Fluid dynamics and Plasma physics AA 363 2:0 credits (Aug-Dec 2023)
Thursday 11:30-12:30 and 14:00-15:00 @ Multimedia room, Physics Department, IISc
Prerequisites: Mathematical physics, vector calculus, complex analysis, thermodynamics and statistical physics, electrodynamics
Recommended resources and books
- Statistical Mechanics, Kerson Huang, John Wiley & Sons, 1987
- Lectures on Theoretical Physics, David Tong, Cambridge University
- Elementary Fluid Dynamics, D. J. Acheson, Clarendon Press, 1990
- An Introduction to Fluid Dynamics, George Batchelor, Cambridge University Press, 1967, 2000
- Fluid Mechanics (2nd edition), Landau & Lifshitz, Course of theoretical physics vol. 6, Butterworth Heineman, 1987
- Geophysical Fluid dynamics, Joseph Pedlosky, Springer-Verlag, 1979
- The physics of fluids and plasmas, A. R. Choudhuri, Cambridge university press, 1998
- Elements of gas dynamics and the classical theory of shock waves. Ya. B. Zeldovich and Yu. P. Raizer, Academic Press, 1968
- Plasma Physics for Astrophysics, Russell M. Kulsrud, Princeton university press, 2004
- Introduction to Plasma Physics with space and laboratory applications, D. Gurnett and A. Bhattacharjee, Cambridge university press, 2005
- Fundamentals of Plasma Physics, Paul Bellan, Cambridge University Press, 2006
Fluids
Grading
- Practice Questions One mid-term (05 Oct, 2023 at 2 PM) and another Final exam (first week of Dec) to be conducted at IIA.
- Lectures-1&2 (03-Aug): Fluid as collection of classical neutral particles (hard spheres) obeying Newton's laws, Hamiltonian equations while undergoing elastic collisions. Derivation of Boltzmann transport equation for single particle distribution function from N-particle distribution function and motivating the collisional integral. BBGKY heirarchy and closure. Formulating the collision integral using scattering propability and two particle distribution function. Boltzmann H-theorem and irreversability in statistical ensembles. Particle collisions as Markov processes and numerical example (distribution of candies amongst two children) of a transition from an initial non-equilibrium state to the equilibrium state to demonstrate molecular chaos [Huang, Tong, personal notes]
- Lectures-3, 4 & 5 (17-Aug): Scattering matrix and collision integral assuming time-reversal and parity invarience. Show that scattering matrix ω(p_{1},p_{1} | p'_{1},p'_{1})= ω(p'_{1},p'_{1} | p_{1},p_{1}). Definition of collisional invarients and expectation values, derivation of conservation equations for mass, momentum and energy using Boltzman transport equation and collisonal invarient quantities.[Huang, Tong, personal notes]
- Lectures-6, 7 & 8 (24-Aug): Derivation of the Boltzman H-theorem and dH/dt<0 if condition of molecular chaos is true, and dH/dt=0 if distribution is Maxwell-Boltzman. MB distribution as the equilibrium distribution for which the collision integral vanishes. Expressions for density, pressure, heat flux for MB distribution (f_{1}). The local MB distrubution does not satisfy the Boltzman equation and a first order correction such that f_{1}=f_{1}^{0}+g needed. Use of relaxation approximation and relaxation time, τ to estimate g. The first order correction to local MB distribution such that Boltzman equation is satisfied leads to appearance of transport quantities like viscosity and heat conductivity. Motivating the stress tensor (proportional to strain rate) from arguments of symmetry and isotropy. Eulerian and Lagrangian formulation of fluid dynamics (finite control volume vs finite mass element). Different approximations like incompressible, Boussinesq, anelastic. Linearizing Euler equation to obtain sound wave propagation. Incompressible and anelastic approximations of fluid equations. [Huang, Tong, Batchelor, Acheson, personal notes]. Click this link for proof of symmetry of the viscous stress tensor wrt indices i,j.(page nos. 8 to 10). Viscous flow around a cylinder, Boundary layer and adverse pressure gradient and vortex formation in the wake due to shear instabilities. Von-Karman vortices and Strouhal number. Technique to break vortex associated oscillation in tall buildings and factory chimneys. [Acheson]
- Lectures-9 & 10 (31-Aug): Derivation of Bernoulli's theorem and function. Complex potentials and method of images for calculation of invisid and incompressible flows. Milne circle theorem for a circular boundary. Use of Milne circle theorem for invisid flow around a circular cylinder, conformal mapping of circle to flat plate, airfoil. Streamfunction for 2D flows: Exact differential, independent of path of integration. Kelvin and Helmholtz Vorticity theorems and vorticity frozen in fluid. [Batchelor Method of images, Complex flow potential for irrotational flows, streamlines for a line vortex against a wall for invisid fluid (coaxal circles) [Personal notes, Acheson] [Note: Presence of viscosity will create vortices (due to interaction with rigid boundaries) in the domain so that complex potential method is rendered ineffective.] Vorticity equation, Streamtubes and "Frozen-in" Vortex tubes, Vortex tubes are bounded by curves which are fluid "material lines". Neither streamtubes nor vortex tubes can arbitraritly end inside the volume of a fluid.
- Lectures 11, 12 (07 Sep): Rotating coordinate frames, Coriolis force. Vorticity equation in rotating coordinate frame, Kelvin's circulation in intertial frame vs in rotating frame. Rossby number. Low and high Rossby number flows, Bathtub vortex experiment in northern and southern hemisphere. Introduce aspect ratio, D/L to compare vertical scales to horizontal scales in planetary atmosphere. Shallow water equations.
- Lectures 15, 16 (14 Sep) : Thermal wind balance equations, Taylor-Proudman constraint (No variation of velocity components perpendicular to Ω in the direction parallel to Ω) Geostrophic balance in the velocity equation is a manifestaion of the "Thermal wind balance" in the vorticity equation for low Rossby number flows. Ertel's potential vorticity (PV) conservation theorem (which is essentially Kelvin's vorticity derived for a very special surface), Rossby waves and concept via Ertel's potential vorticity theorem applied to shallow water equations.[Pedlosky]
- Lectures 17, 18 (21 Sep) : Geostrophic balance, Derivation of dispersion relation for (quasi-geotrophic) Rossby waves and equatorial Rossby waves by linearization of shallow water equations [Pedlosky]. The equator acts as a waveguide for equatorial Rossby waves as their amplitude decreases away from the equator. The eigenfunctions for equatorial Poincare and Rossby wave are given by H_{n}(Y) e^{-Y2/2}, where Y=(β/ √ gH )^{1/2} y and H_{n}(Y) is Hermite polynomial of degree n and y is the length in meridional direction. Dispersion curves for planetary waves at the mid-latitude and equator. Rossby waves always propagate westward and are responsible for weather phenomena like El Nino and southern oscillations (ENSO) and periodicities in monsson (dry and wet spells) Surface gravity wave: dispersion relations for shallow and deep fluids (presence and absence of rotation, presence/absence of surface tension), effect of surface tension on dispersion relation. For unstable equilibrium (like heavy fluid on top of lighter fluid) surface gravity waves in the linear regime will grow into Rayleigh-Taylor instability, the examples of which are finger line projections inside supernova remnants and mushroom cloud of a nuclear bomb explosion. Presence of mean shear at a fluid interface can give rise to Kelvin-Helmholtz Instability even when lighter fluid rests on the top of heavier fluid. Example: patterns observed on surface of Jupiter near the giant red spot. Astrophysical jets, shear instabilities in protoplanetary disks [Acheson, chap 3.1-3.6, 3.8]. Internal gravity modes (g-modes): driven by buoyancy forces, derivation of the Brunt-Vaisala frequency, N. A gravitationally stratified fluid in unstable to convection when N < 0. Convective instability revisited: Schwarzchild criteria - Adiabatic parcel rising under pressure equilibrium. Existence of a critical Rayleigh number. Convection in stars and mixing length theory of Prandtl.
- Homework reading instead of Lecture on 28-Sep (Holiday):
-1. Existence of a critical Rayleigh number
-2. Convection in stars and mixing length theory of Prandtl (section 5.5 till 5.5.3)
- Lecture 19, 20 (12 Oct): Shocks as mathematical discontinuities in the fluid appearing due to flow speed exceeding the characteristic speed (sound speed) in the medium or due to nonlinear distortion of finite amplitude waves. Classification of shocks into travelling or stationary; normal or oblique. Bow shocks between Sun and Earth's magnetosphere or between the heliosphere and the interstellar medium is an example of normal and stationary shock. Derivation of Rankine-Hugoniot jump conditions for normal shocks. Formation of shocks when flow speed changes from supersonic to subsonic, Inflow into shock front is supersonic and outflow is subsonic in the reference frame of the moving shock. Supernova explosion and remnants, Acceleration of stellar/galactic winds (Parker's wind solution). De Lavel Nozzle and jets where flow transitions from subsonic to supersonic. Properties of supersonic flows.
- Lecture 21, 22 (26 Oct): Boltzman transport and collision in case of two-component (electron/ion) plasma. Lorenz gas model and Ohm's law. Hall effect (Chap 6, Gurnett and Bhattacharjee)
- Lecture 23, 24 (2 Nov): Single fluid approximation for Plasma or Magneto hydrodynamics approximation, Lorentz force, decomposition into magnetic pressure and tension. Faraday's induction equation, magnetic Reynolds number. Definition of current in terms of ion and electron velocities. Alfven's theorem of flux freezing. Plasma beta. Force free and pressure balanced plasma, Stability of plasma columns (kink and sausage instability) in presence/absence of axial magnetic field.
- Lecture 25, 26 (9 Nov): MHD waves in homogenous media and their dispersion relation (nature of slow, fast magnetoacoustic waves, Alfven waves). Relation between wave vector, ambient constant mangetic field and the velocity and magnetic field perturbations. Magnetic helicity, gauge freedom, Linkages between flux tubes. Conservation of helicity
- Lecture 27, 28 (16 Nov): Magnetic reconnection. Sweet Parker reconnection, formation of plasmoids (Paul Bellan) and estimation of flare time scale. Mean field MHD and Reynolds averaging and calculation of alpha effect and turbulent diffusion for isotropic plasma (personal motes)