Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid

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February 2012, 32(2): 657-677. doi: 10.3934/dcds.2012.32.657

Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid

Kun Wang ^1, , Yangping Lin ^2, and Yinnian He ^3,

1.	College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2.	Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
3.	Faculty of Science, Xi'an Jiaotong University, Xi'an 710049

Received October 2010 Revised June 2011 Published September 2011

Abstract
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In this paper, the asymptotic analysis of the two-dimensional viscoelastic Oldroyd flows is presented. With the physical constant $\rho/\delta$ approaches zero, where $\rho$ is the viscoelastic coefficient and $1/\delta$ the relaxation time, the viscoelastic Oldroyd fluid motion equations converge to the viscous model known as the famous Navier-Stokes equations. Both the continuous and discrete uniform-in-time asymptotic errors are provided. Finally, the theoretical predictions are confirmed by some numerical experiments.

Keywords: Viscoelastic flows, long time behavior, Oldroyd model, asymptotic behavior, Navier-Stokes equations..

Mathematics Subject Classification: Primary: 35L70, 76D05, 76A1.

Citation: Kun Wang, Yangping Lin, Yinnian He. Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 657-677. doi: 10.3934/dcds.2012.32.657

References:

[1]	Y. Agranovich and P. Sobolevskiĭ, Investigation of a mathematical model of a viscoelastic fluid, (Russian), 1989 (): 3. Google Scholar
[2]	Y. Agranovich and P. Sobolevskiĭ, Motion of non-linear visco-elastic fluid,, Nonlinear Anal., 32 (1998), 755. doi: 10.1016/S0362-546X(97)00519-1. Google Scholar
[3]	M. Akhmatov and A. Oskolkov, On convergence difference schemes for the equations of motion of an Oldroyd fluid,, Zap. Nauchn. Semin. Leningrad. Otdel. Mat. Inst. Steklov., 159 (1987), 143. doi: 10.1007/BF01305224. Google Scholar
[4]	W. Allegretto, Y. P. Lin and A. H. Zhou, Long-time stability of finite element approximations for parabolic equations with memory,, Numer. Methods Partial Differential Equations, 15 (1999), 333. doi: 10.1002/(SICI)1098-2426(199905)15:3<333::AID-NUM5>3.0.CO;2-0. Google Scholar
[5]	G. Araújo, S. Menezes and A. Marinho, Existence of solutions for an Oldroyd model of viscoelastic fluids,, J. Differential Equations, 2009 (). Google Scholar
[6]	R. Bird, R. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids. Vol. 1, Fluid Mechanics,", John Wiley & Sons, (1977). Google Scholar
[7]	J. Cannon, R. Ewing, Y. N. He and Y. P. Lin, A modified nonlinear Galerkin method for the viscoelastic fluid motion equations,, Int. J. Eng. Sci., 37 (1999), 1643. doi: 10.1016/S0020-7225(98)00142-6. Google Scholar
[8]	P. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, 4,, North-Holland Publishing Co., (1978). Google Scholar
[9]	V. Girault and P. Raviart, "Finite Element Approximation of the Navier-Stokes Equations,", Springer-Verlag, (1979). doi: 10.1007/BFb0063447. Google Scholar
[10]	D. Goswami and A. Pani, A priori error estimates for semidiscrete finite element approximations to equations of motion arising in Oldroyd fluids of order one,, Int. J. Numer. Anal. Model., 8 (2011), 324. Google Scholar
[11]	Y. N. He and Y. Li, Asymptotic behavior of linearized viscoelastic flow problem,, Discrete Contin. Dyn. Syst.-Ser. B, 10 (2008), 843. Google Scholar
[12]	Y. N. He and K. T. Li, Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem,, Numer. Math., 98 (2004), 647. doi: 10.1007/s00211-004-0532-y. Google Scholar
[13]	Y. N. He, Y. P. Lin, S. Shen and R. Tait, On the convergence of viscoelastic fluid flows to a steady state,, Adv. Differential Equations, 7 (2002), 717. Google Scholar
[14]	Y. N. He, Y. P. Lin, S. Shen, W. W. Sun and R. Tait, Finite element approximation for the viscoelastic fluid motion problem,, J. Comput. Appl. Math., 155 (2003), 201. Google Scholar
[15]	J. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem part IV: Error analysis for second-order time discretization,, SIAM J. Numer. Anal., 27 (1990), 353. doi: 10.1137/0727022. Google Scholar
[16]	D. Joseph, "Fluid Dynamics of Viscoelastic Liquids,", Applied Mathematical Sciences, 84 (1990). Google Scholar
[17]	A. Kotsiolis and A. Oskolkov, Initial-boundary value problems for equations of slightly compressible Jeffreys-Oldroyd fluids,, Zap. Nauchn. Semin. POMI, 208 (1993), 200. doi: 10.1007/BF02362429. Google Scholar
[18]	J. Oldroyd, On the formulation of the rheological equations of state,, Proc. Roy. Soc. London. Ser. A., 200 (1950), 523. Google Scholar
[19]	A. Oskolkov, Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, Contributions to "Boundary Value Problems of Mathematical Physics"(ed. J. Schulenberger),, Proc. Steklov Inst. Math., 179 (1989), 137. Google Scholar
[20]	A. Oskolkov, The penalty method for equations of viscoelastic media,, Zap. Nauchn. Semin. POMI, 224 (1995), 267. doi: 10.1007/BF02364990. Google Scholar
[21]	A. Pani and J. Yuan, Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model,, IMA J. Numer. Anal., 25 (2005), 750. doi: 10.1093/imanum/dri016. Google Scholar
[22]	A. Pani, J. Yuan and P. Damázio, On a linearized backward Euler method for the equations of motion of Oldroyd fluids of order one,, SIAM J. Numer. Anal., 44 (2006), 804. doi: 10.1137/S0036142903428967. Google Scholar
[23]	P. Sobolevskiĭ, Stabilization of viscoelastic fluid motion (Oldroyd's mathematical model),, Differential Integral Equations, 7 (1994), 1597. Google Scholar
[24]	P. Sobolevskiĭ, Asymptotic of stable viscoelastic fluid motion (Oldroyd's mathematical model),, Math. Nachr., 177 (1996), 281. doi: 10.1002/mana.19961770116. Google Scholar
[25]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", Third edition, 2 (1984). Google Scholar
[26]	R. Temam and X. Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a general 2D domain,, Asymptot. Anal., 14 (1997), 293. Google Scholar
[27]	R. Temam and X. Wang, Boundary layers asscociated with incompressible Navier-Stokes equations: The noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647. doi: 10.1006/jdeq.2001.4038. Google Scholar
[28]	K. Wang, Y. N. He and X. L. Feng, On error estimates of the penalty method for the viscoelastic flow problem I: Time discretization,, Appl. Math. Model., 34 (2010), 4089. doi: 10.1016/j.apm.2010.04.008. Google Scholar
[29]	K. Wang, Y. N. He and Y. Q. Shang, Fully discrete finite element method for the viscoelastic fluid motion equations,, Discrete Contin. Dyn. Syst.-Ser. B, 13 (2010), 665. Google Scholar
[30]	K. Wang, Y. Q. Shang and R. Zhao, Optimal error estimates of the penalty method for the linearized viscoelastic flows,, Int. J. Comput. Math., 87 (2010), 3236. doi: 10.1080/00207160902980500. Google Scholar

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References:

[1]	Y. Agranovich and P. Sobolevskiĭ, Investigation of a mathematical model of a viscoelastic fluid, (Russian), 1989 (): 3. Google Scholar
[2]	Y. Agranovich and P. Sobolevskiĭ, Motion of non-linear visco-elastic fluid,, Nonlinear Anal., 32 (1998), 755. doi: 10.1016/S0362-546X(97)00519-1. Google Scholar
[3]	M. Akhmatov and A. Oskolkov, On convergence difference schemes for the equations of motion of an Oldroyd fluid,, Zap. Nauchn. Semin. Leningrad. Otdel. Mat. Inst. Steklov., 159 (1987), 143. doi: 10.1007/BF01305224. Google Scholar
[4]	W. Allegretto, Y. P. Lin and A. H. Zhou, Long-time stability of finite element approximations for parabolic equations with memory,, Numer. Methods Partial Differential Equations, 15 (1999), 333. doi: 10.1002/(SICI)1098-2426(199905)15:3<333::AID-NUM5>3.0.CO;2-0. Google Scholar
[5]	G. Araújo, S. Menezes and A. Marinho, Existence of solutions for an Oldroyd model of viscoelastic fluids,, J. Differential Equations, 2009 (). Google Scholar
[6]	R. Bird, R. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids. Vol. 1, Fluid Mechanics,", John Wiley & Sons, (1977). Google Scholar
[7]	J. Cannon, R. Ewing, Y. N. He and Y. P. Lin, A modified nonlinear Galerkin method for the viscoelastic fluid motion equations,, Int. J. Eng. Sci., 37 (1999), 1643. doi: 10.1016/S0020-7225(98)00142-6. Google Scholar
[8]	P. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, 4,, North-Holland Publishing Co., (1978). Google Scholar
[9]	V. Girault and P. Raviart, "Finite Element Approximation of the Navier-Stokes Equations,", Springer-Verlag, (1979). doi: 10.1007/BFb0063447. Google Scholar
[10]	D. Goswami and A. Pani, A priori error estimates for semidiscrete finite element approximations to equations of motion arising in Oldroyd fluids of order one,, Int. J. Numer. Anal. Model., 8 (2011), 324. Google Scholar
[11]	Y. N. He and Y. Li, Asymptotic behavior of linearized viscoelastic flow problem,, Discrete Contin. Dyn. Syst.-Ser. B, 10 (2008), 843. Google Scholar
[12]	Y. N. He and K. T. Li, Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem,, Numer. Math., 98 (2004), 647. doi: 10.1007/s00211-004-0532-y. Google Scholar
[13]	Y. N. He, Y. P. Lin, S. Shen and R. Tait, On the convergence of viscoelastic fluid flows to a steady state,, Adv. Differential Equations, 7 (2002), 717. Google Scholar
[14]	Y. N. He, Y. P. Lin, S. Shen, W. W. Sun and R. Tait, Finite element approximation for the viscoelastic fluid motion problem,, J. Comput. Appl. Math., 155 (2003), 201. Google Scholar
[15]	J. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem part IV: Error analysis for second-order time discretization,, SIAM J. Numer. Anal., 27 (1990), 353. doi: 10.1137/0727022. Google Scholar
[16]	D. Joseph, "Fluid Dynamics of Viscoelastic Liquids,", Applied Mathematical Sciences, 84 (1990). Google Scholar
[17]	A. Kotsiolis and A. Oskolkov, Initial-boundary value problems for equations of slightly compressible Jeffreys-Oldroyd fluids,, Zap. Nauchn. Semin. POMI, 208 (1993), 200. doi: 10.1007/BF02362429. Google Scholar
[18]	J. Oldroyd, On the formulation of the rheological equations of state,, Proc. Roy. Soc. London. Ser. A., 200 (1950), 523. Google Scholar
[19]	A. Oskolkov, Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, Contributions to "Boundary Value Problems of Mathematical Physics"(ed. J. Schulenberger),, Proc. Steklov Inst. Math., 179 (1989), 137. Google Scholar
[20]	A. Oskolkov, The penalty method for equations of viscoelastic media,, Zap. Nauchn. Semin. POMI, 224 (1995), 267. doi: 10.1007/BF02364990. Google Scholar
[21]	A. Pani and J. Yuan, Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model,, IMA J. Numer. Anal., 25 (2005), 750. doi: 10.1093/imanum/dri016. Google Scholar
[22]	A. Pani, J. Yuan and P. Damázio, On a linearized backward Euler method for the equations of motion of Oldroyd fluids of order one,, SIAM J. Numer. Anal., 44 (2006), 804. doi: 10.1137/S0036142903428967. Google Scholar
[23]	P. Sobolevskiĭ, Stabilization of viscoelastic fluid motion (Oldroyd's mathematical model),, Differential Integral Equations, 7 (1994), 1597. Google Scholar
[24]	P. Sobolevskiĭ, Asymptotic of stable viscoelastic fluid motion (Oldroyd's mathematical model),, Math. Nachr., 177 (1996), 281. doi: 10.1002/mana.19961770116. Google Scholar
[25]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", Third edition, 2 (1984). Google Scholar
[26]	R. Temam and X. Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a general 2D domain,, Asymptot. Anal., 14 (1997), 293. Google Scholar
[27]	R. Temam and X. Wang, Boundary layers asscociated with incompressible Navier-Stokes equations: The noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647. doi: 10.1006/jdeq.2001.4038. Google Scholar
[28]	K. Wang, Y. N. He and X. L. Feng, On error estimates of the penalty method for the viscoelastic flow problem I: Time discretization,, Appl. Math. Model., 34 (2010), 4089. doi: 10.1016/j.apm.2010.04.008. Google Scholar
[29]	K. Wang, Y. N. He and Y. Q. Shang, Fully discrete finite element method for the viscoelastic fluid motion equations,, Discrete Contin. Dyn. Syst.-Ser. B, 13 (2010), 665. Google Scholar
[30]	K. Wang, Y. Q. Shang and R. Zhao, Optimal error estimates of the penalty method for the linearized viscoelastic flows,, Int. J. Comput. Math., 87 (2010), 3236. doi: 10.1080/00207160902980500. Google Scholar

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