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March  2019, 18(2): 735-750. doi: 10.3934/cpaa.2019036

Steady flows of an Oldroyd fluid with threshold slip

Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, Voronezh, 394018, Russia

Received  February 2018 Revised  May 2018 Published  October 2018

Fund Project: This work was supported by the Russian Foundation for Basic Research, project no. 16-31- 00182 mol_a.

We consider a mathematical model that describes 3D steady flows of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain under mixed boundary conditions, including a threshold-slip boundary condition. Using the concept of weak solutions, we reduce the original slip problem to a coupled system of variational inequalities and equations for the velocity field and stresses. For arbitrary large data (forcing and boundary data) and suitable material constants, we prove the existence of weak solutions and establish some of their properties.

Citation: Evgenii S. Baranovskii. Steady flows of an Oldroyd fluid with threshold slip. Communications on Pure & Applied Analysis, 2019, 18 (2) : 735-750. doi: 10.3934/cpaa.2019036
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$ ^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

Yu. Ya. Agranovich and P. E. Sobolevskii, Motion of nonlinear visco-elastic fluid, Nonlinear Anal., 32 (1998), 755-760.  doi: 10.1016/S0362-546X(97)00519-1.  Google Scholar

[3]

M. A. Artemov and E. S. Baranovskii, Mixed boundary-value problems for motion equations of a viscoelastic medium, Electron. J. Differ. Equ., 2015 (2015), Article No. 252.  Google Scholar

[4]

E. S. Baranovskii, On steady motion of viscoelastic fluid of Oldroyd type, Sb. Math., 205 (2014), 763-776.  doi: 10.1070/SM2014v205n06ABEH004397.  Google Scholar

[5]

E. S. Baranovskii, Optimal control for steady flows of the Jeffreys fluids with slip boundary condition, J. Appl. Ind. Math., 8 (2014), 168-176.  doi: 10.1134/S1990478914020033.  Google Scholar

[6]

E. S. Baranovskii and M. A. Artemov, Global existence results for Oldroyd fluids with wall slip, Acta Appl. Math., 147 (2017), 197-210.  doi: 10.1007/s10440-016-0076-z.  Google Scholar

[7]

E. S. Baranovskii, On weak solutions to evolution equations of viscoelastic fluid flows, J. Appl. Ind. Math., 11 (2017), 174-184.  doi: 10.1134/S199047891702003X.  Google Scholar

[8]

E. S. Baranovskii, On flows of viscoelastic fluids under threshold-slip boundary conditions J. Phys.: Conf. Ser., 973 (2018), Article ID 012051. doi: 10.1088/1742-6596/973/1/012051.  Google Scholar

[9]

O. Bejaoui and M. Majdoub, Global weak solutions for some Oldroyd models, J. Differ. Equ., 254 (2013), 660-685.  doi: 10.1016/j.jde.2012.09.010.  Google Scholar

[10]

S. D. Besbes and C. Guillopé, Non-isothermal flows of viscoelastic incompressible fluids, Nonlinear Anal., 44 (2001), 919-942.  doi: 10.1016/S0362-546X(99)00315-6.  Google Scholar

[11]

H. Brezis, Equations et inéquations non linéaires dans les espaces en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280.  Google Scholar

[12]

J.-Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112.  doi: 10.1137/S0036141099359317.  Google Scholar

[13]

Q. Chen and C. Miao, Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Anal., 68 (2008), 1928-1939.  doi: 10.1016/j.na.2007.01.042.  Google Scholar

[14]

L. Chupin, Some theoretical results concerning diphasic viscoelastic flows of the Oldroyd kind, Adv. Differ. Equ., 9 (2004), 1039-1078.   Google Scholar

[15]

T. M. Elgindi and J. Liu, Global wellposedness to the generalized Oldroyd type models in $ R^3 $, J. Differ. Equ., 259 (2015), 1958-1966.  doi: 10.1016/j.jde.2015.03.026.  Google Scholar

[16]

D. FangM. Hieber and R. Zi, Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters, Math. Ann., 357 (2013), 687-709.  doi: 10.1007/s00208-013-0914-5.  Google Scholar

[17]

D. Fang and R. Zi, Global solutions to the Oldroyd-B model with a class of large initial data, SIAM J. Math. Anal., 48 (2016), 1054-1084.  doi: 10.1137/15M1037020.  Google Scholar

[18]

E. Fernández-CaraF. Guillén and R. Ortega, Some theoretical results concerning non Newtonian fluids of the Oldroyd kind, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 26 (1998), 1-29.   Google Scholar

[19]

H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kokyuroku, 888 (1994), 199-216.   Google Scholar

[20]

C. Guillopé and J. C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990), 849-869.  doi: 10.1016/0362-546X(90)90097-Z.  Google Scholar

[21]

L. He and L. Xi, Global well-posedness for viscoelastic fluid system in bounded domains, SIAM J. Math. Anal., 42 (2010), 2610-2625.  doi: 10.1137/10078503X.  Google Scholar

[22]

M. HieberY. Naito and Y. Shibata, Global existence results for Oldroyd-B fluids in exterior domains, J. Differ. Equ., 252 (2012), 2617-2629.  doi: 10.1016/j.jde.2011.09.001.  Google Scholar

[23]

T. Kashiwabara, On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type, J. Differ. Equ., 254 (2013), 756-778.  doi: 10.1016/j.jde.2012.09.015.  Google Scholar

[24]

C. Le Roux and A. Tani, Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions, Math. Methods Appl. Sci., 30 (2007), 595-624.  doi: 10.1002/mma.802.  Google Scholar

[25]

C. Le Roux, On flows of viscoelastic fluids of Oldroyd type with wall slip, J. Math. Fluid Mech., 16 (2014), 335-350.  doi: 10.1007/s00021-013-0159-9.  Google Scholar

[26]

Z. LeiC. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398.  doi: 10.1007/s00205-007-0089-x.  Google Scholar

[27]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[28]

P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math. Ser. B, 21 (2000), 131-146.  doi: 10.1142/S0252959900000170.  Google Scholar

[29]

V. G. Litvinov, Motion of A Nonlinearly Viscous Fluid, Nauka, Moscow, 1982.  Google Scholar

[30]

S. Maryani, Global well-posedness for free boundary problem of the Oldroyd-B model fluid flow, Math. Methods Appl. Sci., 39 (2016), 2202-2219.  doi: 10.1002/mma.3634.  Google Scholar

[31]

J. T. Oden and N. Kikuchi, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1988. doi: 10.1137/1.9781611970845.  Google Scholar

[32]

J. G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A, 200 (1950), 523-541.  doi: 10.1098/rspa.1950.0035.  Google Scholar

[33]

J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. R. Soc. Lond. Ser. A, 245 (1958), 278-297.  doi: 10.1098/rspa.1958.0083.  Google Scholar

[34]

K. R. Rajagopal, On some unresolved issues in non-linear fluid dynamics, Russian Math. Surveys, 58 (2003), 319-330.  doi: 10.1070/RM2003v058n02ABEH000612.  Google Scholar

[35]

M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech., 65 (1985), 449-451.  doi: 10.1002/zamm.19850650919.  Google Scholar

[36]

M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, 2$ ^{nd} $ edition, Springer-Verlag, New York, 2004.  Google Scholar

[37]

J.-C. Saut, Lectures on the mathematical theory of viscoelastic fluids, in Lectures on the analysis of nonlinear partial differential equations. Part 3 (eds. F. Lin and P. Zhang), Int. Press, Somerville, MA, (2013), 325-393.  Google Scholar

[38]

C. Truesdell, A First Course in Rational Continuum Mechanics, Academic Press, New York, 1977.  Google Scholar

[39]

E. M. Turganbaev, Initial-boundary value problems for the equations of a viscoelastic fluid of Oldroyd type, Sib. Math. J., 36 (1995), 389-403.  doi: 10.1007/BF02110162.  Google Scholar

[40]

D. A. Vorotnikov, On the existence of weak stationary solutions of a boundary value problem in the Jeffreys model of the motion of a viscoelastic medium, Russian Math. (Iz. VUZ), 48 (2004), 10-14.   Google Scholar

[41]

Z. Ye and X. Xu, Global regularity for the 2D Oldroyd-B model in the corotational case, Math. Methods Appl. Sci., 39 (2016), 3866-3879.  doi: 10.1002/mma.3834.  Google Scholar

[42]

V. G. Zvyagin and D. A. Vorotnikov, Approximating-topological methods in some problems of hydrodynamics, J. Fixed Point Theory Appl., 3 (2008), 23-49.  doi: 10.1007/s11784-008-0056-7.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$ ^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

Yu. Ya. Agranovich and P. E. Sobolevskii, Motion of nonlinear visco-elastic fluid, Nonlinear Anal., 32 (1998), 755-760.  doi: 10.1016/S0362-546X(97)00519-1.  Google Scholar

[3]

M. A. Artemov and E. S. Baranovskii, Mixed boundary-value problems for motion equations of a viscoelastic medium, Electron. J. Differ. Equ., 2015 (2015), Article No. 252.  Google Scholar

[4]

E. S. Baranovskii, On steady motion of viscoelastic fluid of Oldroyd type, Sb. Math., 205 (2014), 763-776.  doi: 10.1070/SM2014v205n06ABEH004397.  Google Scholar

[5]

E. S. Baranovskii, Optimal control for steady flows of the Jeffreys fluids with slip boundary condition, J. Appl. Ind. Math., 8 (2014), 168-176.  doi: 10.1134/S1990478914020033.  Google Scholar

[6]

E. S. Baranovskii and M. A. Artemov, Global existence results for Oldroyd fluids with wall slip, Acta Appl. Math., 147 (2017), 197-210.  doi: 10.1007/s10440-016-0076-z.  Google Scholar

[7]

E. S. Baranovskii, On weak solutions to evolution equations of viscoelastic fluid flows, J. Appl. Ind. Math., 11 (2017), 174-184.  doi: 10.1134/S199047891702003X.  Google Scholar

[8]

E. S. Baranovskii, On flows of viscoelastic fluids under threshold-slip boundary conditions J. Phys.: Conf. Ser., 973 (2018), Article ID 012051. doi: 10.1088/1742-6596/973/1/012051.  Google Scholar

[9]

O. Bejaoui and M. Majdoub, Global weak solutions for some Oldroyd models, J. Differ. Equ., 254 (2013), 660-685.  doi: 10.1016/j.jde.2012.09.010.  Google Scholar

[10]

S. D. Besbes and C. Guillopé, Non-isothermal flows of viscoelastic incompressible fluids, Nonlinear Anal., 44 (2001), 919-942.  doi: 10.1016/S0362-546X(99)00315-6.  Google Scholar

[11]

H. Brezis, Equations et inéquations non linéaires dans les espaces en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280.  Google Scholar

[12]

J.-Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112.  doi: 10.1137/S0036141099359317.  Google Scholar

[13]

Q. Chen and C. Miao, Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Anal., 68 (2008), 1928-1939.  doi: 10.1016/j.na.2007.01.042.  Google Scholar

[14]

L. Chupin, Some theoretical results concerning diphasic viscoelastic flows of the Oldroyd kind, Adv. Differ. Equ., 9 (2004), 1039-1078.   Google Scholar

[15]

T. M. Elgindi and J. Liu, Global wellposedness to the generalized Oldroyd type models in $ R^3 $, J. Differ. Equ., 259 (2015), 1958-1966.  doi: 10.1016/j.jde.2015.03.026.  Google Scholar

[16]

D. FangM. Hieber and R. Zi, Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters, Math. Ann., 357 (2013), 687-709.  doi: 10.1007/s00208-013-0914-5.  Google Scholar

[17]

D. Fang and R. Zi, Global solutions to the Oldroyd-B model with a class of large initial data, SIAM J. Math. Anal., 48 (2016), 1054-1084.  doi: 10.1137/15M1037020.  Google Scholar

[18]

E. Fernández-CaraF. Guillén and R. Ortega, Some theoretical results concerning non Newtonian fluids of the Oldroyd kind, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 26 (1998), 1-29.   Google Scholar

[19]

H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kokyuroku, 888 (1994), 199-216.   Google Scholar

[20]

C. Guillopé and J. C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990), 849-869.  doi: 10.1016/0362-546X(90)90097-Z.  Google Scholar

[21]

L. He and L. Xi, Global well-posedness for viscoelastic fluid system in bounded domains, SIAM J. Math. Anal., 42 (2010), 2610-2625.  doi: 10.1137/10078503X.  Google Scholar

[22]

M. HieberY. Naito and Y. Shibata, Global existence results for Oldroyd-B fluids in exterior domains, J. Differ. Equ., 252 (2012), 2617-2629.  doi: 10.1016/j.jde.2011.09.001.  Google Scholar

[23]

T. Kashiwabara, On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type, J. Differ. Equ., 254 (2013), 756-778.  doi: 10.1016/j.jde.2012.09.015.  Google Scholar

[24]

C. Le Roux and A. Tani, Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions, Math. Methods Appl. Sci., 30 (2007), 595-624.  doi: 10.1002/mma.802.  Google Scholar

[25]

C. Le Roux, On flows of viscoelastic fluids of Oldroyd type with wall slip, J. Math. Fluid Mech., 16 (2014), 335-350.  doi: 10.1007/s00021-013-0159-9.  Google Scholar

[26]

Z. LeiC. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398.  doi: 10.1007/s00205-007-0089-x.  Google Scholar

[27]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[28]

P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math. Ser. B, 21 (2000), 131-146.  doi: 10.1142/S0252959900000170.  Google Scholar

[29]

V. G. Litvinov, Motion of A Nonlinearly Viscous Fluid, Nauka, Moscow, 1982.  Google Scholar

[30]

S. Maryani, Global well-posedness for free boundary problem of the Oldroyd-B model fluid flow, Math. Methods Appl. Sci., 39 (2016), 2202-2219.  doi: 10.1002/mma.3634.  Google Scholar

[31]

J. T. Oden and N. Kikuchi, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1988. doi: 10.1137/1.9781611970845.  Google Scholar

[32]

J. G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A, 200 (1950), 523-541.  doi: 10.1098/rspa.1950.0035.  Google Scholar

[33]

J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. R. Soc. Lond. Ser. A, 245 (1958), 278-297.  doi: 10.1098/rspa.1958.0083.  Google Scholar

[34]

K. R. Rajagopal, On some unresolved issues in non-linear fluid dynamics, Russian Math. Surveys, 58 (2003), 319-330.  doi: 10.1070/RM2003v058n02ABEH000612.  Google Scholar

[35]

M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech., 65 (1985), 449-451.  doi: 10.1002/zamm.19850650919.  Google Scholar

[36]

M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, 2$ ^{nd} $ edition, Springer-Verlag, New York, 2004.  Google Scholar

[37]

J.-C. Saut, Lectures on the mathematical theory of viscoelastic fluids, in Lectures on the analysis of nonlinear partial differential equations. Part 3 (eds. F. Lin and P. Zhang), Int. Press, Somerville, MA, (2013), 325-393.  Google Scholar

[38]

C. Truesdell, A First Course in Rational Continuum Mechanics, Academic Press, New York, 1977.  Google Scholar

[39]

E. M. Turganbaev, Initial-boundary value problems for the equations of a viscoelastic fluid of Oldroyd type, Sib. Math. J., 36 (1995), 389-403.  doi: 10.1007/BF02110162.  Google Scholar

[40]

D. A. Vorotnikov, On the existence of weak stationary solutions of a boundary value problem in the Jeffreys model of the motion of a viscoelastic medium, Russian Math. (Iz. VUZ), 48 (2004), 10-14.   Google Scholar

[41]

Z. Ye and X. Xu, Global regularity for the 2D Oldroyd-B model in the corotational case, Math. Methods Appl. Sci., 39 (2016), 3866-3879.  doi: 10.1002/mma.3834.  Google Scholar

[42]

V. G. Zvyagin and D. A. Vorotnikov, Approximating-topological methods in some problems of hydrodynamics, J. Fixed Point Theory Appl., 3 (2008), 23-49.  doi: 10.1007/s11784-008-0056-7.  Google Scholar

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