Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit

Aim: To cast all equations in the same generic form: Reasons: 1.Allows quick identification of conserved quantities 2.This form works best in constructing numerical codes for Computational Fluid Dynamics

Generic Form: Transported quantity is a scalar S, so flux F must be a vector! Component form:

Generic Form: Transported quantity is a vector M, so the flux must be a tensor T. Component form:

Mass conservation: already in conservation form! Continuity Equation: transport of the scalar  Excludes ‘external mass sources’ due to processes like two-photon pair production etc.

Fluxes at four cell boundaries! Density inside a cell

Mass conservation: already in conservation form! Continuity Equation: transport of the scalar  Momentum conservation: transport of a vector! Algebraic Manipulation

Starting point: Equation of Motion

Use: 1. product rule for differentiation 2. continuity equation for density

Use divergence chain rule for dyadic tensors

Rewrite pressure gradient as a divergence

Momentum density Stress tensor = momentum flux Momentum source: gravity

Energy density is a scalar! Kinetic energy density Internal energy density Gravitational potential energy density Irreversibly lost/gained energy per unit volume

Internal energy per unit mass Specific enthalpy Irreversible gains/losses, e.g. radiation losses“Dynamical Friction”

Summary: conservative form of the fluid equations in an ideal fluid: Mass Momentum Energy

ADIABATIC FLUID

Extra mathematical constraints one can put on a flow: 1. Incompressibility: 2.No vorticity (“swirl-free flow”): 3.Steady flow:

Solution:

Far away from sphere: This suggests: m = 1 !

Trial Solution:

 A = U

Trial Solution:

Constant density flow:

Steady constant-density flow around sphere:

PARADOX OF D’ALAMBERT

NO fore-aft symmetry, Now there is a drag force!

Viscosity = internal friction due to molecular diffusion, viscosity coefficient  : Viscous force density: (incompressible flow!) Equation of motion:

Very viscous flow: >> VL, Re << 1 Friction-free flow: > 1

Because of viscosity: no slip, velocity vanishes on sphere!

Automatically satisfied by writing:

Steady flow equation Slow flow approximation of this equation: From:

Steady slow flow equation Take divergence of slow flow equation:

General solution with constant pressure at infinity:

For this particular case: Components of pressure gradient:

Steady slow flow equation Vorticity:

Steady slow flow equation

Trial solution:

Conditions at infinity:

Conditions at surface sphere:

All flow quantities can now be determined:

For this particular flow at r=a :