Abstract
We present and analyze fully discrete fractional time stepping techniques for the solution of the micropolar Navier Stokes equations, which is a system of equations that describes the evolution of an incompressible fluid whose material particles possess both translational and rotational degrees of freedom. The proposed schemes uncouple the computation of the linear and angular velocity and the pressure. We develop a first order scheme which is unconditionally stable and delivers optimal convergence rates, and an almost unconditionally stable second order scheme.
Similar content being viewed by others
References
Amirat, Y., Hamdache, K.: Unique solvability of equations of motion for ferrofluids. Nonlinear Anal. 73(2), 471–494 (2010)
Amirat, Y., Hamdache, K., Murat, F.: Global weak solutions to equations of motion for magnetic fluids. J. Math. Fluid Mech. 10(3), 326–351 (2008)
Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general purpose object oriented finite element library. ACM Trans. Math. Softw. 4(33), 24/1–24/27 (2007)
Bangerth, W., Heister, T., Kanschat, G.: deal.II Differential Equations Analysis Library, Technical Reference. http://www.dealii.org
Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comp. 22, 745–762 (1968)
Chorin, A.J.: On the convergence of discrete approximations to the Navier–Stokes equations. Math. Comp. 23, 341–353 (1969)
Dahler, J.S., Scriven, L.E.: Angular momentum of continua. Nature 192, 36–37 (1961)
Dahler, J.S., Scriven, L.E.: Theory of structured continua. I. General consideration of angular momentum and polarization. Proc. R. Soc. 275(1363), 504–527 (1963)
Eringen, A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)
Eringen, A.C.: Microcontinuum Field Theories I. Foundations and Solids. Springer, New York (1999)
Eringen, A.C.: Microcontinuum Field Theories II. Fluent Media. Springer, New York (2001)
Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)
Fu, L.M., Tsai, C.H., Leong, K.P., Wen, C.Y.: Rapid micromixer via ferrofluids. Phys. Procedia, 9, 270–273 (2010); 12th International Conference on Magnetic Fluids (ICMF12)
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations Theory and Algorithms. Springer, Berlin (1986)
Guermond, J.-L.: Un résultat de convergence d’ordre deux en temps pour l’approximation des équations de Navier–Stokes par une technique de projection incrémentale. M2AN Math. Model. Numer. Anal. 33(1), 169–189 (1999)
Guermond, J.-L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44–47), 6011–6045 (2006)
Guermond, J.-L., Pasquetti, R., Popov, B.: Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230(11), 4248–4267 (2011)
Guermond, J.-L., Quartapelle, L.: On the approximation of the unsteady Navier–Stokes equations by finite element projection methods. Numer. Math. 80(5), 207–238 (1998)
Guermond, J.-L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228(8), 2834–2846 (2009)
Guermond, J.-L., Salgado, A.J.: Error analysis of a fractional time-stepping technique for incompressible flows with variable density. SIAM J. Numer. Anal. 49(3), 917–944 (2011)
He, Y.: The Euler implicit/explicit scheme for the 2D time-dependent Navier–Stokes equations with smooth or non-smooth initial data. Math. Comp. 77(264), 2097–2124 (2008)
Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)
Łukaszewicz, G.: Micropolar fluids. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (1999). Theory and applications
Marion, M., Temam, R.: Navier–Stokes equations: theory and approximation. In: Handbook of Numerical Analysis, vol. VI, Handb. Numer. Anal., VI, pp. 503–688. North-Holland, Amsterdam (1998)
Nochetto, R.H., Salgado, A.J., Tomas, I.: Ferrohydrodynamics: Modeling Issues and Numerical Methods. In preparation (2014)
Nochetto, R.H., Salgado, A.J., Tomas, I.: The micropolar Navier–Stokes equations: a priori error analysis. Math. Models Methods Appl. Sci. 24(7), 1237–1264 (2014)
Ortega-Torres, E., Rojas-Medar, M.: Optimal error estimate of the penalty finite element method for the micropolar fluid equations. Numer. Funct. Anal. Optim. 29(5–6), 612–637 (2008)
Prohl, A.: Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier–Stokes Equations. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1997)
Rosensweig, R.E.: Ferrohydrodynamics. Dover, New York (1997)
Temam, R.: Une méthode d’approximation de la solution des équations de Navier–Stokes. Bull. Soc. Math. France 96, 115–152 (1968)
Temam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires ii. Arch. Rat. Mech. Anal. 33, 377–385 (1969)
Temam, R.: Navier–Stokes Equations. AMS Chelsea Publishing, Providence, RI (2001). Theory and numerical analysis, Reprint of the 1984 edition
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, volume 25 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2006)
Tone, F.: Error analysis for a second order scheme for the Navier–Stokes equations. Appl. Numer. Math. 50(1), 93–119 (2004)
Tsai, T.-H., Liou, D.-S., Kuo, L.-S., Chen, P.-H.: Rapid mixing between ferro-nanofluid and water in a semi-active y-type micromixer. Sens. Actuators A: Phys. 153(2), 267–273 (2009)
Wheeler, M.F.: A priori \(L_{2}\) error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10, 723–759 (1973)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Salgado, A.J. Convergence Analysis of Fractional Time-Stepping Techniques for Incompressible Fluids with Microstructure. J Sci Comput 64, 216–233 (2015). https://doi.org/10.1007/s10915-014-9926-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-014-9926-x