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

Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces

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.
Citation: Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 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 (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. 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, 14 (2008), 211. 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.

[6]

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

[7]

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

[8]

N. Dunford and J. Schwartz, "Linear Operators,", Interscience Publcation, (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, (2004), 26.

[10]

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

[11]

E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids,", Birkhäuser, (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. 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. doi: 10.1080/03605300903380746.

[14]

J. Frehse and M. Růžička, Non-homogenous generalized Newtonian fluids,, Mathematische Zeitschrift, 260 (2008), 355. 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. doi: 10.1137/S0036141002410988.

[16]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen,", Akademie-Verlag, (1974).

[17]

F. Gulilén-González, Density-dependent incompressible fluids with non-Newtonian viscosity,, Czechoslovak Math. J., 54 (2004), 637. 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. 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.

[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. 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.

[25]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", 2nd ed., (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.

[27]

P. L. Lions., "Mathematical Topics in Fluid Mechanics Vol. 1, Incompressible Models,", Oxford Lecture Series in Mathematics and its Applications, (1996).

[28]

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

[29]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-valued Solutions to Evolutionary PDEs,", Chapman & Hall, (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. doi: 10.1142/S0218202595000449.

[31]

J. Musielak, "Orlicz Spaces and Modular Spaces,", 1034 of Lecture Notes in Mathematics, 1034 (1983).

[32]

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

[33]

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

[34]

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

[35]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces,", Monographs and Textbooks in Pure and Applied Mathematics, 146 (1991).

[36]

T. Roubiček, "Nonlinear Partial Differential Equations with Applications,", Birkhäuser, (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, 1748 (2000). doi: 10.1007/BFb0104029.

[39]

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

[40]

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

[41]

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

[42]

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

[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. 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. 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, (2008).

[46]

C. S. Yeh and K. C. Chen, A thermodynamic model for magnetorheological fluids,, Continnum Mech. Thermodyn., 9 (1997), 273. 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 (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. 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, 14 (2008), 211. 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.

[6]

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

[7]

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

[8]

N. Dunford and J. Schwartz, "Linear Operators,", Interscience Publcation, (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, (2004), 26.

[10]

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

[11]

E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids,", Birkhäuser, (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. 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. doi: 10.1080/03605300903380746.

[14]

J. Frehse and M. Růžička, Non-homogenous generalized Newtonian fluids,, Mathematische Zeitschrift, 260 (2008), 355. 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. doi: 10.1137/S0036141002410988.

[16]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen,", Akademie-Verlag, (1974).

[17]

F. Gulilén-González, Density-dependent incompressible fluids with non-Newtonian viscosity,, Czechoslovak Math. J., 54 (2004), 637. 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. 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.

[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. 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.

[25]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", 2nd ed., (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.

[27]

P. L. Lions., "Mathematical Topics in Fluid Mechanics Vol. 1, Incompressible Models,", Oxford Lecture Series in Mathematics and its Applications, (1996).

[28]

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

[29]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-valued Solutions to Evolutionary PDEs,", Chapman & Hall, (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. doi: 10.1142/S0218202595000449.

[31]

J. Musielak, "Orlicz Spaces and Modular Spaces,", 1034 of Lecture Notes in Mathematics, 1034 (1983).

[32]

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

[33]

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

[34]

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

[35]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces,", Monographs and Textbooks in Pure and Applied Mathematics, 146 (1991).

[36]

T. Roubiček, "Nonlinear Partial Differential Equations with Applications,", Birkhäuser, (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, 1748 (2000). doi: 10.1007/BFb0104029.

[39]

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

[40]

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

[41]

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

[42]

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

[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. 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. 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, (2008).

[46]

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

[1]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212

[2]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417

[3]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138

[4]

Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068

[5]

Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231

[6]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503

[7]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483

[8]

Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010

[9]

Horst Heck, Gunther Uhlmann, Jenn-Nan Wang. Reconstruction of obstacles immersed in an incompressible fluid. Inverse Problems & Imaging, 2007, 1 (1) : 63-76. doi: 10.3934/ipi.2007.1.63

[10]

Youcef Amirat, Kamel Hamdache. On a heated incompressible magnetic fluid model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 675-696. doi: 10.3934/cpaa.2012.11.675

[11]

Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255

[12]

D. L. Denny. Existence of solutions to equations for the flow of an incompressible fluid with capillary effects. Communications on Pure & Applied Analysis, 2004, 3 (2) : 197-216. doi: 10.3934/cpaa.2004.3.197

[13]

Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations & Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331

[14]

Youcef Amirat, Kamel Hamdache. Weak solutions to stationary equations of heat transfer in a magnetic fluid. Communications on Pure & Applied Analysis, 2019, 18 (2) : 709-734. doi: 10.3934/cpaa.2019035

[15]

María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012

[16]

Scott W. Hansen, Andrei A. Lyashenko. Exact controllability of a beam in an incompressible inviscid fluid. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 59-78. doi: 10.3934/dcds.1997.3.59

[17]

I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759

[18]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024

[19]

Bo-Qing Dong, Zhi-Min Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 765-784. doi: 10.3934/dcds.2009.23.765

[20]

Emil Novruzov. On existence and nonexistence of the positive solutions of non-newtonian filtration equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 719-730. doi: 10.3934/cpaa.2011.10.719

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]