AMS 691 Special Topics in Applied Mathematics Lecture 5

1. AMS 691Special Topics in Applied MathematicsLecture 5 James Glimm Department of Applied Mathematics and Statistics, Stony Brook University Brookhaven National Laboratory

2. Today Viscosity Ideal gas Gamma law gas Shock Hugoniots for gamma law gas Rarefaction curves fro gamma law gas Solution of Reimann problems

3. Total time derivatives

4. Euler’s Equation

5. Conservation form of equations

6. Momentum flux

7. Viscous Stress Tensor

8. Incompressible Navier-Stokes Equation (3D)

9. Two Phase NS Equationsimmiscible, Incompressible • Derive NS equations for variable density • Assume density is constant in each phase with a jump across the interface • Compute derivatives of all discontinuous functions using the laws of distribution derivatives • I.e. multiply by a smooth test function and integrate formally by parts • Leads to jump relations at the interface • Away from the interface, use normal (constant density) NS eq. • At interface use jump relations • New force term at interface • Surface tension causes a jump discontinuity in the pressure proportional to the surface curvature. Proportionality constant is called surface tension

10. Reference forideal fluid EOS and gamma law EOS @Book{CouFri67, author = "R. Courant and K. Friedrichs", title = "Supersonic Flow and Shock Waves", publisher = "Springer-Verlag", address = "New York", year = "1967", }

11. EOS. Gamma law gas, Ideal EOS

12. Derivation of ideal EOS

13. Gamma

15. Proof

16. Polytropic = gamma law EOS

18. Specific Enthalpy i = e +PV

19. Enthalpy for a gamma law gas

20. Hugoniot curve for gamma law gas Rarefaction waves are isentropic, so to study them we study Isentropic gas dynamics (2x2, no energy equation). is EOS.

21. Characteristic Curves

22. Isentropic gas dynamics, 1D

24. Riemann Invariants

25. Centered Simple Wave