General Description
Many fluid flows are unstable in the sense that small disturbances superimposed on the basic mean flow can amplify and significantly distort the basic state.  In this course we investigate the hydrodynamic stability of a variety of flows ranging from thin layers heated from below, to the flow between rotating cylinders, shear and boundary layer flows.

Aims
The aim of this course unit  is to look at various topics in hydrodynamic stability theory and introduce students to some of the classical as well as  more modern  ideas and techniques.

Learning Outcomes
On successful completion of the course unit students will be able to
· derive linearised stability equations for a given basic state
· perform a normal-mode analysis leading to an eigenvalue problems
· use the ideas of weakly non-linear stability theory in simple flows
· appreciate the different physical mechanisms leading to instability in fluid flows.

Prerequisites
MT3261 (Viscous fluid flow), MATH40201 (Perturbation Methods).

Future topics requiring this course unit
None

Content
Basic concepts of stability theory, stability, instability, normal modes,  marginal stability, neutral curves, temporal/spatial growth.
Rayleigh-Benard instability. Navier-Stokes equations and formulation of the linearised stability problem.  Cell patterns. experimental observations.
Shear Flow boundary layer instability. Stability of parallel flows. Inviscid  stability theory and properties of Rayleigh equation. Inflexion point criteria, Fjortoft’s theorem. Howard’s semi-circle theorem. Viscous/Tollmien-Schlichting instability. Orr-Sommerfeld equation. Parallel flow approximation and application to boundary layers. Gaster Transformation.
Introduction to nonlinear stability theory.  The Stuart-Landau equation.  Local bifurcation theory:  Saddle-node, Pitchfork, Hopf and transcritical bifurcations.  Structural (topological) stability.  The Ginsberg-Landau equation and modulation.
Benjamin–Feir instability.
Time–dependent flows – Mathieu’s equation and the parametric pendulum.

Teaching and learning methods
Three lectures and an examples class each week for nine weeks.

The classes will all take place in the nine weeks prior to the Easter vacation.  The examination will take place soon after the Easter vacation before the normal examination period.
 
 

Learning hours

Activity	Hours
Staff/student contact	35
Private study	112
Total hours	147

 

Assessment

Activity	Length	Weighted within unit
Coursework		20%
End of semester examination	2 hours	80%

Core learning materials
Textbooks
P.G. Drazin & W. Reid, Hydrodynamic Stability, C.U.P. 1982.
J.T. Stuart (ed. L. Rosenhead),  Laminar Boundary Layers, Dover paperback. Chap IX . 1988.
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,  Dover paperback. 1981.