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General Relativistic Hydrodynamics with Viscosity

General Relativistic Hydrodynamics with Viscosity. Phys. Rev. D 69 , 104030 (2004). Presented by Yuk Tung Liu. Collaborators: Matthew D. Duez Stuart L. Shapiro Branson C. Stephens. 14 th Midwest Relativity Meeting October 15, 2004. Motivation.

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General Relativistic Hydrodynamics with Viscosity

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  1. General Relativistic Hydrodynamics with Viscosity Phys. Rev. D 69, 104030 (2004) Presented by Yuk Tung Liu Collaborators: Matthew D. Duez Stuart L. Shapiro Branson C. Stephens 14th Midwest Relativity Meeting October 15, 2004

  2. Motivation • Viscosity can have significant effects on relativistic stars - suppress gravitational-wave driven (CFS) instabilities

  3. Motivation • Viscosity can have significant effects on relativistic stars • - suppress gravitational-wave driven (CFS) instabilities • - drive a secular (Jacobi) bar-mode instability

  4. Motivation • Viscosity can have significant effects on relativistic stars • - suppress gravitational-wave driven (CFS) instabilities • - drive a secular (Jacobi) bar-mode instability • - destroy differential rotation  secular evolution of • hypermassive neutron stars

  5. Formalism • Evolve the metric using BSSN formulation • Gauge choices Lapse: K-driver (approximate maximal slicing) Shift: Gamma-driver (approximate “Gamma-freezing” condition)

  6. Hydrodynamic Variables Stress-energy tensor Shear tensor: Specific enthalpy: Rest-mass density: 0 Pressure: P Coefficient of shear viscosity: Specific internal energy:  4-velocity: u4-acceleration: a -law equation of state:

  7. Hydrodynamic Equations Define new hydrodynamic variables: Baryon number conservation Energy equation Navier-Stokes equation

  8. Viscosity Law • Want to explore point of principle: evolve general relativistic hydrodynamics with viscosity • Not interested in the details of viscosity in neutron stars • Assume simple viscosity of the form =P P (P : positive constant) • Choose P such that the viscous timescale vis= a few dynamical times (long enough for the system to be evolved quasi-statically, but short enough to make numerical treatment trackable) • This viscosity law is consistent with a “turbulent viscosity”

  9. Code Test – Evolution of a stable, uniformly rotating star R/M = 4 During the entire simulation, M / M < 0.1%; J / J < 1.5% ; Violation of Hamiltonian and momentum constraints< 1%

  10. Evolution of a Differentially Rotating Star vis = 5.5Prot During the entire simulation, M / M < 0.4%; J / J < 0.4% ; Violation of Hamiltonian and momentum constraints< 1%

  11. Scaling of Secular Evolution with Viscosity Parameter

  12. Conclusion • We have developed a hydrodynamic code to solve the fully-relativistic Navier-Stokes equation • Our code is able to evolve relativistic stars for dozens of rotation periods • We studied the secular evolution of hypermassive neutron stars (next talk) • We will use this code to study the viscosity-driven (Jacobi) bar-mode instability

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