September  2014, 19(7): 2111-2132. doi: 10.3934/dcdsb.2014.19.2111

Uniqueness and stability results for non-linear Johnson-Segalman viscoelasticity and related models

1. 

Department of Mathematics, University of Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy, Italy, Italy

Received  April 2013 Revised  July 2013 Published  August 2014

In this paper we have proved exponential asymptotic stability for the corotational incompressible diffusive Johnson-Segalman viscolelastic model and a simple decay result for the corotational incompressible hyperbolic Maxwell model. Moreover we have established continuous dependence and uniqueness results for the non-zero equilibrium solution.
    In the compressible case, we have proved a Hölder continuous dependence theorem upon the initial data and body force for both models, whence follows a result of continuous dependence on the initial data and, therefore, uniqueness.
    For the Johnson-Segalman model we have also dealt with the case of negative elastic viscosities, corresponding to retardation effects. A comparison with other type of viscoelasticity, showing short memory elastic effects, is given.
Citation: Franca Franchi, Barbara Lazzari, Roberta Nibbi. Uniqueness and stability results for non-linear Johnson-Segalman viscoelasticity and related models. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2111-2132. doi: 10.3934/dcdsb.2014.19.2111
References:
[1]

R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol 1. Fluid Mechanics, John Wiley & Sons, New York, 1987.

[2]

T. Bodnár and L. Pirkl, A Remark on the Deviatoric Decomposition of Oldroyd Type Models, Colloquium FLUID DYNAMICS 2011, Institute of Thermomechanics AS CR, v.v.i., Prague, October 19 - 21, 2011. Available from: http://www.it.cas.cz/en/colloqium-fluid-dynamics-2011

[3]

R. M. Christensen, Theory of Viscoelasticity, $2^{nd}$ edition, Dover Publications, New York, 2010. doi: 10.1115/1.3408900.

[4]

C. I. Christov, Frame indifferent formulation of Maxwell's elastic-fluid model and the rational continuum mechanics of the electromagnetic field, Mech. Res. Comm., 38 (2011), 334-339. doi: 10.1016/j.mechrescom.2011.03.002.

[5]

M. Fabrizio, Dario graffi in a complex historical period, in Mathematicians in Bologna 1861-1960, (ed. S. Coen), Springer Basel AG, 2012, 179-195. doi: 10.1007/978-3-0348-0227-7_7.

[6]

M. Fabrizio and F. Franchi, Delayed thermal models. Stability and thermodynamics, J. Thermal Stresses, 37 (2014), 160-173. doi: 10.1080/01495739.2013.839619.

[7]

F. Franchi, On the behaviour of one-dimensional waves in thermo-viscoelastic fluids, Meccanica, 17 (1982), 3-10. doi: 10.1007/BF02156001.

[8]

R. J. Gordon and W. R. Schowalter, Anisotropic fluid theory: A different approach to the dumbbell theory of dilute polymer solutions, Trans. Soc. Rheo., 16 (1972), 79-97. doi: 10.1122/1.549256.

[9]

D. Graffi, On a method for proving uniqueness theorems in mathematical physics, Atti. Sem. Mat. Fis. Univ. Modena, (Italian) [On a method to prove uniqueness theorems in mathematical physics], 37 (1989), 259-284.

[10]

A. Guaily and M. Epstein, A unified hyperbolic model for viscoelastic liquids, Mech. Res. Comm., 37 (2010), 158-163. doi: 10.1016/j.mechrescom.2009.12.004.

[11]

T. Gültop, B. Alyavuz and M. Kopaç, Propagation of acceleration waves in the viscoelastic Johnson-Segalman fluids, Mech. Res. Comm., 37 (2010), 153-157.

[12]

S. J. Haward, Buckling instabilities in dilute polymer solution elastic strands, Rheol. Acta., 49 (2010), 1219-1225. doi: 10.1007/s00397-010-0467-4.

[13]

T. Hayat, A. Afsar and N. Ali, Peristaltic transport of a Johnson-Segalman fluid in an asymmetric channel, Math. Comput. Modelling, 47 (2008), 380-400. doi: 10.1016/j.mcm.2007.04.012.

[14]

T. Hayat, S. Hina and A. A. Hendi, Slip effects on peristaltic transport of a Maxwell fluid with heat and mass transfer, J. Mech. Med. Biol., 12 (2012), [22 pages], 1250001. doi: 10.1142/S0219519412004375.

[15]

S. Hinaa, T. Hayat and S. Asghard, Peristaltic transport of Johnson-Segalman fluid in a curved channel with compliant walls, Nonlinear Anal. Model. Control, 17 (2012), 297-311.

[16]

D. Hu and T. Leliévre, New entropy estimates for the Oldroyd-B model and related models, Commun. Math. Sci., 5 (2007), 909-916. doi: 10.4310/CMS.2007.v5.n4.a9.

[17]

M. W. Johnson and D. Segalman, A model for viscoelastic fluid behavior which allows nonaffine deformation, J. Non-Newtionan Fluid Mech., 4 (1977), 255-270.

[18]

D. D. Joseph, M. Renardy and J. C. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluids, Arch. Rat. Mech. Anal., 87 (1985), 213-251. doi: 10.1007/BF00250725.

[19]

R. W. Kolkka, D. S. Malkus, M. G. Hansen and G. R. Ierley, Spurt phenomena of the Johnson-Segalman fluid and related models, J. Non-Newtonian Fluid Mech., 29 (1988), 303-335. doi: 10.1016/0377-0257(88)85059-6.

[20]

H. V. J. Le Meur, Well-posedness of surface wave equations above a viscoelastic fluid, J. Math. Fluid Mech., 13 (2011), 481-514. doi: 10.1007/s00021-010-0029-7.

[21]

V. Y. Liapidevskii, V. V. Pukhnachev and A. Tani, Nonlinear waves in incompressible viscoelastic Maxwell medium, Wave Motion, 48 (2011), 727-737. doi: 10.1016/j.wavemoti.2011.04.002.

[22]

A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105. doi: 10.1007/s10659-010-9292-3.

[23]

C. J. Pipe, N. J. Kim, P. A. Vasquez, L. P. Cook and G. H. McKinley, Wormlike micellar solutions: II. Comparison between experimental data and scission model predictions, Journal of Rheology, 54 (2010), 881-913. doi: 10.1122/1.3439729.

[24]

L. E. Payne and B. Straughan, Convergence of the equations for a maxwell fluid, Stud. Appl. Math., 103 (1999), 267-278.

[25]

M. Renardy, Similarity solutions for jet breakup for various models of viscoelastic fluids, J. Non-Newtonian Fluid Mech., 104 (2002), 65-74. doi: 10.1016/S0377-0257(02)00016-2.

[26]

M. Renardy, On control of shear flow of an upper convected Maxwell fluid, ZAMM Z. Angew. Math. Mech., 87 (2007), 213-218. doi: 10.1002/zamm.200610313.

[27]

C. E. Seyler and M. R. Martin, Relaxation model for extended magnetohydrodynamics: Comparison to magnetohydrodynamics for dense Z-pinches, Phys. Plasmas, 18 (2011), [13 pages], 012703. doi: 10.1063/1.3543799.

[28]

B. Straughan, The Energy Method, Stability and Nonlinear Convection, $2^{nd}$ edition, Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21740-6.

[29]

B. Straughan, Heat Waves, Applied Mathematical Sciences 177, Springer New York, 2011. doi: 10.1007/978-1-4614-0493-4.

[30]

D. Y. Tzou, Macro-to-Microscale Heat Transfer: The Lagging Behavior, Taylor and Francis, Washington, 1997.

[31]

L. Wilson, H. Zhou, W. Kang and H. Wang, Controllability of non-newtonian fluids under homogeneous extensional flow, Appl. Math. Sci., 2 (2008), 2145-2156.

show all references

References:
[1]

R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol 1. Fluid Mechanics, John Wiley & Sons, New York, 1987.

[2]

T. Bodnár and L. Pirkl, A Remark on the Deviatoric Decomposition of Oldroyd Type Models, Colloquium FLUID DYNAMICS 2011, Institute of Thermomechanics AS CR, v.v.i., Prague, October 19 - 21, 2011. Available from: http://www.it.cas.cz/en/colloqium-fluid-dynamics-2011

[3]

R. M. Christensen, Theory of Viscoelasticity, $2^{nd}$ edition, Dover Publications, New York, 2010. doi: 10.1115/1.3408900.

[4]

C. I. Christov, Frame indifferent formulation of Maxwell's elastic-fluid model and the rational continuum mechanics of the electromagnetic field, Mech. Res. Comm., 38 (2011), 334-339. doi: 10.1016/j.mechrescom.2011.03.002.

[5]

M. Fabrizio, Dario graffi in a complex historical period, in Mathematicians in Bologna 1861-1960, (ed. S. Coen), Springer Basel AG, 2012, 179-195. doi: 10.1007/978-3-0348-0227-7_7.

[6]

M. Fabrizio and F. Franchi, Delayed thermal models. Stability and thermodynamics, J. Thermal Stresses, 37 (2014), 160-173. doi: 10.1080/01495739.2013.839619.

[7]

F. Franchi, On the behaviour of one-dimensional waves in thermo-viscoelastic fluids, Meccanica, 17 (1982), 3-10. doi: 10.1007/BF02156001.

[8]

R. J. Gordon and W. R. Schowalter, Anisotropic fluid theory: A different approach to the dumbbell theory of dilute polymer solutions, Trans. Soc. Rheo., 16 (1972), 79-97. doi: 10.1122/1.549256.

[9]

D. Graffi, On a method for proving uniqueness theorems in mathematical physics, Atti. Sem. Mat. Fis. Univ. Modena, (Italian) [On a method to prove uniqueness theorems in mathematical physics], 37 (1989), 259-284.

[10]

A. Guaily and M. Epstein, A unified hyperbolic model for viscoelastic liquids, Mech. Res. Comm., 37 (2010), 158-163. doi: 10.1016/j.mechrescom.2009.12.004.

[11]

T. Gültop, B. Alyavuz and M. Kopaç, Propagation of acceleration waves in the viscoelastic Johnson-Segalman fluids, Mech. Res. Comm., 37 (2010), 153-157.

[12]

S. J. Haward, Buckling instabilities in dilute polymer solution elastic strands, Rheol. Acta., 49 (2010), 1219-1225. doi: 10.1007/s00397-010-0467-4.

[13]

T. Hayat, A. Afsar and N. Ali, Peristaltic transport of a Johnson-Segalman fluid in an asymmetric channel, Math. Comput. Modelling, 47 (2008), 380-400. doi: 10.1016/j.mcm.2007.04.012.

[14]

T. Hayat, S. Hina and A. A. Hendi, Slip effects on peristaltic transport of a Maxwell fluid with heat and mass transfer, J. Mech. Med. Biol., 12 (2012), [22 pages], 1250001. doi: 10.1142/S0219519412004375.

[15]

S. Hinaa, T. Hayat and S. Asghard, Peristaltic transport of Johnson-Segalman fluid in a curved channel with compliant walls, Nonlinear Anal. Model. Control, 17 (2012), 297-311.

[16]

D. Hu and T. Leliévre, New entropy estimates for the Oldroyd-B model and related models, Commun. Math. Sci., 5 (2007), 909-916. doi: 10.4310/CMS.2007.v5.n4.a9.

[17]

M. W. Johnson and D. Segalman, A model for viscoelastic fluid behavior which allows nonaffine deformation, J. Non-Newtionan Fluid Mech., 4 (1977), 255-270.

[18]

D. D. Joseph, M. Renardy and J. C. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluids, Arch. Rat. Mech. Anal., 87 (1985), 213-251. doi: 10.1007/BF00250725.

[19]

R. W. Kolkka, D. S. Malkus, M. G. Hansen and G. R. Ierley, Spurt phenomena of the Johnson-Segalman fluid and related models, J. Non-Newtonian Fluid Mech., 29 (1988), 303-335. doi: 10.1016/0377-0257(88)85059-6.

[20]

H. V. J. Le Meur, Well-posedness of surface wave equations above a viscoelastic fluid, J. Math. Fluid Mech., 13 (2011), 481-514. doi: 10.1007/s00021-010-0029-7.

[21]

V. Y. Liapidevskii, V. V. Pukhnachev and A. Tani, Nonlinear waves in incompressible viscoelastic Maxwell medium, Wave Motion, 48 (2011), 727-737. doi: 10.1016/j.wavemoti.2011.04.002.

[22]

A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105. doi: 10.1007/s10659-010-9292-3.

[23]

C. J. Pipe, N. J. Kim, P. A. Vasquez, L. P. Cook and G. H. McKinley, Wormlike micellar solutions: II. Comparison between experimental data and scission model predictions, Journal of Rheology, 54 (2010), 881-913. doi: 10.1122/1.3439729.

[24]

L. E. Payne and B. Straughan, Convergence of the equations for a maxwell fluid, Stud. Appl. Math., 103 (1999), 267-278.

[25]

M. Renardy, Similarity solutions for jet breakup for various models of viscoelastic fluids, J. Non-Newtonian Fluid Mech., 104 (2002), 65-74. doi: 10.1016/S0377-0257(02)00016-2.

[26]

M. Renardy, On control of shear flow of an upper convected Maxwell fluid, ZAMM Z. Angew. Math. Mech., 87 (2007), 213-218. doi: 10.1002/zamm.200610313.

[27]

C. E. Seyler and M. R. Martin, Relaxation model for extended magnetohydrodynamics: Comparison to magnetohydrodynamics for dense Z-pinches, Phys. Plasmas, 18 (2011), [13 pages], 012703. doi: 10.1063/1.3543799.

[28]

B. Straughan, The Energy Method, Stability and Nonlinear Convection, $2^{nd}$ edition, Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21740-6.

[29]

B. Straughan, Heat Waves, Applied Mathematical Sciences 177, Springer New York, 2011. doi: 10.1007/978-1-4614-0493-4.

[30]

D. Y. Tzou, Macro-to-Microscale Heat Transfer: The Lagging Behavior, Taylor and Francis, Washington, 1997.

[31]

L. Wilson, H. Zhou, W. Kang and H. Wang, Controllability of non-newtonian fluids under homogeneous extensional flow, Appl. Math. Sci., 2 (2008), 2145-2156.

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