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

2. 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

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

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

6. 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.

8. Fluxes at four cell boundaries! Density inside a cell

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

10. Starting point: Equation of Motion

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

13. Use divergence chain rule for dyadic tensors

14. Rewrite pressure gradient as a divergence

16. Momentum density Stress tensor = momentum flux Momentum source: gravity

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

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

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

20. ADIABATIC FLUID

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

24. Solution:

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

26. Trial Solution:

27.  A = U

28. Trial Solution:

31. Constant density flow:

32. Steady constant-density flow around sphere:

34. PARADOX OF D’ALAMBERT

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

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

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

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

42. Automatically satisfied by writing:

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

44. Steady slow flow equation Take divergence of slow flow equation:

45. General solution with constant pressure at infinity:

46. For this particular case: Components of pressure gradient:

47. Steady slow flow equation Vorticity:

48. Steady slow flow equation

50. Trial solution:

52. Conditions at infinity:

53. Conditions at surface sphere:

54. All flow quantities can now be determined:

57. For this particular flow at r=a :