# American Institute of Mathematical Sciences

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

## Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid

 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

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.
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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