Class 24. Fluid Dynamics, Part 1• The equations of fluid dynamics are coupled PDEs that form an IVP (hyperbolic).• Use the techniques described so far, plus additions.Fluid Dynamics in Astrophysics• Whenever mean free path λ  problem scale L in a plasma, can use continuum equa-tions to describe evolution o f macroscopic variables, e.g., density, pressure, etc.• Mathematically,λ '1σn∼1016[n/1 cm−3]cm,where σ = classical cross-section of atom or ion (∼ πr2Bohr).• Where is λ  L in astrophysics?Medium ∼ n (cm−3) ∼ λ (cm) ∼ L (cm) Scaleplanetary atmosphere 102010−4102–31–10 mstellar interior 102410−810111 Rprotoplanetary disk 101010610131 AUGMC 10 1015101910 pcdiffuse ISM 1 10161020100 pccluster gas 0.1 1017102210 kpcuniverse 10−61022> 1024> 1 Mpc• What would we like to learn from studying fluid dynamics?1. Steady-state structure of certain fluid flows, e.g., C-shocks (“continuous”).2. Time evolution of system, e.g.,– Propagation of shock through clumpy medium.– Accretion flow onto proto star or black hole.– Formation of structure in universe.3. Growth and saturation of instabilities, e.g.,– Rayleigh-Taylor:heavy fluidlight fluidg∗ Important in SN explosions, ISM, etc.– Kelvin-Helmholtz:fastslow∗ Important in jets and outflows in ISM.• To study t hese phenomena, must use equations of fluid dynamics.1Equations of Fluid Dynamics1. Continuity equation:∂ρ∂t+ ∇·(ρv) = 0, (1)where ρ = mass density, v = velocity, and ∇ = (∂∂x,∂∂y,∂∂z).• Sometimes see this written as:DρDt= −ρ∇·v,whereDD t≡∂∂t+v·∇ = Lagrangian or co-moving or substantive derivative (ra te ofchange of ρ in fluid frame, as opposed to∂∂t= Eulerian derivat ive, rate of changein lab fra me).• For an incompressible fluid, ρ is constant in space and time, so the continuityequation reduces to:∇·v = 0.• The continuity equation is a statement of mass conservation.2. Euler’s equation (equation of motion):∂v∂t+ (v·∇)v =Fρ−1ρ∇p, (2)where p = pressure and F = any external force (other than gas pressure) acting on aunit volume.• More compactly,ρDvDt= F − ∇p.• For gravity, have F = −ρ∇φ, where ∇2φ = 4πGρ. In hydrostatic equilibrium,F = ∇p, so there is no mass flow. E.g., in 1-D, have d p/dr = −ρ GM(r)/r2=−gρ, where g = gravitational acceleration.• For viscosity, F = µ∇2v, where µ = coefficient of dynamical viscosity, assumingρ = constant (incompressible fluid). If there are no other force terms in F, thisgives the Navier-Stokes equation.• Similarly, can add force terms for electric and/or magnetic fields.• For the steady flow of a gas, ∂v/∂t = 0 and, if there are no external forces, getρ v·∇v = −∇p ,which is Bernoulli’s equation for compressible flow.• Euler’s equation is a statement of momentum conservation.23. Energy equation:∂e∂t+ ∇· [(e + p)v] = 0, (3)where e ≡ ρ(ε+12v2) = energy density (energy/volume) and ε = specific internal energy(energy/mass).• In L agrange form,DeDt= −e(∇·v) − ∇·(pv),or, more compactly,DεDt= −pρ(∇·v).• The energy equation is a statement of energy conservation (there are many alter-native ways to write t he energy equation, depending on the context, e.g., usingspecific enthalpy (= ε + p/ρ), specific entropy combined with temperature andheat transfer, etc.).4. Equation of state:p = p(ρ, ε). (4)• Needed to close system.• E.g., for ideal gas, p = (γ − 1)ρε, where γ = adiabatic index (= ratio of specificheats at constant volume and pressure).1Fo r ideal monatomic, diatomic, andpolyatomic gases, γ = 5/3, 7/5, and 4/3, respectively.Solving the Equations of Fluid Dynamics• There are many choices one can make when adopting a numerical algorithm t o solvethe equations of fluid dynamics, e.g.,1. Finite differencing methods, including:(a) Flux-conservative form.(b) Operator splitting.2. Particle methods (e.g., smoothed particle hydrodynamics, or SPH).• Schematically (will discuss methods in italics),HDfinitedifferencingparticlemethodsoperatorsplit FDE formGodunovschemesetc.L−WSPHvortexmethodsconservative1Also have pVγ= constant, T Vγ−1= constant, T p(1−γ)/γ=