AMS 691 Special Topics in Applied Mathematics Lecture 5 James Glimm Department of Applied Mathematics and Statistics, Stony Brook University Brookhaven National Laboratory Total time derivatives Eulers Equation Conservation form of equations Momentum flux Viscous Stress Tensor Incompressible Navier-Stokes Equation (3D) Two Phase NS Equations immiscible, 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 Reference for ideal fluid and gamma law author = "R. Courant and K. Friedrichs", title = "Supersonic Flow and Shock Waves", publisher = "Springer-Verlag", address = "New York", year = "1967", } EOS. Gamma law gas, Ideal EOS Derivation of EOS Gamma Proof Polytropic = gamma law EOS Specific Enthalpy i = e +PV Enthalpy for a gamma law gas 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. Characteristic Curves Isentropic gas dynamics, 1D Riemann Invariants Centered Simple Wave