| Title:
|
Planar flows of incompressible heat-conducting shear-thinning fluids — existence analysis (English) |
| Author:
|
Bulíček, Miroslav |
| Author:
|
Ulrych, Oldřich |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
56 |
| Issue:
|
1 |
| Year:
|
2011 |
| Pages:
|
7-38 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier's slip boundary conditions, and on the so-called $L^\infty $-truncation method, used to obtain the strong convergence of the velocity gradient. The important point of the approach consists in the choice of an appropriate form of the balance of energy. (English) |
| Keyword:
|
heat-conducting fluid |
| Keyword:
|
non-Newtonian fluid |
| Keyword:
|
shear-thinning fluid |
| Keyword:
|
existence |
| Keyword:
|
weak solution |
| Keyword:
|
suitable weak solution |
| Keyword:
|
$L^{\infty }$-truncation method |
| Keyword:
|
balance of energy |
| MSC:
|
35A01 |
| MSC:
|
35D30 |
| MSC:
|
35Q30 |
| MSC:
|
35Q35 |
| MSC:
|
35Q80 |
| MSC:
|
76A05 |
| MSC:
|
76D03 |
| idZBL:
|
Zbl 1224.35312 |
| idMR:
|
MR2807424 |
| DOI:
|
10.1007/s10492-011-0007-2 |
| . |
| Date available:
|
2011-01-03T14:47:17Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141404 |
| . |
| Reference:
|
[1] Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der mathematischen Wissenschaften, No. 223.Springer Berlin-Heidelberg-New York (1976). MR 0482275 |
| Reference:
|
[2] Boccardo, L., Dall'Aglio, A., Gallouët, T., Orsina, L.: Nonlinear parabolic equations with measure data.J. Funct. Anal. 147 (1997), 237-258. MR 1453181, 10.1006/jfan.1996.3040 |
| Reference:
|
[3] Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data.J. Funct. Anal. 87 (1989), 149-169. MR 1025884, 10.1016/0022-1236(89)90005-0 |
| Reference:
|
[4] Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations.Nonlinear Anal., Theory Methods Appl. 19 (1992), 581-597. Zbl 0783.35020, MR 1183665, 10.1016/0362-546X(92)90023-8 |
| Reference:
|
[5] Bulíček, M., Feireisl, E., Málek, J.: A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients.Nonlinear Anal., Real World Appl. 10 (2009), 992-1015. Zbl 1167.76316, MR 2474275 |
| Reference:
|
[6] Bulíček, M., Málek, J., Rajagopal, K. R.: Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries.SIAM J. Math. Anal. 41 (2009), 665-707. Zbl 1195.35239, MR 2515781, 10.1137/07069540X |
| Reference:
|
[7] Bulíček, M., Málek, J., Rajagopal, K. R.: On the need for compatibility of thermal and mechanical data in flow problems.Int. J. Eng. Sci Submitted. |
| Reference:
|
[8] Bulíček, M., Málek, J., Rajagopal, K. R.: Navier's slip and evolutionary Navier-Stokes like systems with pressure and shear-rate dependent viscosity.Indiana Univ. Math. J. 56 (2007), 51-85. Zbl 1129.35055, MR 2305930, 10.1512/iumj.2007.56.2997 |
| Reference:
|
[9] Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations.Commun. Pure Appl. Math. 35 (1982), 771-831. Zbl 0509.35067, MR 0673830, 10.1002/cpa.3160350604 |
| Reference:
|
[10] Consiglieri, L.: Weak solutions for a class of non-Newtonian fluids with energy transfer.J. Math. Fluid Mech. 2 (2000), 267-293. Zbl 0974.35090, MR 1781916, 10.1007/PL00000952 |
| Reference:
|
[11] Consiglieri, L., Rodrigues, J. F., Shilkin, T.: A limit model for unidirectional non-Newtonian flows with nonlocal viscosity.In: Trends in Partial Differential Equations of Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications 61 Birkhäuser Basel (2005), 37-44. Zbl 1080.35081, MR 2129608 |
| Reference:
|
[12] Feireisl, E., Málek, J.: On the Navier-Stokes equations with temperature-dependent transport coefficients.Differ. Equ. Nonlinear Mech., Art. ID 90616 (electronic only) (2006). Zbl 1133.35419, MR 2233755 |
| Reference:
|
[13] Frehse, J., Málek, J., Růžička, M.: Large data existence result for unsteady flows of inhomogeneous shear-thickening heat-conducting incompressible fluids.Commun. Partial Differ. Equations 35 (2010), 1891-1919. Zbl 1213.35348, MR 2754072, 10.1080/03605300903380746 |
| Reference:
|
[14] Frehse, J., Málek, J., Steinhauer, M.: On existence results for fluids with shear dependent viscosity---unsteady flows.In: Partial Differential Equations. Proc. ICM'98 Satellite Conference, Prague, Czech Republic, August 10-16, 1998. Res. Notes Math. 406 Chapman & Hall/CRC Boca Raton (2000), 121-129. MR 1713880 |
| Reference:
|
[15] Grisvard, P.: Elliptic Problems in Nonsmooth Domains.Monographs and Studies in Mathematics 24. Pitman Adv. Publishing Program Pitman Boston-London-Melbourne (1985). Zbl 0695.35060, MR 0775683 |
| Reference:
|
[16] Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs.Chapman & Hall London (1996). MR 1409366 |
| Reference:
|
[17] Málek, J., Rajagopal, K. R., Růžička, M.: Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity.Math. Models Methods Appl. Sci. 5 (1995), 789-812. MR 1348587, 10.1142/S0218202595000449 |
| Reference:
|
[18] Rajagopal, K. R.: Mechanics of non-Newtonian fluids.In: Recent Developments in Theoretical Fluid Mechanics (Paseky, 1992). Pitman Res. Notes Math. Ser. 291 Longman Scientific & Technical Harlow (1993), 129-162. Zbl 0818.76003, MR 1268237 |
| Reference:
|
[19] Tartar, L.: Compensated compactness and applications to partial differential equations.In: Nonlinear Analysis and Mechanics: Heriot-Watt Symp. Res. Notes Math. 39 Pitman Boston (1979), 121-129. Zbl 0437.35004, MR 0584398 |
| . |
