self-similar solution of the three dimensional compressible navier-stokes equation s
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Self-Similar Solution of the three dimensional compressible Navier-Stokes Equation s. Imre Ferenc Barna. Center for Energy Research (EK) of the Hungarian Academy of Sciences. Outline. Solutions of PDEs self-similar , traveling wave non-compressible Navier - Stokes e quation - PowerPoint PPT PresentationTRANSCRIPT
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Self-Similar Solution of the three dimensional
compressible Navier-Stokes Equations
Imre Ferenc Barna
Center for Energy Research (EK) of the Hungarian Academy of Sciences
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Outline
• Solutions of PDEs self-similar, traveling wave
• non-compressible Navier-Stokes equation
with my 3D Ansatz & geometry my solution + other
solutions, replay from last year
• compressible Navier-Stokes equation with the same Ansatz, some part of the solutions, traveling wave analysis
• Summary & Outlook more EOS & viscosity functions
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Physically important solutions of PDEs
- Travelling waves: arbitrary wave fronts u(x,t) ~ g(x-ct), g(x+ct) - Self-similar
Sedov, Barenblatt, Zeldovich
in Fourier heat-conduction
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The non-compressible Navier-Stokes equation
3 dimensional cartesian coordinates, Euler description v velocity field, p pressure, a external field kinematic viscosity, constant density Newtonian fluid
just to write outall the coordinates
Consider the most general case
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My 3 dimensional Ansatz
A more general function does not work for N-S
The final applied forms
Geometrical meaning: all v components with coordinate constrain x+y+z=0lie in a plane = equivalent
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The obtained ODE system
as constraints we got for the exponents:
universality relations
Continuity eq. helps us to get an additional constraint:
c is prop. to mass flow rate
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Solutions of the ODE
a single Eq. remains
Kummer is spec.
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Solutions of N-S
I.F. Barna http://arxiv.org/abs/1102.5504Commun. Theor. Phys. 56 (2011) 745-750
analytic only for one velocity component
Geometrical explanation: Naver-Stokes makes a getting a multi-valued surfacedynamics of this planeall v components with
coordinate constrain x+y+z=0lie in a plane = equivalent
for fixed space it decays in timet^-1/2 KummerT or U(1/t)
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Other analytic solutions Without completeness, usually from Lie algebra studies all are for non-compressible N-S
Presented 19 various solutionsone of them is:
Sedov, stationary N-S, only the angular part
Solutions are Kummer functions as well
Ansatz:
“Only” Radial solution for 2 or 3 D
Ansatz:
Ansatz:
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The compressible Navier-Stokes eq.
3 dimensional cartesian coordinates, Euler description, Newtonian fluid, politropic EOS (these can be
changed later) v velocity field, p pressure, a external field viscosities, density
just write outall the coordinates:
Consider the most general case:
(No temperature at this point)
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The applied Ansatz & Universality Relations
A more general function does not work for N-S
Note, that n remains free, presenting some physics in the system, polytropic EOS
as constraints we got for the exponents:universality relations
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The obtained ODE system
The most general case, n is free
Continuity can be integrated
N-S can be intergated once, after some algebragetting an ODE of:
No analytic solutions exist , but the direction field can be investigated for reasonable parameters
This is for the density
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The properties of the solutions
From the universality relations the global properties of the solutions are known
has decay & spreading just spreading in time for fixed space
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General properties of the solutions for other
exponents
`There are different regimes for different ns
n > 1 all exponents are positive decaying, spreading solutions for speed and densityn = 1 see above-1 ≤ n ≤ +1 decaying and spreading density & enhancing velocity in timen ≠ -1n ≤ -1 sharpening and enhancing density & decaying and sharpening velocity
Relevant physics is for n >1 the analysis is in progress to see the shape functions
en
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Traveling wave solutions
where C is the wave velocity
After some algebra the next ODE can be obtained: (for n = 1)
Detailed analysis is in progress
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Summary & Outlook• The self-similar Ansatz is presented as a tool for non-
linear PDA• The non-compressible & compressible N-S eq. is
investigated and the results are discussed
• An in-depth analysis is in progress for further EOS, more general viscosity functions could be analysed like the Ostwald-de Waele power law
• To investigate some relativistic cases, which may attracts the interest of the recent community
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Questions, Remarks, Comments?…