Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces

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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

Abstract
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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 and 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.
[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).
[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.
[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.
[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.
[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.
[7]	T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations, 16 (1974), 201-256.
[8]	N. Dunford and J. Schwartz, "Linear Operators," Interscience Publcation, New York, 1958.
[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.
[10]	A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., 72 (2005), 410-428. doi: 10.1112/S0024610705006630.
[11]	E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.
[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.
[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.
[14]	J. Frehse and M. Růžička, Non-homogenous generalized Newtonian fluids, Mathematische Zeitschrift, 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1.
[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.
[16]	H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen," Akademie-Verlag, Berlin, 1974.
[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.
[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.
[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.
[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.
[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.
[22]	M. Krasnosel'skiĭ and Ya. Rutickiĭ, "Convex Functions and Orlicz Spaces," Groningen (translation), 1961.
[23]	A. Kufner, O. John and S. Fučik, "Function Spaces," Noordhoff International Publishing. Prague: Publishing House of the Czechoslovak Academy of Sciences, 1977.
[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.
[25]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd ed., Gordon and Breach, New York, 1969.
[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.
[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.
[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.
[29]	J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996.
[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.
[31]	J. Musielak, "Orlicz Spaces and Modular Spaces," 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983.
[32]	V. Mustonen and M. Tienari, On monotone-like mappings in Orlicz-Sobolev spaces, Math. Bohem., 124 (1999), 255-271.
[33]	K. R. Rajagopal and M. Růžička, On the modeling of electrorheological materials, Mech. Research Comm., 23 (1996), 401-407.
[34]	K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Continuum Mechanics and Thermodynamics, 13 (2001), 59-78.
[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.
[36]	T. Roubiček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 2005.
[37]	R. T. Rockaffellar, "Convex Analysis," Princton University Press, 1970.
[38]	M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.
[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.
[40]	M. S. Skaff, Vector valued Orlicz spaces generalized N-Functions, I, Pacific Journal of Mathematics, 28 (1969), 193-206.
[41]	M. S. Skaff, Vector valued Orlicz spaces, II, Pacific Journal of Mathematics, 28 (1969), 413-430.
[42]	B. Turett, Fenchel-Orlicz spaces, Dissertationes Math. (Rozprawy Mat.), 181 (1980), 55.
[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.
[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.
[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.
[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.

show all references

References:

[1]	R. A. Adams and J. J. A. Fournier, "Sobolev Spaces," Academic Press, 2003.
[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).
[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.
[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.
[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.
[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.
[7]	T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations, 16 (1974), 201-256.
[8]	N. Dunford and J. Schwartz, "Linear Operators," Interscience Publcation, New York, 1958.
[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.
[10]	A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., 72 (2005), 410-428. doi: 10.1112/S0024610705006630.
[11]	E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.
[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.
[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.
[14]	J. Frehse and M. Růžička, Non-homogenous generalized Newtonian fluids, Mathematische Zeitschrift, 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1.
[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.
[16]	H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen," Akademie-Verlag, Berlin, 1974.
[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.
[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.
[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.
[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.
[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.
[22]	M. Krasnosel'skiĭ and Ya. Rutickiĭ, "Convex Functions and Orlicz Spaces," Groningen (translation), 1961.
[23]	A. Kufner, O. John and S. Fučik, "Function Spaces," Noordhoff International Publishing. Prague: Publishing House of the Czechoslovak Academy of Sciences, 1977.
[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.
[25]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd ed., Gordon and Breach, New York, 1969.
[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.
[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.
[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.
[29]	J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996.
[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.
[31]	J. Musielak, "Orlicz Spaces and Modular Spaces," 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983.
[32]	V. Mustonen and M. Tienari, On monotone-like mappings in Orlicz-Sobolev spaces, Math. Bohem., 124 (1999), 255-271.
[33]	K. R. Rajagopal and M. Růžička, On the modeling of electrorheological materials, Mech. Research Comm., 23 (1996), 401-407.
[34]	K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Continuum Mechanics and Thermodynamics, 13 (2001), 59-78.
[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.
[36]	T. Roubiček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 2005.
[37]	R. T. Rockaffellar, "Convex Analysis," Princton University Press, 1970.
[38]	M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.
[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.
[40]	M. S. Skaff, Vector valued Orlicz spaces generalized N-Functions, I, Pacific Journal of Mathematics, 28 (1969), 193-206.
[41]	M. S. Skaff, Vector valued Orlicz spaces, II, Pacific Journal of Mathematics, 28 (1969), 413-430.
[42]	B. Turett, Fenchel-Orlicz spaces, Dissertationes Math. (Rozprawy Mat.), 181 (1980), 55.
[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.
[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.
[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.
[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.

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