American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - B, 2014, 19 (7) : 2111-2132. doi: 10.3934/dcdsb.2014.19.2111
References:
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Coen), Springer Basel AG, 2012, 179-195. doi: 10.1007/978-3-0348-0227-7_7.  Google Scholar [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.  Google Scholar [7] F. Franchi, On the behaviour of one-dimensional waves in thermo-viscoelastic fluids, Meccanica, 17 (1982), 3-10. doi: 10.1007/BF02156001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105. doi: 10.1007/s10659-010-9292-3.  Google Scholar [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.  Google Scholar [24] L. E. Payne and B. Straughan, Convergence of the equations for a maxwell fluid, Stud. Appl. Math., 103 (1999), 267-278. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] B. Straughan, Heat Waves, Applied Mathematical Sciences 177, Springer New York, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar [30] D. Y. Tzou, Macro-to-Microscale Heat Transfer: The Lagging Behavior, Taylor and Francis, Washington, 1997. Google Scholar [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.  Google Scholar

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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. Google Scholar [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 Google Scholar [3] R. M. Christensen, Theory of Viscoelasticity, $2^{nd}$ edition, Dover Publications, New York, 2010. doi: 10.1115/1.3408900.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] F. Franchi, On the behaviour of one-dimensional waves in thermo-viscoelastic fluids, Meccanica, 17 (1982), 3-10. doi: 10.1007/BF02156001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105. doi: 10.1007/s10659-010-9292-3.  Google Scholar [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.  Google Scholar [24] L. E. Payne and B. Straughan, Convergence of the equations for a maxwell fluid, Stud. Appl. Math., 103 (1999), 267-278. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] B. Straughan, Heat Waves, Applied Mathematical Sciences 177, Springer New York, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar [30] D. Y. Tzou, Macro-to-Microscale Heat Transfer: The Lagging Behavior, Taylor and Francis, Washington, 1997. Google Scholar [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.  Google Scholar
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