4A10 Flow instability Fluid-structure interaction CUED IIB Lecture notes Shreyas Mandre Department of Engineering, University of Cambridge sdm63@cam.ac.uk ©2026 Shreyas Mandre CC-BY-NC-ND (Last updated: 15 March 2026 ) Contents 1 Introduction to stability 1.1 Stability in terms of breaking symmetry 1.2 The role of positive feedback in symmetry breaking 1.3 The state and state variables 1.4 Dynamics governing the evolution of state 1.5 The steady state 1.6 Perturbation of the steady state 1.7 Linearization 1.8 A word about non-dimensionalization 1.9 Normal modes and exponential growth 1.10 Sufficient condition for instability 1.11 Conclusion 2 Simple models for structures 2.1 Lumped mass-spring-damper 2.2 Pitch and heave 2.3 A general structure with a finite number of degrees of freedom 2.4 A stretched string or a one-dimensional membrane 2.5 Euler beam 2.6 Conclusion 3 Simple models for fluids 3.1 Parametrization in the viscous and inviscid limit 3.1.1 Low Reynolds number flow 3.1.2 High Reynolds number flow 3.1.3 Steady lift on thin airfoils 3.1.4 Steady bluff body drag 3.2 Simplified versions of Navier-Stokes 3.2.1 Force on an unsteady curved fluid-carrying conduit 3.2.2 Thin layer flow 3.3 Conclusion 4 Galloping and flutter 4.1 Galloping 4.1.1 Translational galloping 4.1.2 Rotational galloping 4.1.3 Suppressing galloping 4.2 Flutter 4.2.1 Suppressing flutter 4.3 Conclusion 5 Vortex induced vibrations 5.1 Fluid dynamical instability 5.2 Vortex-induced vibrations 5.3 Mode locking 5.4 Response to vortex shed by identical neighbours 5.5 Suppressing vortex-induced vibrations 5.6 Conclusion 6 Conduits and channels 6.1 An infinite firehose 6.2 An infinite channel with flexible walls 6.3 Conclusion Bibliography