1. General Relativistic Hydrodynamics with Viscosity Collaborators: Matthew D. Duez Stuart L. Shapiro Branson C. Stephens Phys. Rev. D 69, 104030 (2004) Presented by Yuk Tung Liu 14 th 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 = 5.5P rot 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