Advanced Transport

Welcome to the study of advanced transport phenomena in Fall 2026!

The principles of mass, momentum, and energy transport are relevant across scales—from sub-cellular to atmospheric—and across disciplines. This one-semester graduate course provides students with the tools to understand and describe such a diverse set of phenomena. Relevant ideas from vector and tensor calculus are reviewed, as partial differential equations are the natural language of transport phenomena. General concepts from continuum mechanics are introduced, and the balance laws of mass, momentum, and energy are rigorously derived from first principles. Applications of the balance laws are studied, with the goal of developing a physical understanding of various phenomena. The concepts of non-dimensionalization and scaling are emphasized throughout.

Office Hours

• Amaresh Sahu: Tuesday, 1–3 pm in CPE 3.462
• David Pinegar: Monday, 1–2 pm in BUR 128 and Wednesday, 10–11 am in CPE 2.222

Problem Sets

• [01]: Due 03 September
• [02]: Due 10 September
• [03]: Due 17 September
• [04]: Due 24 September
• [05]: Due 01 October
• [06]: Due 22 October
• [07]: Due 29 October

Exams

• [Midterm 1]: 8 October (in lecture)
• Midterm 2: 12 November (6–8 pm)
• Final: 10 December @ 13:00

Lectures

01: Review of mathematics, prerequisite material

We expect you are familiar with standard mathematical concepts from an undergraduate degree in chemical engineering. A partial list of topics is presented below:

• vector and multivariable calculus
  • dot product, cross product, magnitude of a vector, gradient of a scalar function, divergence and curl of a vector field, Taylor series in multiple variables and dimensions, cylindrical and spherical coordinate systems
• linear algebra
  • eigenvalues and eigenvectors, symmetric and antisymmetric matrices, matrices as linear transformations
• ordinary and partial differential equations

Resources

The following may be helpful in reviewing material:

• David Leighton Jr.’s The Mathematical Language of Transport [link]
• my linear algebra notes for undergraduate numerical methods [link]
• Grant Sanderson’s series on [calculus], [linear algebra], and [differential equations]

02: Intro, motivation, and dimensional analysis (simple treatment)

• examples:
  • dolphins and bubble rings [link]
  • forming marbling art patterns [link]
  • instabilities and pattern formation in clouds [link]
  • drug delivery [link]
  • medication [link]
• reading (intro):
  • [has] §1
  • [pp] §1
  • [lgl] Ch.1: §A,§B
• reading (dimensional analysis):
  • [has] §§3.1, 3.2, 3.4, 3.7, 3.10
  • [as] Ch.2: §3

03: Dimensional analysis (formal treatment)

• examples:
  • nucleolar surface fluctuations: data collapse in Fig. 4 [link] [news]
  • Ilya Sutskever on scaling in AI [link]
  • vortices in flowing river [link]
• required reading:
  • [as] Ch.2
• optional reading:
  • [has] §3

04–06: Mathematical preliminaries

• examples:
  • flow visualization (see vector velocities) [link]
  • complex transport phenomena [link]
  • critical opalescence: a different scaling in physics [link]
  • Richard Feynman on mathematics vs physics [link]
    • bonus: the pleasure of finding things out [short] [long]
• reading:
  • [as] Ch.3
  • [has] §2
  • [kkm] §1
  • [pp] §2

07–10: Kinematics

• examples:
  • deformations of continuous media [link]
  • flow visualization around objects [link] [link]
  • vesicle in flow [tumbling] [pearling]
  • flow through a constriction [link]
  • reversible flow [link]
• required reading:
  • [as] Ch.4
• optional reading:
  • [pp] §3 (detailed)
  • [kkm] §2 (detailed)
  • [has] §§4.5,4.6

11: Preparation for the balance laws

• required reading:
  • [as] Ch.5
• optional reading:
  • [has] §§4.1,4.2 (one dimension), §§4.7,4.8
  • [kkm] §§3.2,3.3 (note Theorem 3.2.3)
  • [pp] §§4.1,4.3,4.4
  • [lgl] Ch.2§B

12: Balance of mass (continuity equation)

• examples:
  • blood flow in the body [link]
  • arteriovenous fistula [link]
  • bacterial networks [link]
• reading:
  • [as] Ch.6
  • [has] §§4.1–4.8
  • [kkm] §3.3
  • [pp] §4.4

13–15: Balance of linear and angular momentum

• examples:
  • turbulence (William Irvine) [article] [manuscript]
  • colloidal chiral fluids (William Irvine) [link]
  • laminar vs turbulent flow [link]
  • angular momentum of continua (Dahler and Scriven) [link]
• reading:
  • [as] Ch.8
  • [has] §4.9
  • [kkm] §3.4
  • [lgl] Ch.2§C
  • tractions and stress: [link]
  • [as-thesis] Ch.III

16: Isothermal Newtonian fluids: Navier–Stokes

• examples:
  • hydrodynamic starvation in larval fishes: video SM02 [link]
  • eating marshmallows underwater [video]
• reading:
  • [kkm] §4
  • [has] §4.12, §5.5.2
  • [lgl] Ch.2§F,§H

17: Start-up of Couette flow

• examples:
  • experimental realization [link]
• reading:
  • [kkm] §4.3 (steady state), §4.6 (start-up)
  • [as] Ch.9

18–19: Boundary conditions, Gravity-driven flow down an incline, the Reynolds number

• reading:
  • [has] §§1.6,4.19,5.2.5
  • [kkm] §§4.4,4.7
  • [lgl] Ch.2§L,§M (the latter is advanced but quite interesting)

20–21: Balance of energy and the heat equation

• reading:
  • [has] §4.13
  • [kkm] §5.1
  • [lgl] Ch.2§D,§E