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Astrophysical Fluid Dynamics

Astrophysical Fluid Dynamics

Course Outline: 

A very large number of astrophysical systems can be modelled as fluids that move under the action of two external forces (gravity and electromagnetism) as well as the forces exerted internally on the fluid by its adjacent parts as they jostle one another in their motion. An added complication is that the fluid material is a thermodynamic system: it is able to absorb, store, and expel energy as it moves. A fluid is therefore a complex system whose motion is governed by the laws of gravity, electrodynamics and thermodynamics.

The fluid flows observed here on earth occur everywhere in the universe, but on a grander scale and under far more extreme conditions. In this course, we study the elements of fluid dynamics and learn how to describe simple flows. For simplicity, we ignore electromagnetic forces. The study of their effects on flow is the subject of magnetohydrodynamics, which will be studied in the Masters course. This course concentrates of flows that are driven by gravity, pressure and thermodynamics. The theory developed in the lectures is illustrated by examples both of terrestrial and astrophysical flows.

Fluid dynamical processes are the driving force behind most fundamental processes in the universe, i.e. spiral density waves in galaxies, triggering bursts of star formation in the spiral arms as it passes through a region, solar and stellar flares, stellar evolution, instabilities in stars giving rise to stellar pulsation, accretion processes in binary systems, as well as the super-relativistic jets ejected by black holes in the heart of galaxies, and many others. Fluid dynamics is thus central to a good understanding of astrophysics and astrophysical processes.
The aim of this course is to provide aspiring astrophysicists with a broad introduction to the fundamental fluid processes that you can expect to feature in a wide range of astrophysical environments.

Syllabus: 

  • Introduction to Fluids

  • Basic Concepts

  1. Physical Nature of a Fluid

  2. Two Ways to Model a Fluid

  3. Continuum Model

  4. Some Terminology

  • Fluid Kinematics

  • The Velocity Field

  1. Description of Fluid Velocity

  2. Visualising flows

  3. Equations for streamlines

  4. Pathlines

  5. Relation Between Streamlines and Pathlines

  6. Streaklines

  • Fluid Acceleration

  1. Acceleration of a Fluid Point

  2. Physical interpretations

  3. Convective Derivative

  4. Dry and Wet Observers

  5. Advective term in different systems of coordinates

  • Motion of a Fluid Element

  1. Introduction

  2. Relative motion of fluid points

  3. Linear strain rate

  4. Volume strain rate

  5. Shearing Rate

  6. Pure Shear

  7. Rotation Rate

  8. General Motion of a Fluid Element

  • Specialised Flows

  1. One, two and three dimensional flows

  2. Steady and unsteady flows

  3. Viscous and inviscid flows

  4. Laminar and turbulent flows

  5. Compressible and incompressible flows

  6. Rectilinear and circular flows

  • Rotation of the Milky Way

  1. Differential Rotation

  2. Measuring of the Velocity of a Nearby Star

  3. The Model

  4. Velocity of Nearby Stars Relative to the Sun

  5. Radial and Transverse Velocity

  6. Discussion

  • Forces on Fluids

  • Forces on continuous media

  1. Forces

  2. Volume and surface forces

  3. Description of volume forces

  4. Gravitational force density

  5. Stress

  • Governing Equations

  1. Principles that Govern Fluid Flows

  2. Equations of Motion

  3. Forces on Fluids

  4. Definition of Stress

  5. Internal Stress

  6. The Stress Tensor

  7. Isotropic Stress

  8. Homogeneous stress

  9. Contact Force on a Fluid Element

  10. Symmetry of the stress tensor

  11. Equations of Motion

  12. Cauchy’s equations in a general system of coordinates

  • Conservation Principles

  1. Introduction

  2. Continua

  3. Densities

  4. Time evolution of densities

  5. Current densities

  6. The Continuity Equation

  7. Differential form of the continuity equation

  8. Conservation Equations

  • Conservation of Mass

  1. The principle of conservation of mass

  2. Alternative derivation

  3. Summary

  • Conservation of Momentum

  1. Fluid momentum

  2. Momentum density in a fluid

  3. Continuity equation for fluid momentum

  4. Physical interpretation

  5. Alternative forms of the momentum continuity equation

  • Conservation of energy

  1. Energy Transport in Fluids

  2. Work done by Applied Forces

  3. Kinetic Energy Equation

  4. Internal Energy Equation

  5. Example: Conducting Ideal Gas

  • Hydrostatics

  • Fluids at Rest

  1. The Hydrostatic Equation

  2. Fluid at Rest in an Uniform Gravitational Field

  3. Archimedes’ Principle

  • Plane-Parallel Atmospheres

  1. Plane Parallel Atmosphere in Uniform Gravitational Field

  2. Isothermal Atmosphere

  3. Adiabatic atmosphere

  4. Polytropic atmosphere

  • Stellar Structure

  1. Barotropic systems

  2. Polytropic systems

  • Solar Corona

  1. Brief Description of the Solar Corona

  2. The Model

  3. Discussion of results

  • Inviscid Fluids

  • Euler’s Equations

  1. Frictionless Fluids

  2. Equation of motion for inviscid fluids

  3. Incompressibility

  4. Vorticity

  • Bernoulli’s Equation

  1. Introduction

  2. General principle

  3. Steady flow of a fluid with uniform density

  4. Steady flow of an incompressible fluid

  5. Irrotational flow with constant density

  6. Examples

  • Supersonic Jet Flows

  1. Bernoulli’s equation for compressible flows

  2. Nozzle flow

  3. de Laval’s Nozzle

  4. Supersonic jets in astrophysics

  • Viscous Fluids

  • Newton’s Law of Viscosity

  1. Fluid Friction

  2. Newton’s Law of Viscosity

  3. Mathematical formulation of Newton’s law

  4. Important property of Newton’s flow

  • Newtonian Fluids

  1. Newtonian and non-Newtonian fluids

  2. Stress-Rate of Strain Relations in General

  3. Newtonian Fluids

  4. Physical Significance of α and β

  5. Coefficient of bulk viscosity

  • Newtonian Flows

  1. Equation of Motion for Newtonian Fluids

  2. Kinetic Energy Equation

  • Navier-Stokes Equation

  1. Stokes' Assumption

  2. Navier-Stokes' Equation

  • Simple Solutions of the Navier-Stokes Equations

  1. Channel flow

    1. Channel flow without gravity

    2. Channel flow with gravity

  2. Flow through a finite rectangular slit

  3. Pipe flow

  • Accretion Discs

  • Perturbation Theory

  • Perturbation Equations

  1. The Equilibrium State

  2. Equations for the Perturbed Flow

  3. Adiabatic Perturbations

  4. Isothermal Perturbations

  • Star and Galaxy Formation

  1. Stability of Gas Clouds

  2. Stability of Rotating Gas Clouds

  • Appendix
  1. Rotating Frames of Reference


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Programmes | by Dr. Radut