AMS 691 Special Topics in Applied Mathematics Lecture 5

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AMS 691Special Topics in Applied
MathematicsLecture 5

• James Glimm
• Department of Applied Mathematics and Statistics,
• Stony Brook University
• Brookhaven National Laboratory

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Total time derivatives
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Eulers Equation
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Conservation form of equations
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Momentum flux
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Viscous Stress Tensor
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Incompressible Navier-Stokes Equation (3D)
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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

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Reference for ideal fluid andgamma law EOS
_at_BookCouFri67, author "R. Courant and
K. Friedrichs", title "Supersonic Flow
and Shock Waves", publisher
"Springer-Verlag", address "New York",
year "1967",
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EOS. Gamma law gas, Ideal EOS
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Derivation of EOS
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Gamma
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Proof
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Polytropic gamma law EOS
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Specific Enthalpy i e PV
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Enthalpy for a gamma law gas
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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.
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Characteristic Curves
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Isentropic gas dynamics, 1D
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Riemann Invariants
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Centered Simple Wave