Fluid dynamics and Plasma physics AA 363 2:0 credits (Aug-Dec 2022) Friday 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
- Homework Questions - submission DEADLINE Oct 15, 2022
- Homework assignments carrying a weightage of 30%. HW assignments will be roughly 1-2 problems per lecture which will be a combination of reading, derivations, concepts and numerics, plotting etc. I prefer that assignments are always turned in as hard copies including plots.
- [Friendly advice] I strongly recommend you keep working consistently on the homework given in the class rather than working on all of it at the end of the semester.
- Two exams - One on Fluids and other on Plasmas conducted in Oct and Dec, 2022, respectively, at IIA.
- Lectures-1&2 (05-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 (12-Aug): Scattering matrix and collision integral assuming time-reversal and parity invarience. Show that scattering matrix ω(p1,p1 | p'1,p'1)= ω(p'1,p'1 | p1,p1). Relation between scattering cross section, impact parameter and collision integral. 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-5, 6 & 7 (19-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 (f1). The local MB distrubution does not satisfy the Boltzman equation and a first order correction such that f1=f10+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). [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).
- (26-Aug): No class. Organizing IUSSTF network center meeting at IIA (22 Aug - 02 Sep, 2022).
- (02-Sep): No class. Organizing IUSSTF network center meeting at IIA (22 Aug - 02 Sep, 2022).
- Lectures-8&9 (09-Sep): Linearizing Euler equation to obtain sound wave propagation. Incompressible and anelastic approximations of fluid equations. Streamlines and flow visualization techniques like streaklines, pathlines and timelines. Bernoulli's equation along streamlines, Stream function for 2D. Streamlines and streaklines are snapshots in time. Streamlines, Particle pathlines lines and streaklines are different for unsteady flows. Streamlines do not cross in space whereas streaklines can intersect when the origin point for one streakline lies on another streakline. Streamfunction for 2D flows: Exact differential, independent of path of integration. [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]
- Lectures-10, 11, & 12 (16-Sep): Milne circle theorem as method of images for a circular boundary. Use of Milne cirlce theorrem for invisid flow around a circular cylinder, conformal mapping of circle to flat plate, airfoil. Singularities of flow appear at cusps where the bluff body contour is not differentiable. Viscous flow around a cylinder, Boundary layer and estimation of BL width using scaling arguments, 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] [Note: Presence of viscosity will create vortices (due to interaction with rigid boundaries) in the domain so that complex potential method is rendered ineffective.]
- Lectures 13, 14 (23 Sep): Vorticity equation, Streamtubes and "Frozen-in" Vortex tubes, Kelvin's circulation theorem and Helmholtz vortex theorems. 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. Rotating coordinate frames, Coriolis force. Vorticity equation in rotating coordinate frame, Kelvin's circulation in intertial frame vs in rotating frame.
- Lectures 15, 16, & 17 (30 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. 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) [Pedlosky]
- Lectures 18, 19 (07 Oct): Surface gravity wave (f-modes): dispersion relations for shallow and deep fluids, 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]
- Lectures 20, 21 & 22 (14 Oct) : 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.
- Lectures 23, 24 (21 Oct) : Energy cascading from larger to smaller scales in 3D turbulence due to the effect of "vortex stretching" and "dissipation term" in the vorticity equation. Absence of vortex stretching in 2D leads to inverse cascade. Transition to turbulence at high Reynolds number by mechanism of fluid instabilities like KH instability, Rayleigh-Taylor instability or shock instability. Turbulent boundary layer has greater mixing due to greater momentum transfer between layers at molecular scales as opposed to laminar boundary layers where flow remains in layers or "lamina". Concepts of injection scale (largest length scale), inertial scale (inertial term in NS equation dominate) and dissipation scale (viscous term dominates) in fully developed turbulence in statistically steady state with constant energy injection rate. Derivation of the Kolmogorov (dissipation) length scale as a function of Reynold's number and -5/3 law for kinetic energy density. Turbulent eddies defined as correlation length scales in the fluid. Lighthill mechanism and fluid flow as source for sound waves.
- Lectures 25, 26 (28 Oct): Introduction to helioseismoligy:
Gravity (g) modes, acoustic (p) modes and surface gravity (f) standing wave modes on the Sun. Only p and f modes are observable using "Dopplerograms" where as g-modes are trapped inside the solar radiative zone. How helioseismology played a key role in resolution of the solar neutrino problem. Helioseismology is used as a probe of the internal structure and flows in the interior of the Sun. In presence of rotation, the acoustic waves travelling in prograde and retrograde directions have different frequencies (splitting) and speeds.link to presentation 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, attached or detached shock fronts. 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. 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.