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Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid

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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.
    Mathematics Subject Classification: Primary: 35L70, 76D05, 76A10.

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  • [1]

    Y. Agranovich and P. SobolevskiĭInvestigation of a mathematical model of a viscoelastic fluid (Russian), Dokl. Akad. Nauk Ukrain. SSR Ser. A, 1989, 3-6, 86.

    [2]

    Y. Agranovich and P. Sobolevskiĭ, Motion of non-linear visco-elastic fluid, Nonlinear Anal., 32 (1998), 755-760.doi: 10.1016/S0362-546X(97)00519-1.

    [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), Chisl. Metody i Voprosy Organiz. Vychisl. 8, 143-152, 179, translation in J. Soviet. Math., 47 (1989), 2926-2933.doi: 10.1007/BF01305224.

    [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), 333-354.doi: 10.1002/(SICI)1098-2426(199905)15:3<333::AID-NUM5>3.0.CO;2-0.

    [5]

    G. Araújo, S. Menezes and A. MarinhoExistence of solutions for an Oldroyd model of viscoelastic fluids, J. Differential Equations, 2009, 16 pp.

    [6]

    R. Bird, R. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids. Vol. 1, Fluid Mechanics," John Wiley & Sons, New York, 1977.

    [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), 1643-1662.doi: 10.1016/S0020-7225(98)00142-6.

    [8]

    P. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

    [9]

    V. Girault and P. Raviart, "Finite Element Approximation of the Navier-Stokes Equations," Springer-Verlag, Berlin, 1979.doi: 10.1007/BFb0063447.

    [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), 324-352.

    [11]

    Y. N. He and Y. Li, Asymptotic behavior of linearized viscoelastic flow problem, Discrete Contin. Dyn. Syst.-Ser. B, 10 (2008), 843-856.

    [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), 647-673.doi: 10.1007/s00211-004-0532-y.

    [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), 717-742.

    [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), 201-222.

    [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), 353-384.doi: 10.1137/0727022.

    [16]

    D. Joseph, "Fluid Dynamics of Viscoelastic Liquids," Applied Mathematical Sciences, 84, Springer-Verlag, New York, 1990.

    [17]

    A. Kotsiolis and A. Oskolkov, Initial-boundary value problems for equations of slightly compressible Jeffreys-Oldroyd fluids, Zap. Nauchn. Semin. POMI, 208 (1993), 200-218, 223, translation in J. Math. Sci., 81 (1996), 2578-2588.doi: 10.1007/BF02362429.

    [18]

    J. Oldroyd, On the formulation of the rheological equations of state, Proc. Roy. Soc. London. Ser. A., 200 (1950), 523-541.

    [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), 137-182.

    [20]

    A. Oskolkov, The penalty method for equations of viscoelastic media, Zap. Nauchn. Semin. POMI, 224 (1995), 267-278, 340-341, translation in J. Math. Sci. (New York), 88 (1998), 283-291.doi: 10.1007/BF02364990.

    [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), 750-782.doi: 10.1093/imanum/dri016.

    [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), 804-825.doi: 10.1137/S0036142903428967.

    [23]

    P. Sobolevskiĭ, Stabilization of viscoelastic fluid motion (Oldroyd's mathematical model), Differential Integral Equations, 7 (1994), 1597-1612.

    [24]

    P. Sobolevskiĭ, Asymptotic of stable viscoelastic fluid motion (Oldroyd's mathematical model), Math. Nachr., 177 (1996), 281-305.doi: 10.1002/mana.19961770116.

    [25]

    R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," Third edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.

    [26]

    R. Temam and X. Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a general 2D domain, Asymptot. Anal., 14 (1997), 293-321.

    [27]

    R. Temam and X. Wang, Boundary layers asscociated with incompressible Navier-Stokes equations: The noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686.doi: 10.1006/jdeq.2001.4038.

    [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), 4089-4105.doi: 10.1016/j.apm.2010.04.008.

    [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), 665-684.

    [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), 3236-3253.doi: 10.1080/00207160902980500.

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