TAM 531

Dynamics of fluids in the limit of zero viscosity: governing equations of motion, kinematics, and vorticity transport; general theory of irrotational flow, including two-dimensional potential flow, the complex potential, and three-dimensional potential flow; applications to thin airfoil theory and free streamline theory; inviscid flows with vorticity; vortex dynamics; water wave theory; aspects of inviscid compressible flow. Course Information: 4 graduate hours. No professional credit. Prerequisite: MATH 285 OR MATH 286 OR MATH 441; TAM 435.

Dynamics of fluids in the limit of zero viscosity: governing equations of motion, kinematics, and vorticity transport; general theory of irrotational flow, including two-dimensional potential flow, the complex potential, and three-dimensional potential flow; applications to thin airfoil theory and free streamline theory; inviscid flows with vorticity; vortex dynamics; water wave theory; and aspects of inviscid compressible flow. Prerequisite: TAM 435 or equivalent; MATH 380; MATH 385, MATH 386, or MATH 441. 4 graduate hours.

Textbook:

No Textbook Used

Topics:

Governing equations (2 hr)
General equations of inviscid, compressible flow of mass, momentum and energy; inviscid approximation and effects of vanishing viscosity; general equation for inviscid, incompressible flow

Kinematics and general theorems of fluid motion (10 hr)
Eulerian and Lagrangian descriptions; streamlines, stream functions, vortex lines, vortex tubes; analysis of deformation; transport theorems; transport equations for an ideal fluid; Bernoulli|#39;s equation; circulation and Kelvin|#39;s theorem; compressible and incompressible vorticity equation

Irrotational flow theory and applications (16 hr)
General features of irrotational flow and the velocity potential; complex potential for two-dimensional ideal flow; Blasius theorem for force and moment; Joukowski airfoils; three-dimensional flow past moving bodies; singularity and image methods; slender body theory; lifting-line theory; free-streamline theory; motion of a solid body through ideal fluid

Inviscid flows with vorticity (10 hr)
Self induced motion of a line vortex; instability of a vortex sheet; integrals of the vorticity distribution; motion of a group of point vortices; steady axisymmetric flow with swirl; flows in rotating systems, Coriolis forces; motion of a thin layer on a sphere

Waves in layered fluids (10 hr)
Irrotational waves at a free surface; linearized theory and instabilities; shallow water wave theory; hydraulic jumps and tidal bores; waves with capillarity

Aspects of inviscid compressible flow (10 hr)
Review of the equations of compressible flow; characteristic form and Riemann invariants; weak formulation; small amplitude acoustics; simple waves; normal shock waves and the shock tube problem; wave interactions; similarity solutions, Taylor blast wave, Guderly implosion; oblique shocks and Mach waves

TOTAL HOURS: 58

9/20/2018