Relativistic Hydrodynamics L3&4/SS14/ZAH
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1 Conservation form: Remember: [ q] 0 conservative div Flux t f non-conservative 1. Euler equations: are the hydrodynamical equations describing the time-evolution of ideal fluids/plasmas, i.e., frictionless flows [ q] div Flux fint f t ext 2. Navier-Stokes = Euler frictional forces dissipative flows Frictional forces = f(second order operators) (,, ) 2 2 x x y y [ q] t div Flux=f +f L viscous int ext 2 Q1: Re-write the continuity and the momentum equations in cartesian, cylindrical and spherical coordinates. Q2: Show that for r >> 1 the equations in cylindrical and spherical reduces to their Cartesian form. Q3: Prove that the equations in cylindrical and spherical coordinates diverge from their cartesian counterpart for r << 1. L3_4 / S. 1
2 Weakly compressible Navier-Stokes equations: Assume we are given a chamber filled with a gas. According to Boyle's law: PV = Const., the pressure increases with decreasing the volume (i.e., moving the piston to the left, see Figure!). For simplicity, let us decrease the volume of the gas to become smaller than a critical volume V c, i.e., if the mean free path between two arbitrary particles becomes smaller than a critical length, c then with the help of other mechanisms, the gas may condensate and undergo a phase transition, beyond which any additional pressure would have a negligible effect on the density of the enclosed medium. In this case, the medium is said to be incompressible. For example: Let us fill the chamber with Hydrogen and Oxygen molecules and try to compress them. Under certain conditions (of course with the help of other mechanisms), the molecules dissociate and undergo phase transition to produce water. Once water is produced, it becomes insensitive to increasing/decreasing the pressure. The pressure at 1600 m deep in the ocean is about 150 atmospheres, while the densiy increase is less than 1 percent. From the HD point of view, this corresponds to setting the density constant. Thus: V 0 t V 0 E t d int (E V) P V d int ρv (ρv V) P L t vis 2 L3_4 / S. 2 V 0 d E = const. int V (V )V t 1 1 P + L vis Lagrangian 2
3 The vorticity equation: Relativistic Hydrodynamics Assuming the flow to be incompressible, the momentum equation in Euler hydrodynamics reads: V 1 (V )V P + f t ρ Take the curl operator ( ) and substitute the identity: we obtain: 1 (V )V (V V) V ( V) 2 curl of a gradient = ( V) (V V) V ( V) [ P] + f t 2 ρ curl of a conservative force = 0 1 V P, 2 t ρ The vorticity equation Where V vorticity L3_4 / S. 3
4 Kelvin theorem: Let ρ P. V 0 ds 0 t t advected t Stokes theorem Verify! S The flux of the vorticity is conserved and that vortices conserve their form as they move with the fluid 1. If the flow is ideal and contains no vortices, then vortices will never develop. 2. If the flow is rotational, incompressible and diffusive, i.e., V t Then, the Kelvin theorem does not apply here as vortices can be generated and/or annihilated equally. L3_4 / S. 4
5 Self-gravitating systems: Relativistic Hydrodynamics Consider a spherically symmetric object enclosed in a sphere of radius R, with the following density distribution: In this case the gravitational potential, φ satisfies the Poisson equatio n: ϕ (r) In the case that all the mass concentrated just in one point, then: ϕ = GM/r. In both cases, the gravitational force is calculated as follows: f = -ϕ. Boundary conditions: Given is an equation of the type: L(u) = F, where L is a combination of various differential operators acting on u. There are three types of boundary condition to be considered: 1. Dirichlet: u = f on Ω 2. Neuman: u/ n = f on Ω 3. and mixed BCs: u/ n + ku = f on Ω Well-posedness of a mathematical problem A mathematical problem is said to be well-posed, if: 1. it has (numerical) solutions, and 2. the (numerical) solution is unique, and 3. the (numerical) solution depends continuously on the initial and boundary conditions (i.e., ICs & BCs). L3_4 / S. 5
6 Important fluid numbers: We saw that the hydro-dynamical equations in the conservative formulation have the following form: Rate of change of the conservative quantity q + Flux through + = the boundaries Sum over all internal and external sources/forces Equivalently: [ q] t div Flux = f + f L viscous int ext 2 By taking the ratio of one term to another, one may measure the relative importance of terms that can be used to characterize/ distinguish one flow from another. Consider for example the time-evolution of a physical variable q subject to the following internal and external forces: Rate of change Pressure Gravity Vis Rad Mag of q with time Advection f f f f f fi =Inertia [1] [1] [1] [ 2] [1,2] [1] [1] Mag [2] q + F fp fg fc fvis fdiffusion fheat fmag t The different forces operating here give rise to the following subset of known fluid numbers: Re = fi / fvis Reynolds number mag Mag Re fi / fdiffusion Mag. Reynolds number Prm ercury Pr = fvis / fl2t Prantl number Pr oil > 100 engine M = fi / fp Mach number Fr = fi / fgravity Froude number Ro = fi / fc Rossby number (Inertia/Coriolis) L3_4 / S. 6
7 Here are typical values for the Mach number versus Reynolds numbers: Other typical values: Re mag >> 1 Fr < 1 (confined flows general) Ro(tornadoes) 10 3 Ro(low atmospheric pressure regions) 10-1 Ro(ocean systems) 1 L3_4 / S. 7
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