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June  2013, 33(6): 2565-2592. doi: 10.3934/dcds.2013.33.2565

Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces

Aneta Wróblewska-Kamińska 1,

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland

Received  December 2011 Revised  October 2012 Published  December 2012

Our purpose is to show existence of weak solutions to unsteady flow of non-Newtonian incompressible nonhomogeneous fluids with nonstandard growth conditions of the stress tensor. We are motivated by the fluids of anisotropic behaviour and characterised by rapid shear thickening. Since we are interested in flows with the rheology more general than power-law-type, we describe the growth conditions with the help of an $x$--dependent convex function and formulate our problem in generalized Orlicz spaces.
Keywords: incompressible fluid, generalized Minty method, Non-Newtonian fluid, weak solutions, smart fluids., Orlicz spaces, nonhomogeneous fluid.
Mathematics Subject Classification: 35Q35, 46E30, 76D0.
Citation: Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2565-2592. doi: 10.3934/dcds.2013.33.2565
References:
[1]

R. A. Adams and J. J. A. Fournier, "Sobolev Spaces," Academic Press, 2003.  Google Scholar

[2]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneus fluids, Studies in Mathematics and its Applications (translated from Russian), 22, North-Holland, Amsterdam, (1990).  Google Scholar

[3]

L. Boccardo, F. Murat and J. P. Puel, Existence of bounded solutions for non linear elliptic unilateral problems, Annali di Matematica Pura ed Applicata, 152 (1988), 183-196. doi: 10.1007/BF01766148.  Google Scholar

[4]

L. Diening, J. Málek and M. Steinhauer, On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications, ESAIM, Control, Optimisation and Calculus of Variations, 14 (2008), 211-232. doi: 10.1051/cocv:2007049.  Google Scholar

[5]

L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motion of generalized Newtonian fluids, Ann. Scuola Norm. Sup. Pisa., 9 (2010), 1-46. Google Scholar

[6]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventionse Mathematicae, 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[7]

T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations, 16 (1974), 201-256.  Google Scholar

[8]

N. Dunford and J. Schwartz, "Linear Operators," Interscience Publcation, New York, 1958.  Google Scholar

[9]

R. G. Egres Jr, Y. S. Lee, J. E. Kirkwood, K. M. Kirkwood, E. D. Wetzl and N. J. Wagner., "Liquid armour": Protective fabrics utilising shear thickening fluids, in "Proceedings of the 4th International Conference of Safety and Protective Fabrics" Pittsburg, PA, (2004), 26-27. Google Scholar

[10]

A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., 72 (2005), 410-428. doi: 10.1112/S0024610705006630.  Google Scholar

[11]

E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.  Google Scholar

[12]

E. Fernández-Cara, F. Guillén-Gonzalez and R. R. Ortega, Some theoretical results for viscoplastic and dilatant fluids with variable density, Nonlinear Anal., 28 (1997), 1079-1100. doi: 10.1016/S0362-546X(97)82861-1.  Google Scholar

[13]

J. Frehse, J. Málek and M. Růžička, Large data existence results for unsteady flows of inhomogeneus heat-conducting incompressible fluids, Communication in Partial Differential Equations, 35 (2010), 1891-1919. doi: 10.1080/03605300903380746.  Google Scholar

[14]

J. Frehse and M. Růžička, Non-homogenous generalized Newtonian fluids, Mathematische Zeitschrift, 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1.  Google Scholar

[15]

J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM Journal on Mathematical Analysis, 34 (2003), 1064-1083. doi: 10.1137/S0036141002410988.  Google Scholar

[16]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen," Akademie-Verlag, Berlin, 1974.  Google Scholar

[17]

F. Gulilén-González, Density-dependent incompressible fluids with non-Newtonian viscosity, Czechoslovak Math. J., 54 (2004), 637-656. doi: 10.1007/s10587-004-6414-8.  Google Scholar

[18]

P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 18 (2008), 1073-1092. doi: 10.1142/S0218202508002954.  Google Scholar

[19]

P. Gwiazda and A. Świerczewska-Gwiazda, On steady non-Newtonian fluids with growth conditions in generalized Orlicz spaces, Topological Methods in Nonlinear Analysis, 32 (2008), 103-114.  Google Scholar

[20]

P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Mathematical Methods in the Applied Sciences, 33 (2010), 125-137. doi: 10.1002/mma.1155.  Google Scholar

[21]

J. M. Houghton, B. A. Schiffman, D. P. Kalman, E. D. Wetzel and N. J. Wagner, "Hypodermic Needle Puncture of Shear Thickening Fluid (STF)-Treated Fabrics," Proceedings of SAMPE. Baltimore MD, 2007. Google Scholar

[22]

M. Krasnosel'skiĭ and Ya. Rutickiĭ, "Convex Functions and Orlicz Spaces," Groningen (translation), 1961.  Google Scholar

[23]

A. Kufner, O. John and S. Fučik, "Function Spaces," Noordhoff International Publishing. Prague: Publishing House of the Czechoslovak Academy of Sciences, 1977.  Google Scholar

[24]

O. A. Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Stek. Inst. Math., 102 (1967), 95-118. Google Scholar

[25]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd ed., Gordon and Breach, New York, 1969.  Google Scholar

[26]

Young S. Lee, E. D. Wetzel and N. J. Wagner, The ballistic impact characteristics of Kevlar woven fabrics impregnated with a colloidal shear thickening fluid, Journal of Materials Science, 38 (2004), 2825-2833. Google Scholar

[27]

P. L. Lions., "Mathematical Topics in Fluid Mechanics Vol. 1, Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, Oxford Science Publications, New York, 1996.  Google Scholar

[28]

J. Málek, J. Nečas and M. Růžička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci., 3 (1993), 35-63. doi: 10.1142/S0218202593000047.  Google Scholar

[29]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996. Google Scholar

[30]

J. Málek, K. R. Rajagopal and M. Růžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models and Methods in Applied Sciences, 5 (1995), 789-812. doi: 10.1142/S0218202595000449.  Google Scholar

[31]

J. Musielak, "Orlicz Spaces and Modular Spaces," 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983.  Google Scholar

[32]

V. Mustonen and M. Tienari, On monotone-like mappings in Orlicz-Sobolev spaces, Math. Bohem., 124 (1999), 255-271.  Google Scholar

[33]

K. R. Rajagopal and M. Růžička, On the modeling of electrorheological materials, Mech. Research Comm., 23 (1996), 401-407. Google Scholar

[34]

K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Continuum Mechanics and Thermodynamics, 13 (2001), 59-78. Google Scholar

[35]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991.  Google Scholar

[36]

T. Roubiček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 2005.  Google Scholar

[37]

R. T. Rockaffellar, "Convex Analysis," Princton University Press, 1970.  Google Scholar

[38]

M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[39]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat Pura Appl., VI Serie., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[40]

M. S. Skaff, Vector valued Orlicz spaces generalized N-Functions, I, Pacific Journal of Mathematics, 28 (1969), 193-206. Google Scholar

[41]

M. S. Skaff, Vector valued Orlicz spaces, II, Pacific Journal of Mathematics, 28 (1969), 413-430.  Google Scholar

[42]

B. Turett, Fenchel-Orlicz spaces, Dissertationes Math. (Rozprawy Mat.), 181 (1980), 55.  Google Scholar

[43]

J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mechanics, 9 (2007), 104-138. doi: 10.1007/s00021-006-0219-5.  Google Scholar

[44]

A. Wróblewska, Steady flow of non-Newtonian fluids - monotonicity methods in generalized Orlicz spaces, Nonlinear Analysis, 72 (2010), 4136-4147. doi: 10.1016/j.na.2010.01.045.  Google Scholar

[45]

A. Wróblewska, "Steady Flow of Non-Newtonian Fluids with Growth Conditions in Orlicz Spaces," Diploma theses at Faculty of Mathematics, Informatics and Mechanics University of Warsaw, 2008. Google Scholar

[46]

C. S. Yeh and K. C. Chen, A thermodynamic model for magnetorheological fluids, Continnum Mech. Thermodyn., 9 (1997), 273-291. doi: 10.1007/s001610050071.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. A. Fournier, "Sobolev Spaces," Academic Press, 2003.  Google Scholar

[2]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneus fluids, Studies in Mathematics and its Applications (translated from Russian), 22, North-Holland, Amsterdam, (1990).  Google Scholar

[3]

L. Boccardo, F. Murat and J. P. Puel, Existence of bounded solutions for non linear elliptic unilateral problems, Annali di Matematica Pura ed Applicata, 152 (1988), 183-196. doi: 10.1007/BF01766148.  Google Scholar

[4]

L. Diening, J. Málek and M. Steinhauer, On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications, ESAIM, Control, Optimisation and Calculus of Variations, 14 (2008), 211-232. doi: 10.1051/cocv:2007049.  Google Scholar

[5]

L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motion of generalized Newtonian fluids, Ann. Scuola Norm. Sup. Pisa., 9 (2010), 1-46. Google Scholar

[6]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventionse Mathematicae, 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[7]

T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations, 16 (1974), 201-256.  Google Scholar

[8]

N. Dunford and J. Schwartz, "Linear Operators," Interscience Publcation, New York, 1958.  Google Scholar

[9]

R. G. Egres Jr, Y. S. Lee, J. E. Kirkwood, K. M. Kirkwood, E. D. Wetzl and N. J. Wagner., "Liquid armour": Protective fabrics utilising shear thickening fluids, in "Proceedings of the 4th International Conference of Safety and Protective Fabrics" Pittsburg, PA, (2004), 26-27. Google Scholar

[10]

A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., 72 (2005), 410-428. doi: 10.1112/S0024610705006630.  Google Scholar

[11]

E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.  Google Scholar

[12]

E. Fernández-Cara, F. Guillén-Gonzalez and R. R. Ortega, Some theoretical results for viscoplastic and dilatant fluids with variable density, Nonlinear Anal., 28 (1997), 1079-1100. doi: 10.1016/S0362-546X(97)82861-1.  Google Scholar

[13]

J. Frehse, J. Málek and M. Růžička, Large data existence results for unsteady flows of inhomogeneus heat-conducting incompressible fluids, Communication in Partial Differential Equations, 35 (2010), 1891-1919. doi: 10.1080/03605300903380746.  Google Scholar

[14]

J. Frehse and M. Růžička, Non-homogenous generalized Newtonian fluids, Mathematische Zeitschrift, 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1.  Google Scholar

[15]

J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM Journal on Mathematical Analysis, 34 (2003), 1064-1083. doi: 10.1137/S0036141002410988.  Google Scholar

[16]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen," Akademie-Verlag, Berlin, 1974.  Google Scholar

[17]

F. Gulilén-González, Density-dependent incompressible fluids with non-Newtonian viscosity, Czechoslovak Math. J., 54 (2004), 637-656. doi: 10.1007/s10587-004-6414-8.  Google Scholar

[18]

P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 18 (2008), 1073-1092. doi: 10.1142/S0218202508002954.  Google Scholar

[19]

P. Gwiazda and A. Świerczewska-Gwiazda, On steady non-Newtonian fluids with growth conditions in generalized Orlicz spaces, Topological Methods in Nonlinear Analysis, 32 (2008), 103-114.  Google Scholar

[20]

P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Mathematical Methods in the Applied Sciences, 33 (2010), 125-137. doi: 10.1002/mma.1155.  Google Scholar

[21]

J. M. Houghton, B. A. Schiffman, D. P. Kalman, E. D. Wetzel and N. J. Wagner, "Hypodermic Needle Puncture of Shear Thickening Fluid (STF)-Treated Fabrics," Proceedings of SAMPE. Baltimore MD, 2007. Google Scholar

[22]

M. Krasnosel'skiĭ and Ya. Rutickiĭ, "Convex Functions and Orlicz Spaces," Groningen (translation), 1961.  Google Scholar

[23]

A. Kufner, O. John and S. Fučik, "Function Spaces," Noordhoff International Publishing. Prague: Publishing House of the Czechoslovak Academy of Sciences, 1977.  Google Scholar

[24]

O. A. Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Stek. Inst. Math., 102 (1967), 95-118. Google Scholar

[25]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd ed., Gordon and Breach, New York, 1969.  Google Scholar

[26]

Young S. Lee, E. D. Wetzel and N. J. Wagner, The ballistic impact characteristics of Kevlar woven fabrics impregnated with a colloidal shear thickening fluid, Journal of Materials Science, 38 (2004), 2825-2833. Google Scholar

[27]

P. L. Lions., "Mathematical Topics in Fluid Mechanics Vol. 1, Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, Oxford Science Publications, New York, 1996.  Google Scholar

[28]

J. Málek, J. Nečas and M. Růžička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci., 3 (1993), 35-63. doi: 10.1142/S0218202593000047.  Google Scholar

[29]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996. Google Scholar

[30]

J. Málek, K. R. Rajagopal and M. Růžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models and Methods in Applied Sciences, 5 (1995), 789-812. doi: 10.1142/S0218202595000449.  Google Scholar

[31]

J. Musielak, "Orlicz Spaces and Modular Spaces," 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983.  Google Scholar

[32]

V. Mustonen and M. Tienari, On monotone-like mappings in Orlicz-Sobolev spaces, Math. Bohem., 124 (1999), 255-271.  Google Scholar

[33]

K. R. Rajagopal and M. Růžička, On the modeling of electrorheological materials, Mech. Research Comm., 23 (1996), 401-407. Google Scholar

[34]

K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Continuum Mechanics and Thermodynamics, 13 (2001), 59-78. Google Scholar

[35]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991.  Google Scholar

[36]

T. Roubiček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 2005.  Google Scholar

[37]

R. T. Rockaffellar, "Convex Analysis," Princton University Press, 1970.  Google Scholar

[38]

M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[39]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat Pura Appl., VI Serie., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[40]

M. S. Skaff, Vector valued Orlicz spaces generalized N-Functions, I, Pacific Journal of Mathematics, 28 (1969), 193-206. Google Scholar

[41]

M. S. Skaff, Vector valued Orlicz spaces, II, Pacific Journal of Mathematics, 28 (1969), 413-430.  Google Scholar

[42]

B. Turett, Fenchel-Orlicz spaces, Dissertationes Math. (Rozprawy Mat.), 181 (1980), 55.  Google Scholar

[43]

J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mechanics, 9 (2007), 104-138. doi: 10.1007/s00021-006-0219-5.  Google Scholar

[44]

A. Wróblewska, Steady flow of non-Newtonian fluids - monotonicity methods in generalized Orlicz spaces, Nonlinear Analysis, 72 (2010), 4136-4147. doi: 10.1016/j.na.2010.01.045.  Google Scholar

[45]

A. Wróblewska, "Steady Flow of Non-Newtonian Fluids with Growth Conditions in Orlicz Spaces," Diploma theses at Faculty of Mathematics, Informatics and Mechanics University of Warsaw, 2008. Google Scholar

[46]

C. S. Yeh and K. C. Chen, A thermodynamic model for magnetorheological fluids, Continnum Mech. Thermodyn., 9 (1997), 273-291. doi: 10.1007/s001610050071.  Google Scholar

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