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September  2014, 7(3): 493-507. doi: 10.3934/krm.2014.7.493

## Hysteretic behavior of a moment-closure approximation for FENE model

 1 Division of Mathematical Models, National Institute for Mathematical Sciences, Daejeon, 305-811, South Korea

Received  February 2014 Revised  May 2014 Published  July 2014

We discuss hysteretic behaviors of dilute viscoelastic polymeric fluids with moment-closure approximation approach in extensional/enlongational flows. Polymeric fluids are modeled by the finite extensible nonlinear elastic (FENE) spring dumbbell model. Hysteresis is one of key features to describe FENE model. We here investigate the hysteretic behavior of FENE-D model introduced in [Y. Hyon et al., Multiscale Model. Simul., 7(2008), pp.978--1002]. The FENE-D model is established from a special equilibrium solution of the Fokker-Planck equation to catch extreme behavior of FENE model in large extensional flow rates. Since the hysteresis of FENE model can be seen during a relaxation in simple extensional flow employing the normal stress/the elongational viscosity versus the mean-square extension, we simulate FENE-D in simple extensional flows to investigate its hysteretic behavior comparing to FENE-P, FENE-L [G. Lielens et al., J. Non-Newtonian Fluid Mech., 76(1999), pp.249--279]. The FENE-P is a well-known pre-averaged approximated model, and it shows a good agreement to macroscopic induced stresses. However, FENE-P does not catch any hysteretic phenomenon. In contrast, the FENE-L shows a better hysteretic behavior than the other models to FENE, but it has a limitation for macroscopic induced stresses in large shear rates. On the other hand, FENE-D presents a good agreement to macroscopic induced stresses even in large shear rates, and moreover, it shows a hysteretic phenomenon in certain large flow rates.
Citation: YunKyong Hyon. Hysteretic behavior of a moment-closure approximation for FENE model. Kinetic and Related Models, 2014, 7 (3) : 493-507. doi: 10.3934/krm.2014.7.493
##### References:
 [1] R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Fluids, Vol. 1, Fluid Mechanics, John Wiley & Sons, New York, 1977. [2] R. B. Bird, O. Hassager, R. C. Armstrong and C. F. Curtiss, Dynamics of Polymeric Fluids, Vol. 2, Kinetic Theory, John Wiley & Sons, New York, 1977. [3] C. Chauviere and A. Lozinski, Simulation of dilute polymer solutions using a Fokker-Planck equation, Computers & fluids, 33 (2004), 687-696. doi: 10.1016/j.compfluid.2003.02.002. [4] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, UK, 1986. [5] P. S. Doyle, E. S. G. Shaqfeh, G. H. McKinley and S. H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech., 76 (1998), 79-110. doi: 10.1016/S0377-0257(97)00113-4. [6] Q. Du, C. Liu and P. Yu, FENE Dumbbell model and its several linear and nonlinear closure approximations, Multiscale Model. Simul., 4 (2005), 709-731. doi: 10.1137/040612038. [7] M. Gurtin, An Introduction to Continuum Mechanics, Academic Press, New York, 1981. [8] M. Herrchen and H. C. Öttinger, A detail comparison of various FENE dumbbell models, J. Non-Newtonian Fluid Mech., 68 (1997), 17-42. doi: 10.1016/S0377-0257(96)01498-X. [9] Y. Hyon, Q. Du and C. Liu, An enhanced macroscopic closure approximation to the micro-macro FENE model for polymeric materials, Multiscale Model. Simul., 7 (2008), 978-1002. doi: 10.1137/070708287. [10] M. Hulsen, A. van Heel and B. van dent Brule, Simulation of viscoelatsic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech., 70 (1997), 79-101. [11] B. Jourdain, C. Le Bris and T. Lelievre, On a variance reduction technique for the micro-macro simulations of polymeric fluids, J. Non-Newtonian Fluid Mech., 122 (2004), 91-106. doi: 10.1016/j.jnnfm.2003.09.006. [12] B. Jourdain, T. Lelievre and C. Le Bris, Existence of solution for a micro-macro model of polymeric fluid: The FENE model, J. Funct. Anal., 209 (2004), 162-193. doi: 10.1016/S0022-1236(03)00183-6. [13] R. Keunings, On the Peterlin approximation for finitely extensible dumbbells, J. Non-Newtonian Fluid Mech., 68 (1997), 85-100. doi: 10.1016/S0377-0257(96)01497-8. [14] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [15] M. Laso and H. C. Ottinger, Calculation of viscoelastic flow using molecular models: The connffessit approach, J. Non-Newtonian Fluid Mech., 47 (1993), 1-20. doi: 10.1016/0377-0257(93)80042-A. [16] G. Lielens, P. Halin, I. Jaumain, R. Keunings and V. Legat, New closure approximations for the kinetic theory of finitely extensible dumbbells, J. Non-Newtonian Fluid Mech., 76 (1998), 249-279. doi: 10.1016/S0377-0257(97)00121-3. [17] G. Lielens, R. Keunings and V. Legat, The FENE-L and FENE-LS closure approximations to the kinetic theory of finitely extensible dumbbells, J. Non-Newtonian Fluid Mech., 87 (1999), 179-196. doi: 10.1016/S0377-0257(99)00063-4. [18] F. H. Lin, C. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math., 60 (2007), 838-866. doi: 10.1002/cpa.20159. [19] H. C. Öttinger, Stochastic Processes in Polymeric Fluids, Tools and Examples for Developing Simulation Algorithms, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-58290-5. [20] R. Owens and T. Phillips, Computational Rheology, Imperial College Press, London, 2002. doi: 10.1142/9781860949425. [21] R. Prabhakar and J. R. Prakash, Gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction, J. Rheol., 50 (2006), 561-593. doi: 10.1122/1.2206715. [22] R. Sizaire, G. Lielens, I. Jaumain, R. Keunings and V. Legat, On the hysteretic behaviour of dilute polymer solutions in relaxation following extensional flow, J. Non-Newtonian Fluid Mech., 82 (1999), 233-253. doi: 10.1016/S0377-0257(98)00164-5. [23] H. Wang, K. Li and P. Zhang, Crucial properties of the moment closure model FENE-QE, J. Non-Newtonian Fluid Mech., 150 (2008), 80-92. doi: 10.1016/j.jnnfm.2007.10.013. [24] P. Yu, Q. Du and C. Liu, From micro to macro dynamics via a new closure approximation to the FENE model of polymeric fluids, Multiscale Model. Simul., 3 (2005), 895-917. doi: 10.1137/030602794.

show all references

##### References:
 [1] R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Fluids, Vol. 1, Fluid Mechanics, John Wiley & Sons, New York, 1977. [2] R. B. Bird, O. Hassager, R. C. Armstrong and C. F. Curtiss, Dynamics of Polymeric Fluids, Vol. 2, Kinetic Theory, John Wiley & Sons, New York, 1977. [3] C. Chauviere and A. Lozinski, Simulation of dilute polymer solutions using a Fokker-Planck equation, Computers & fluids, 33 (2004), 687-696. doi: 10.1016/j.compfluid.2003.02.002. [4] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, UK, 1986. [5] P. S. Doyle, E. S. G. Shaqfeh, G. H. McKinley and S. H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech., 76 (1998), 79-110. doi: 10.1016/S0377-0257(97)00113-4. [6] Q. Du, C. Liu and P. Yu, FENE Dumbbell model and its several linear and nonlinear closure approximations, Multiscale Model. Simul., 4 (2005), 709-731. doi: 10.1137/040612038. [7] M. Gurtin, An Introduction to Continuum Mechanics, Academic Press, New York, 1981. [8] M. Herrchen and H. C. Öttinger, A detail comparison of various FENE dumbbell models, J. Non-Newtonian Fluid Mech., 68 (1997), 17-42. doi: 10.1016/S0377-0257(96)01498-X. [9] Y. Hyon, Q. Du and C. Liu, An enhanced macroscopic closure approximation to the micro-macro FENE model for polymeric materials, Multiscale Model. Simul., 7 (2008), 978-1002. doi: 10.1137/070708287. [10] M. Hulsen, A. van Heel and B. van dent Brule, Simulation of viscoelatsic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech., 70 (1997), 79-101. [11] B. Jourdain, C. Le Bris and T. Lelievre, On a variance reduction technique for the micro-macro simulations of polymeric fluids, J. Non-Newtonian Fluid Mech., 122 (2004), 91-106. doi: 10.1016/j.jnnfm.2003.09.006. [12] B. Jourdain, T. Lelievre and C. Le Bris, Existence of solution for a micro-macro model of polymeric fluid: The FENE model, J. Funct. Anal., 209 (2004), 162-193. doi: 10.1016/S0022-1236(03)00183-6. [13] R. Keunings, On the Peterlin approximation for finitely extensible dumbbells, J. Non-Newtonian Fluid Mech., 68 (1997), 85-100. doi: 10.1016/S0377-0257(96)01497-8. [14] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [15] M. Laso and H. C. Ottinger, Calculation of viscoelastic flow using molecular models: The connffessit approach, J. Non-Newtonian Fluid Mech., 47 (1993), 1-20. doi: 10.1016/0377-0257(93)80042-A. [16] G. Lielens, P. Halin, I. Jaumain, R. Keunings and V. Legat, New closure approximations for the kinetic theory of finitely extensible dumbbells, J. Non-Newtonian Fluid Mech., 76 (1998), 249-279. doi: 10.1016/S0377-0257(97)00121-3. [17] G. Lielens, R. Keunings and V. Legat, The FENE-L and FENE-LS closure approximations to the kinetic theory of finitely extensible dumbbells, J. Non-Newtonian Fluid Mech., 87 (1999), 179-196. doi: 10.1016/S0377-0257(99)00063-4. [18] F. H. Lin, C. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math., 60 (2007), 838-866. doi: 10.1002/cpa.20159. [19] H. C. Öttinger, Stochastic Processes in Polymeric Fluids, Tools and Examples for Developing Simulation Algorithms, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-58290-5. [20] R. Owens and T. Phillips, Computational Rheology, Imperial College Press, London, 2002. doi: 10.1142/9781860949425. [21] R. Prabhakar and J. R. Prakash, Gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction, J. Rheol., 50 (2006), 561-593. doi: 10.1122/1.2206715. [22] R. Sizaire, G. Lielens, I. Jaumain, R. Keunings and V. Legat, On the hysteretic behaviour of dilute polymer solutions in relaxation following extensional flow, J. Non-Newtonian Fluid Mech., 82 (1999), 233-253. doi: 10.1016/S0377-0257(98)00164-5. [23] H. Wang, K. Li and P. Zhang, Crucial properties of the moment closure model FENE-QE, J. Non-Newtonian Fluid Mech., 150 (2008), 80-92. doi: 10.1016/j.jnnfm.2007.10.013. [24] P. Yu, Q. Du and C. Liu, From micro to macro dynamics via a new closure approximation to the FENE model of polymeric fluids, Multiscale Model. Simul., 3 (2005), 895-917. doi: 10.1137/030602794.
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