# TAM 532: Viscous Flow

## Class Description:

Dynamics of flow in which viscosity is significant or dominant, and the development and use of theoretical and numerical tools for practitioners of modern fluid mechanics; physics of viscous layers that arise in both high- and low-Reynolds-number flows; dimensional analysis, exact solutions to the Navier-Stokes equations; jets and wakes; microhydrodynamics; fluid stability; and an introduction to turbulence. Prerequisite: TAM 435 or equivalent; MATH 380; MATH 385, MATH 386, or MATH 441. 4 graduate hours.

Textbooks: No textbook required.

Topics:

Governing equations for viscous, heat-conducting fluids (6 hr)
Continuity, momentum, and energy equations; elementary thermodynamics, simple constitutive theory; boundary conditions; dimensionless form of governing equations; identification of group parameters

Exact solutions of the Navier-Stokes equations (8 hr)

Boundary-layer theory (10 hr)
Inviscid limit and the singular character of the large-Reynolds-number limit; mathematical example of a boundary layer; matched asymptotic expansions; two-dimensional steady flow past a body as a singular perturbation problem; boundary-layer equations, flat-plate boundary layers, displacement thickness, pressure gradients and flow separation; difficulties with boundary-layer theory; approximate techniques (von Kn integrals, Thwaites's method; comparison with numerical solutions); free boundary layers

Low-Reynolds-number flow (6 hr)
The Stokes equations, general methods of solution; axisymmetric flow past a circular cylinder; Whitehead|rsquo;s and Stokes|rsquo;s paradox|mdash;the Oseen equation; lubrication theory

Stability of fluid motion (8 hr)
Definition of stability; linear stability of parallel shear flow, the Orr|ndash;Sommerfeld equation; Squire|rsquo;s theorem; stability of Poiseuille flow

Introduction to turbulent flow (10 hr)
Transition; Reynolds averaging; statistical hydromechanics; turbulent shear flows; turbulent boundary layers; Kolmogorov's theory; coherent structures

Numerical solution of the Navier-Stokes equations (6 hr)
Finite differencing; numerical solution of viscous flow problems; examples and demonstrations

Additional topics (instructor's option) (4 hr)

TOTAL HOURS: 60