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

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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

Franca Franchi ^1, , Barbara Lazzari ^1, and Roberta Nibbi ^1,

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

Abstract
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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.

Keywords: Maxwell viscoelastic fluids, continuous dependence, stability, relaxation/retardation effects., Johnson-Segalman viscolestic fluids.

Mathematics Subject Classification: Primary: 76A05, 35B30; Secondary: 76A1.

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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