# MATH 6750-001/6880-002

## Fluid Dynamics 1

with Aaron Fogleson

- MWF 10:45-11:35am in JWB 208
- Office hours: Wednesday 12-1pm, Friday 1-2pm, or by appointment
- Flyer
- Syllabus

**Schedule**

- Week 1: Mo: no class, solar eclipse, We-Fr: Tensor algebra and calculus
- Week 2: Mo-Fr: Basic definitions, flow, Conservation of mass, Conservation of linear momentum
- Week 3: Mo: no class, Labor Day, We-Fr: Conservation of angular momentum, Conservation of energy, Constitutive equations
- Week 4: Mo-Fr: Constitutive laws, Stress tensor, Newtonian fluid, Boundary conditions, Navier-Stokes equations
- Week 5: Mo-Fr: Unidirectional flow, Couette flow, Poiseuille flow, Rayleigh's problem, Non-Newtonian fluids
- Week 6: Mo-Fr: Power-law fluids, Bingham fluids, Maxwell model, Kelvin-Voigt model
- Week 7: Mo-Fr: Linear viscoelasity, frame invariance, mechanical analogy, Oldroyd-B
- Week 8: Mo-Fr: Numerical methods for incompressible Navier-Stokes for a Newtonian fluid
- Week 9: Mo: Numerical methods, We-Fr: Continuum theory of polymeric fluids at equilibrium, Rigid rod models
- Week 10: Mo, Fr: Continuum theory of polymeric fluids at equilibrium, Gaussian chains, Dumbbell models under flow (distribution function), We: class canceled
- Week 11: Mo-We: Dumbbell models, stress contribution, Oldroyd-B, Fr: Continuum model of platlets in flow
- Week 12: Mo-We: Continuum model of platelets in flow
- Week 13: Mo-We: Two phase fluids model, Fr: no class (Thanksgiving)
- Week 14: Mo-We: Two phase fluids models, Fr: Perturbation theory
- Week 15: Mo-We: Boundary layer past a flat plate

**Lecture Notes**

- Tensor algebra and calculus for fluid dynamics: Notes
- Frames, Kinematics, Conservation of mass, Reynolds' Transport Theorem, Conservation of linear momentum, Conservation of angular momentum, Conservation of energy, Constitutive laws, Stress tensor, Newtonian fluid, Navier-Stoeks, Boundary conditions: Notes
- Unidirectional flow, flow between parallel plates, Raygleigh's problem, Start-up flow between plates, Flow between cylinders: Notes
- Generalized Newtonian fluid, Power Law fluid in pipe flow, Bingham fluid in a Couetter device, General strategy: Notes
- Linear Viscoelasticity, Frame Invariance, Mechanical Models, Maxwell and Kevin-Voigt models: Notes
- Numerical methods for Navier-Stokes: Notes
- Continuum platelets model in flow, Two phase fluid model: Notes
- Perturbation theory, Laminar boundary layer past a flat plate: Notes

**Further references**

- For a discussion of linear viscoelasticity, see Chapters 1 and 3 in
*Constitutive Equations for Polymer Melts and Solutions*by Larson. - For a general discussion of complex fluids (Generalized Newtonian, Mechanical Models), see Chapter 1 (PDF) in
*Complex Fluids in Biological Systems*by Spagnolie - Papers on numerical methods for Newtonian flow: Bell, Colella, Glaz (JCP 1989) and Brown, Minion, Cortez (JCP 2000)
- For a discussion of continuum model of polymers, see Chapters 11 and 13 in
*Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory*by Bird, Armstrong, Bird and Hassager - Papers on platelets model and two phase fluid model: Fogelson (CM 1993), Fogelson (SIAP 1992), Fogelson, Guy (MMB 2004), Du, Fogelson (MMB 2017)
- For a discussion on regular and singular perturbations for differential equations, see Chapter 9 in
*Mathematics Applied to Deterministic Problems in the Natural Sciences*by Lin and Segel

**Homework**

- Homework 1: Due September 8, 2017 at the beginning of class. Problem 6 has been postponed, the new homework only has 5 problems.
- Homework 2: Due October 6, 2017 at the beginning of class. You might find reading Chapter 2 in Acheson useful for solving the last two problems.
- Homework 3: Due December 15, 2017 at Noon in Aaron's mailbox. In problems 1 and 3, the components of the velocity vector are v
_{x}, v_{y}, v_{z}. There was a typo in the original problem 3, there should be no convective derivative on the RHS. The hw has been corrected.