Astrophysical Fluid Dynamics (2) (Draine Chap 35;;Clarke & Carswell Chaps 1-4) Last class: fluid dynamics equations apply in astrophysical situations fluid approximation holds Maxwellian velocity distribution mass, momentum and energy must be conserved reference frames: Eulerian or Lagrangian d/dt following fluid D Dt t u. Lagrangian Eulerian convective
Conservation of mass equation of continuity. u 0 t D Dt. u Eulerian continuity equation Langrangian form Conservation of momentum equation of motion Du Dt P u t u. u Lagrangian equation of motion since D/Dt = / t + u. u t u. u P Eulerian form 2 u t.( u P ) - 1/ 4 ( x B)xB (including viscous and magnetic field effects) Navier-Stokes equation
gas fluid when particles move collectively incompressible fluid? incompressible particles maintain constant volume in phase space may have density fluctuations within this volume e.g. ketchup/mustard packets incompressible shape may change but mass and volume remain same From Lagrangian form of continuity equation D Dt t u. For incompressible, D /Dt = 0,.u = 0
Viscous flows: Clarke & Carswell 11.1 & 11.2 Initially assume momentum of moving fluid element does not change due to differential motion of its neighbors. wwwwwwwwwwwwwwwwwwwwwwww If velocity differences between elements transfers momentum Viscosity ( & included in stress tensor) (if smooth laminar flow, viscosity = 0 ) Linear shear flow shown;; parallel streamlines to velocity gradient From thermal motion, particles have velocity component in j direction Incompressible flows can transition from smooth laminar to turbulent
ij shear viscosity + bulk viscosity pressure term ij ij Ps ij and, from Navier Stokes Equation 2 u t.( u P ) - 1/ 4 ( x B)xB ij S (.u + ( B +1/3 S ))(.u) ij ~ ( 2.u) for viscous stress where S. B are coefficients of shear and bulk viscosity i.e. u/ t ~ / ( 2.u) for viscous stress only And viscosity diffusion of momentum number, Re importance of viscous forces in flow relative to inertial forces likelihood of turbulence
Navier- Stokes: 2 u t.( u P ) - 1/ 4 ( x B)xB Define kinematic viscosity, k = / And k ~ v T (random walk (thermal) velocity x mfp) Reynolds Number R e inertial forces/viscous forces u 2 / ij ~ u 2 / ul -1 = where L is a scale length Thus, R e = ul/ k (dimensionless L/T xl x T/L x 1/L) = ul/v T, v T is thermal velocity For u = v T, R e ~ L/ >> 1 if fluid approximation holds At R e ~ 3000 turbulence
In molecular clouds: u ~ 1 km/s, L ~ 10 pc ul = 10 5 x 3 x 10 19 = 3 x 10 24 k = v T = 1/n kt/m k = T/n (10-8 /10-15 x 10-12 ) ~ 10 19 T 1/2 /n For T ~ 100K, k ~ 10 20 /n since R e ~ ul/v T = 3 x 10 24 x n/10 20 = 3 x 10 4 x n For n = 100, R e ~ 3 x 10 6 i.e. >> 3000 ISM is likely to become turbulent
Conservation of energy (Clarke & Carswell 4.1, 4.2, 4.3) Equation of state P in terms of, T P = (R/ ) T (R gas constant, mean molecular weight) or PV = NkT, (V = vol, N = particle number, k + Boltzmann constant) Assume: barotropic P is a function only of and adiabatic no radiative losses, no conduction For energy conservation, dq = d +PdV dq = heat absorbed by unit mass of fluid, d = change in internal energy/unit mass of fluid, PdV = work done by unit mass of fluid if volume changes by dv For an ideal gas, = (T) dq = (d /dt )dt +PdV = CvdT + (R/ V)TdV where Cv specific heat at constant volume
dq = CvdT + (R/ V)TdV = 0 for adiabatic change Cvd lnt + R/ x d lnv = 0 Cv V T R/, and similar expressions for P from ideal gas equn Can demonstrate that Cp Cv = R/ And = Cp/Cv (ratio of specific heats) V T -1/ -1, P T / -1, P V - Hence equn of state for gas undergoing adiabatic change: P = K K is const and ~ 1/V
Energy equation Clarke & Carswell 4.3 Rewrite energy conservation: D /Dt = dw/dt + DQ/Dt and DW/Dt = -PxD(1/ )/dt = P/ 2 D /Dt D /Dt = P/ 2 D /Dt Q cool, cooling function Define total energy/unit volume = E E = (½u 2 + + ) K E PE internal energy de/dt = E/ D /Dt + (u.du/dt + D /Dt + P/ 2 D /Dt Q cool continuity & motion equns dp/dt, du/dt E/ t +.[(E +P)u] =- Q cool + / t With enthalpy/unit mass, h = + P/ D(h+ ½u 2 + )/Dt = - P/ t + / t - energy equn
D (h+ ½u 2 + ) /Dt = - P/ t + / t - energy equn In steady flow with P/ t = 0 and / t = 0 D(h+ ½u 2 + )/Dt = 0 h+ ½u 2 + is constant along stream lines (streamlines = paths traversed by fluid elements) h + ½u 2 + = constant ( theorem) i.e. P/ + ½u 2 + = constant Dripping faucet: P/ const balances atmospheric pressure - u 2, decreases as flow continues (gravitl pull) Continuity equn: ua = constant (A = x-section of flow at height z) As drops, u increases, A increases droplets not stream
So far assumed an incompressible fluid - density gradients smoothed out, D /Dt = 0 for any fluid element ALSO fluid motions must be lower than sound speed sound-crossing time time for a disturbance to propagate across a region (Is cossing time short enough that region can respond dynamically before affected by another effect?) Astrophysical flows often compressible Pressure information takes finite time to propagate In a uniform medium disturbances propagate as acoustic waves
Recall equations of continuity and motion: t. u 0 ( u t u.. u) P For a 1-d flow, no external forces: du x /dt + u x du x /dx = -dp/dx d(p + u x2 )/dx = 0 for a steady flow Adiabatic constant entropy ds/dt= 0 Consider a small disturbance in fluid: 0 0 + 1 and P 0 P 0 + P 1, 0, P 0 constant Substitute in fluid equns and drop second order etc terms 0 u/ t = - P 1 = - P/ S 1. 2 1 / t 2 = ( P/ ) 2 1 / x 2, for 1d- flow;; see also C&C (6.9)
We have 2 1 / t 2 = ( P/ ) 2 1 / x 2, From equation of state: P= K and dp = ( P/ )d adopting 1- flow, could also express as: 1/ 0 (d 2 1 /dt 2 ) - 1/ 0 ( P/ )d 2 1 /dx 2 = 0 i.e. (d 2 1 /dt 2 ) - ( P 0 / 0 )d 2 1 /dx 2 = 0 This is a wave equation w. solution = 0 e i(ks- t) with angular frequency = 2, and wave number k = 2 / Propagation speed of points of const phase ~ /k And 2 /k 2 = ( P/ ) from wave equation Sound speed c s = ( P/ ) ½ i.e. small adiabatic disturbances travel at sound speed in the medium