# Article

Full entry | PDF   (0.3 MB)
Keywords:
existence; weak solutions; incompressible fluids; non-Newtonian fluids; pressure dependent viscosity; shear dependent viscosity; inflow/outflow boundary conditions; pressure boundary conditions; filtration boundary conditions
Summary:
We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. Suitable boundary conditions on the traction at the inflow/outflow part of boundary are given. As an advantage of this, the mean value of the pressure over the domain is no more a free parameter which would have to be prescribed otherwise. We prove the existence and uniqueness of weak solutions (the latter for small data) and discuss particular applications of the results.
References:
[1] Amrouche, C., Girault, V.: Decomposition of vector spaces and application of the Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44 (1994), 109-140. MR 1257940
[2] Bear, J.: Dynamics of Fluids in Porous Media. Elsevier New York (1972). Zbl 1191.76001
[3] Bogovskiĭ, M. E.: Solution of some problems of vector analysis associated with the operators div and grad. Tr. Semin. S. L. Soboleva 1 (1980), 5-40 Russian. MR 0631691
[4] Boyer, F., Fabrie, P.: Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete Contin. Dyn. Syst., Ser. B 7 (2007), 219-250 (electronic). MR 2276406
[5] Bruneau, C.-H.: Boundary conditions on artificial frontiers for incompressible and compressible Navier-Stokes equations. M2AN, Math. Model. Numer. Anal. 34 (2000), 303-314. DOI 10.1051/m2an:2000142 | MR 1765661 | Zbl 0954.76014
[6] Bruneau, C.-H., Fabrie, P.: Effective downstream boundary conditions for incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 19 (1994), 693-705. DOI 10.1002/fld.1650190805 | Zbl 0816.76024
[7] Bruneau, C.-H., Fabrie, P.: New efficient boundary conditions for incompressible Navier-Stokes equations: A well-posedness result. RAIRO, Modélisation Math. Anal. Numér. 30 (1996), 815-840. DOI 10.1051/m2an/1996300708151 | MR 1423081 | Zbl 0865.76016
[8] Bulíček, M., Fišerová, V.: Existence theory for steady flows of fluids with pressure and shear rate dependent viscosity, for low values of the power-law index. Z. Anal. Anwend. 28 (2009), 349-371. DOI 10.4171/ZAA/1389 | MR 2506365 | Zbl 1198.35174
[9] 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. DOI 10.1137/07069540X | MR 2515781 | Zbl 1195.35239
[10] 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. DOI 10.1512/iumj.2007.56.2997 | MR 2305930 | Zbl 1129.35055
[11] Bulíček, M., Málek, J., Rajagopal, K. R.: Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling $\nu(p,\cdot)\to+\infty$ as $p\to+\infty$. Czechoslovak Math. J. 59 (2009), 503-528. DOI 10.1007/s10587-009-0034-2 | MR 2532387 | Zbl 1224.35311
[12] Conca, C., Murat, F., Pironneau, O.: The Stokes and Navier-Stokes equations with boundary conditions involving the pressure. Japan J. Math., New Ser. 20 (1994), 279-318. DOI 10.4099/math1924.20.279 | MR 1308419 | Zbl 0826.35093
[13] Feistauer, M., Neustupa, T.: On non-stationary viscous incompressible flow through a cascade of profiles. Math. Methods Appl. Sci. 29 (2006), 1907-1941. DOI 10.1002/mma.755 | MR 2259990 | Zbl 1124.35054
[14] Filo, J., Zaušková, A.: 2D Navier-Stokes equations in a time dependent domain with Neumann type boundary conditions. J. Math. Fluid Mech. 10 (2008), 1-46. MR 2602913
[15] Forchheimer, P.: Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing. 45 (1901), 1781-1788 German.
[16] Franta, M., Málek, J., Rajagopal, K. R.: On steady flows of fluids with pressure- and shear-dependent viscosities. Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 461 (2005), 651-670. DOI 10.1098/rspa.2004.1360 | MR 2121929 | Zbl 1145.76311
[17] Hämäläinen, J., Mäkinen, R. A., Tarvainen, P.: Optimal design of paper machine headboxes. Int. J. Numer. Methods Fluids 34 (2000), 685-700. DOI 10.1002/1097-0363(20001230)34:8<685::AID-FLD75>3.0.CO;2-O
[18] Haslinger, J., Málek, J., Stebel, J.: Shape optimization in problems governed by generalised Navier-Stokes equations: existence analysis. Control Cybern. 34 (2005), 283-303. MR 2211072 | Zbl 1167.49328
[19] Hassanizadeh, S. M., Gray, W. G.: High velocity flow in porous media. Transp. in Porous Media 2 (1987), 521-531. DOI 10.1007/BF00192152
[20] Heywood, J. G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 22 (1996), 325-352. DOI 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y | MR 1380844 | Zbl 0863.76016
[21] Hron, J., Málek, J., Nečas, J., Rajagopal, K. R.: Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities. Math. Comput. Simul. 61 (2003), 297-315. DOI 10.1016/S0378-4754(02)00085-X | MR 1984133 | Zbl 1205.76159
[22] Hron, J., Málek, J., Rajagopal, K. R.: Simple flows of fluids with pressure-dependent viscosities. Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 457 (2001), 1603-1622. DOI 10.1098/rspa.2000.0723 | Zbl 1052.76017
[23] Kračmar, S., Neustupa, J.: A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions. Nonlinear Anal., Theory Methods Appl. 47 (2001), 4169-4180. DOI 10.1016/S0362-546X(01)00534-X | MR 1972357 | Zbl 1042.35605
[24] Kučera, P.: Solutions of the stationary Navier-Stokes equations with mixed boundary conditions in bounded domain. In: Analysis, Numerics and Applications of Differential and Integral Equations M. Bach et al. Pitman Res. Notes Math. Ser. 379 (1998), 127-131. MR 1606691
[25] Kučera, P., Skalák, Z.: Local solutions to the Navier-Stokes equations with mixed boundary conditions. Acta Appl. Math. 54 (1998), 275-288. DOI 10.1023/A:1006185601807 | MR 1671783
[26] Lanzendörfer, M.: On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate. Nonlinear Anal., Real World Appl. 10 (2009), 1943-1954. MR 2508405 | Zbl 1163.76335
[27] Málek, J., Nečas, J., Rajagopal, K. R.: Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165 (2002), 243-269. DOI 10.1007/s00205-002-0219-4 | MR 1941479 | Zbl 1022.76011
[28] Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Appl. Math. Math. Comput., Vol. 13. Chapman & Hall London (1996). MR 1409366
[29] Málek, J., Rajagopal, K. R.: Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. In: Evolutionary equations, Vol. II. Handbook Differential Equtions Elsevier/North Holland Amsterdam (2005), 371-459. MR 2182831 | Zbl 1095.35027
[30] Málek, J., Rajagopal, K. R.: Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate dependent viscosities. In: Handbook Math. Fluid Dyn., Vol. IV Elsevier Amsterdam (2006). MR 3929620
[31] Mikeli'c, A.: Homogenization theory and applications to filtration through porous media. Filtration in Porous Media and Industrial Application. Lect. Notes Math. Vol. 1734 A. Fasano Springer Berlin (2000). MR 1816145
[32] Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and its Applications, Vol. 27. Oxford University Press Oxford (2004). MR 2084891
[33] Quarteroni, A.: Fluid structure interaction for blood flow problems. Lecture Notes on Simulation of Fluid and Structure Interaction. AMS-AMIF Summer School, Prague European Mathematical Society Prague (2001).
[34] Shopov, P. J., Iordanov, Y. I.: Numerical solution of Stokes equations with pressure and filtration boundary conditions. J. Comput. Phys. 112 (1994), 12-23. DOI 10.1006/jcph.1994.1078 | Zbl 0798.76042

Partner of