Title: AMS 691 Special Topics in Applied Mathematics Lecture 5
1AMS 691Special Topics in Applied
MathematicsLecture 5
- James Glimm
- Department of Applied Mathematics and Statistics,
- Stony Brook University
- Brookhaven National Laboratory
2Total time derivatives
3Eulers Equation
4Conservation form of equations
5Momentum flux
6Viscous Stress Tensor
7Incompressible Navier-Stokes Equation (3D)
8Two 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
9Reference 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",
10EOS. Gamma law gas, Ideal EOS
11Derivation of EOS
12Gamma
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14Proof
15Polytropic gamma law EOS
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17Specific Enthalpy i e PV
18Enthalpy for a gamma law gas
19Hugoniot curve for gamma law gas
Rarefaction waves are isentropic, so to study
them we study Isentropic gas dynamics (2x2, no
energy equation). is EOS.
20Characteristic Curves
21Isentropic gas dynamics, 1D
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23Riemann Invariants
24Centered Simple Wave