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 and Continuous Dynamical Systems, 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), 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. Marinho, Existence 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.

show all references

References:
[1]

Y. Agranovich and P. Sobolevskiĭ, Investigation 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. Marinho, Existence 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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