by stopping by the office on the 5th floor of Unity Hall.
Week 1: Preliminary ideas, ideal fluid flow, vorticity equation
Week 2: Elementary viscous flow, circular streamlines, convection/diffusion of vorticity
Week 3: Irrotational free-surface flows, capillary waves, effects of finite depth
Week 4: Sound waves, internal gravity waves, hydraulic jumps and shock waves.
Week 5: Vortex motion, Kelvin circulation theorem, Helmholtz vortex theorem, von Kaarman vorticies, Prandtl-Batchelor theorem
Week 6: Derivation of Navier-Stokes Equations
Week 7: Low Reynolds number past a sphere, Moffatt vortices, reversibility of Stokes flow, swimming at low Reynolds numbers
Week 8: Thin-film flows, Hele-Shaw flows, porous media flow, lubrication theory
Week 9: Green's functions, Green's theorem, Stokeslets
Week 10: Boundary-layer theory
Week 11: Instability: Kelvin-Helmholtz, Rayleigh-Benard convection
Week 12: Rotating flows, instability in shear flows
Week 13: Special Topics
Week 14: Final Presentations