1. Self-similar solutions for the two dimensional non- Newtonian Navier-Stokes equation Imre Ferenc Barna Wigner Research Center of the Hungarian Academy of Sciences 2 Dec. 2013 Zimányi School’13

2. Outline Solutions of PDEs self-similar, traveling wave Equations of various dissipative flows & my2/ 3D Ansatz and geometr y my solutions, replay from last years non-newtonian Navier-Stokes e quation with the same Ansatz, some part of the solutions Summary & Outlook

3. 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

4. Physically important solutions of PDEs II - generalised self-similar: - one case: blow-up solution: goes to infinity in finite time other forms are available too - additional: additive separable multiplicative separable generalised separable eg.

5. Various dissipative fluid equations Non-compressible and Newtonian (I) Compressible and Newtonian (II) Non-compressible and Non- Newtonian (III) Compressible and Non-Newtonian tooo complicated, we will see… 0 Euler description; Cartesian coordinate; v velocity field, p pressure, a external field viscosity, density, kompressibility, E stress tensor

6. 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=0 lie in a plane = equivalent

7. The non-compressible Newtonian Navier-Stokes equation(I) just to write out all the coordinates: Considering the most general Cartesian case: as constraints the exponents: universality relations, are all fixed no free paramater(s)

8. Solutions of the ODE a single Eq. remains Kummer is spec. func.

9. Solutions of N-S I.F. Barna http://arxiv.org/abs/1102.5504 Commun. Theor. Phys. 56 (2011) 745-750 analytic only for one velocity component  Geometrical explanation: Naver-Stokes makes a getting a multi-valued surface dynamics of this plane all v components with coordinate constrain x+y+z=0 lie in a plane = equivalent for fixed space it decays in time t^-1/2 KummerT or U(1/t)

10. Other analytic solutions Without completeness, usually from Lie algebra studies all are for non-compressible N-S Presented 19 various solutions one 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:

11. The compressible Newtonian Navier-Stokes eq. (II) most general case: polytropic EOS as constraints the exponents are not fixed: universality relations are depend on the EOS exponent ` There are different regimes for different ns n > 1 all exponents are positive decaying, spreading solutions for speed and density n = 1 see above -1 ≤ n ≤ +1 decaying and spreading density & enhancing velocity in time n ≠ -1 n ≤ -1 sharpening and enhancing density & decaying and sharpening velocity Relevant physics is for n >1 en `

12. The compressible Newtonian Navier-Stokes eq. (II) there is an ODE for the density with analytic solutions for any n BUT more important the ODE for the velocity field: For n =1/2 and n= 3/2 HeunT functions tooo elaborate BUT for n=2 we have the Whitakker functions The solution for the non-compressible case There is no kappa 0 limit to compare the results

13. The compressible Newtonian Navier-Stokes eq. (II) Solutions for the same parameters red is the non/compressible and blue is the compresible solution nu = 0.1, kappa =1, c1 = 1 the compresible solutions for kappa = 0.1,1,2 values red, blue green I.F. Barna http://arxiv.org/abs/1309.0703 & under publicationhttp://arxiv.org/abs/1309.0703

14. The non-compressible non-newtonian Navier-Stokes eq. for 2D (III) Abs value of the tensor in 2D, in 3D much longer There are existence theorems for weak solutions for some rs Our Ansatz:

15. The non-compressible non-newtonian Navier-Stokes eq. for 2D (III) For C1 =0, mu_1 = 0.1 under heavy investigation Large number of solutions (20), some of them are explicit with compact support

16. The non-compressible non-newtonian Navier-Stokes eq. for 2D (III) for r = 0 we get back the non-newtonian ODE with the Kummer M/W solutions but for 2D An important message: the non-Newtonian contribution gives „more localised” velocity distributions then tha Newtonian

17. Summary & Outlook The self-similar Ansatz is presented as a tool for non-linear PDA The non-compressible N-S eq. & compressible N-S eq. & non-Newtonian non-compressible N-S are investigated The non-compressible N-S is the „simplest case” The other two systems have more „localized” or even compact support solutions which are important Clear out the dark points, include heat conduction

18. Questions, Remarks, Comments?…