AMS 691 Special Topics in Applied Mathematics Lecture 5 - PowerPoint PPT Presentation

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AMS 691 Special Topics in Applied Mathematics Lecture 5

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Title: 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
Total time derivatives
3
Eulers Equation
4
Conservation form of equations
5
Momentum flux
6
Viscous Stress Tensor
7
Incompressible Navier-Stokes Equation (3D)
8
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

9
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",
10
EOS. Gamma law gas, Ideal EOS
11
Derivation of EOS
12
Gamma
13
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14
Proof
15
Polytropic gamma law EOS
16
(No Transcript)
17
Specific Enthalpy i e PV
18
Enthalpy for a gamma law gas
19
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.
20
Characteristic Curves
21
Isentropic gas dynamics, 1D
22
(No Transcript)
23
Riemann Invariants
24
Centered Simple Wave
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