Stationary flows of shear thickening fluids in 2D. (English) Zbl 1294.35097
Summary: We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on a domain in \({\mathbb{R}^{2}}\). We assume that the stress tensor is generated by a potential of the form \({H = h (|{\varepsilon} (u)|)}\), \(\varepsilon (u)\) denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions \(h\) having the property that \(h'(t)/t\) increases (shear thickening case).
Keywords:
generalized Newtonian fluids; equations of Navier-Stokes type; steady flows; non-uniformly elliptic systems; korn inequalities in Orlicz spacesReferences:
| [1] | Adams R.A.: Sobolev Spaces. Academic Press, New York (1975) · Zbl 0314.46030 |
| [2] | Anzellotti G., Giaquinta M.: Existence of the displacement field for an elasto-plastic body subject to Hencky’s law and von Mises yield condition. Manuscr. Math. 32, 101–136 (1980) · Zbl 0465.73022 · doi:10.1007/BF01298185 |
| [3] | Apushkinskaya D., Bildhauer M., Fuchs M.: Steady states of anisotropic generalized Newtonian fluids. J. Math. Fluid Mech. 7(2), 261–297 (2005) · Zbl 1162.76375 · doi:10.1007/s00021-004-0118-6 |
| [4] | Beirão da Veiga H.: On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip and non-slip boundary conditions. Commun. Pure Appl. Math. 58, 552–577 (2005) · Zbl 1075.35045 · doi:10.1002/cpa.20036 |
| [5] | Beirão da Veiga, H.: Navier–Stokes equations with shear thickening viscosity. Regularity up to the boundary. J. Math. Fluid Mech. (2008). doi: 10.1007/s00021-008-0257-2 · Zbl 1213.76047 |
| [6] | Beirão da Veiga, H.: A note on the global integrability, for any finite power, of the full gradient for a class of power models, p < 2. In: Advances in Mathematical Fluid Mechanics, pp. 37–42. Springer, Berlin (2010) · Zbl 1374.35278 |
| [7] | Beirãoda Veiga H., Kaplický P., Ružička M.: Regularity theorems, up to the boundary, for shear thickening fluids . Comp. Rend. Math. 348(9–10), 541–544 (2010) · Zbl 1352.35085 · doi:10.1016/j.crma.2010.04.010 |
| [8] | Bildhauer M., Fuchs M.: Differentiability and higher integrability results for local minimizers of splitting type variational integrals in 2D with applications to nonlinear Hencky-materials. Calc. Var. 37(1–2), 167–186 (2010) · Zbl 1189.49057 · doi:10.1007/s00526-009-0257-y |
| [9] | Bildhauer M., Fuchs M., Zhong X.: A lemma on the higher integrability of functions with applications to the regularity theory of two dimensional generalized Newtonian fluids. Manuscr. Math. 116(2), 135–156 (2005) · Zbl 1116.49018 · doi:10.1007/s00229-004-0523-4 |
| [10] | Breit, D., Fuchs, M.: The nonlinear Stokes problem with general potentials having superquadratic growth. J. Math. Fluid Mech. (2010). doi: 10.1007/s00021-010-0023-0 · Zbl 1270.35335 |
| [11] | Byun S.-S., Yao F., Zhou S.: Gradient estimates in Orlicz space for nonlinear elliptic equations. J. Funct. Anal. 255, 1851–1873 (2008) · Zbl 1156.35038 · doi:10.1016/j.jfa.2008.09.007 |
| [12] | Frehse J., Málek J., Steinhauer M.: On analysis of steady flows of fluids with shear dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34(5), 1064–1083 (2003) · Zbl 1050.35080 · doi:10.1137/S0036141002410988 |
| [13] | Fuchs M.: A note on non-uniformly elliptic Stokes-type systems in two variables. J. Math. Fluid Mech. 12, 266–279 (2010) · Zbl 1195.76148 · doi:10.1007/s00021-008-0285-y |
| [14] | Fuchs M.: Korn inequalities in Orlicz spaces. Irish Math. Soc. Bull. 65, 5–9 (2010) · Zbl 1237.26013 |
| [15] | Galdi, G.: An introduction to the mathematical theory of the Navier-Stokes equations vol. I. Springer Tracts in Natural Philosophy vol. 38. Springer, Berlin (1994) · Zbl 0949.35004 |
| [16] | Galdi, G.: An introduction to the mathematical theory of the Navier–Stokes equations vol. II. Springer Tracts in Natural Philosophy, vol. 39. Springer, Berlin (1994) · Zbl 0949.35004 |
| [17] | Jia H., Li D., Wang L.: Regularity theory in Orlicz spaces for elliptic equations in Reifenberg domains. J. Math. Anal. Appl. 334, 804–817 (2007) · Zbl 1178.35154 · doi:10.1016/j.jmaa.2006.12.081 |
| [18] | Kaplický P.: Regularity of flow of anisotropic fluid. J. Math. Fluid Mech. 10, 71–88 (2008) · Zbl 1162.76304 · doi:10.1007/s00021-006-0217-7 |
| [19] | Kaplický P., Málek J., Stará J.: C 1,{\(\alpha\)} -solutions to a class of nonlinear fluids in two dimensions–stationary Dirichlet problem. Zapiski Nauchnyh Seminarow POMI 259, 122–144 (1999) |
| [20] | Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. Gordon And Breach, New York (1969) · Zbl 0184.52603 |
| [21] | Málek, J., Necas, J., Rokyta, M., Ružička, M.: Weak and Measure Valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996) |
| [22] | Mosolov P.P., Mjasnikov V.P.: On the correctness of boundary value problems in the mechanics of continuous media. Math. USSR Sbornik 17(2), 257–267 (1972) · Zbl 0264.73011 · doi:10.1070/SM1972v017n02ABEH001503 |
| [23] | Rao M.M., Ren Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991) · Zbl 0724.46032 |
| [24] | Reshetnyak Y.G.: Estimates for certain differential operators with finite dimensional kernel. Sibirsk. Mat. Zh. 11, 414–428 (1970) · Zbl 0233.35010 |
| [25] | Wolf J.: Interior C 1, {\(\alpha\)}-regularity of weak solutions to the equations of stationary motion to certain non-Newtonian fluids in two dimensions. Boll U.M.I.(8) 10B, 317–340 (2007) · Zbl 1140.76007 |
| [26] | Zhou, S.: Private communication |
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