Part IB - Fluid Dynamics Based on lectures by P. F. Linden Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Parallel viscous flow Plane Couette flow, dynamic viscosity. Momentum equation and boundary conditions. Steady flows including Poiseuille flow in a channel. Unsteady flows, kinematic viscosity, brief description of viscous boundary layers (skin depth). [3] Kinematics Material time derivative. Conservation of mass and the kinematic boundary condition. Incompressibility; streamfunction for two-dimensional flow. Streamlines and path lines. [2] Dynamics Statement of Navier-Stokes momentum equation. Reynolds number. Stagnation- point flow; discussion of viscous boundary layer and pressure field. Conservation of momentum; Euler momentum equation. Bernoulli's equation. Vorticity, vorticity equation, vortex line stretching, irrotational flow remains irrotational. 4] Potential flows Velocity potential; Laplace's equation, examples of solutions in spherical and cylindrical geometry by separation of variables. Translating sphere. Lift on a cylinder with circulation. Expression for pressure in time-dependent potential flows with potential forces. Oscil- lations in a manometer and of a bubble. [3] Geophysical flows Linear water waves: dispersion relation, deep and shallow water, standing waves in a container, Rayleigh-Taylor instability. Euler equations in a rotating frame. Steady geostrophic flow, pressure as streamfunction. Motion in a shallow layer, hydrostatic assumption, modified continuity equation. Conservation of potential vorticity, Rossby radius of deformation. [4] 1
Contents IB Fluid Dynamics Contents 0 Introduction 3 1 Parallel viscous flow 4 1.0 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Steady parallel viscous flow 6 1.3 Derived properties of a flow 10 1.4 More examples 11 2 Kinematics 15 2.1 Material time derivative 15 2.2 Conservation of mass 16 2.3 Kinematic boundary conditions 17 2.4 Streamfunction for incompressible flow 18 3 Dynamics 22 3.1 Navier-Stokes equations 22 3.2 Pressure 23 3.3 Reynolds number 24 3.4 A case study: stagnation point flow (u = 0) 26 3.5 Momentum equation for inviscid (v = 0) incompressible fluid 29 3.6 Linear f lows 33 3.7 Vorticity equation . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Inviscid irrotational flow 37 4.1 Three-dimensional potential flows 37 . . . . . . . . . . . . . . . . . 41 4.2 Potential flow in two dimensions . 4.3 Time dependent potential flows 44 5 Water waves 48 5.1 Dimensional analysis 48 5.2 Equation and boundary conditions 49 5.3 Two-dimensional waves (straight crested waves)