# American Institute of Mathematical Sciences

October  2014, 7(5): 1101-1109. doi: 10.3934/dcdss.2014.7.1101

## Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow

 1 Institute of Mathematics, University of Paderborn, D-33095 Paderborn, Germany

Received  March 2013 Revised  November 2013 Published  May 2014

For the transition from slip- to no-slip fluid flow, we establish lower and upper bounds to the resulting change of the $L^2$-norm of the vorticity. Moreover we present a transport-diffusion splitting scheme, built up solely by a transport step and subsequent diffusion step (without any additional vorticity creation operator as introduced in former studies by Lighthill, Marsden, and Chorin), the splitting scheme being consistent with the Navier-Stokes equations with no-slip condition.
Citation: Reimund Rautmann. Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1101-1109. doi: 10.3934/dcdss.2014.7.1101
##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [2] G. Alessandrini, A. Douglis and E. Fabes, An approximate layering method for the Navier-Stokes equations in bounded cylinders,, Ann. Mat. Pura Appl., 135 (1983), 329. doi: 10.1007/BF01781075. Google Scholar [3] H. Amann, Linear and Quasilinear Parabolic Problems,, Vol. 1, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar [4] M. Bause, Optimale Konvergenzraten F"ur voll Diskretisierte Navier-Stokes Approximationen H"oherer Ordnung in Gebieten mit Lipschitz-Rand,, Dissertation, (1997). Google Scholar [5] J. T. Beale and A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations,, Math. Comp., 37 (1981), 243. doi: 10.1090/S0025-5718-1981-0628693-0. Google Scholar [6] J. T. Beale and C. Greengard, Convergence of Euler-Stokes splitting of the Navier-Stokes equations,, IBM Research Report RC 18072 (1992), 47 (1992), 1083. doi: 10.1002/cpa.3160470805. Google Scholar [7] G. Benfatto and M. Pulvirenti, Generation of vorticity near the boundary in planar Navier-Stokes flows,, Commun. Math. Phys., 96 (1984), 59. doi: 10.1007/BF01217348. Google Scholar [8] G. Benfatto and M. Pulvirenti, Convergence of Chorin-Marsden product formula in the half-plane,, Commun. Math. Phys., 106 (1986), 427. doi: 10.1007/BF01207255. Google Scholar [9] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, Rend. Mat. Sem. Univ. Padova, 31 (1961), 308. Google Scholar [10] A. J. Chorin, Numerical study of slightly viscous flow,, J. Fluid Mech., 57 (1973), 785. doi: 10.1017/S0022112073002016. Google Scholar [11] A. J. Chorin, Vortex sheet approximation of boundary layers,, J. Comput. Phys., 27 (1978), 428. doi: 10.1016/0021-9991(78)90019-0. Google Scholar [12] A. J. Chorin, T. J. R. Hughes, M. F. McCracken and J. E. Marsden, Product formulas and numerical algorithms,, Comm. Pure Appl. Math., 31 (1978), 205. doi: 10.1002/cpa.3160310205. Google Scholar [13] P. Deuring and W. von Wahl, Strong solutions of the Navier-Stokes system in Lipschitz bounded domains,, Math. Nachr., 171 (1995), 111. doi: 10.1002/mana.19951710108. Google Scholar [14] A. Douglis and E. Fabes, A layering method for viscous, incompressible $L_p$ flows occupying $\mathbbR^n$,, Research Notes in Math., 108 (1984). Google Scholar [15] D. Fujiwara and H. Morimoto, An $L_r$-theorem of the Helmholtz decomposition of vector fields,, J. Fac. Sci. Tokyo U., 24 (1977), 685. Google Scholar [16] M.J. Lighthill, Introductions. Real and ideal fluids, II. Introduction. Boundary layer theory,, in Laminar Boundary Layers, (1963), 1. Google Scholar [17] J. Marsden, On product formulas for nonlinear semi-groups,, J. Funct. Anal., 13 (1973), 51. doi: 10.1016/0022-1236(73)90066-9. Google Scholar [18] J. Marsden, Applications of Global Analysis in Mathematical Physics,, Publish/Perish, (1974). Google Scholar [19] V. G. Maz'ja and B. A. Plamenevskii, First boundary value problem for the equation of hydrodynamic in a domain with a piecewise-smooth boundary,, J. Soviet. Math., (1983), 777. doi: 10.1007/BF01094440. Google Scholar [20] O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equation,, Numer. Math., 38 (1982), 309. doi: 10.1007/BF01396435. Google Scholar [21] L. Prandtl, Die Entstehung von Wirbeln in einer Flüssigkeit mit kleiner Reibung,, Zeitschrift für Flugtechnik und Motorluftschiffahrt, 18 (1927), 489. Google Scholar [22] L. Prandtl, The generation of vortices in fluids of small viscosity,, Journal of the Royal Aeronautical Society, 31 (1927), 720. Google Scholar [23] R. Rautmann, Ein Vektorpotentialmodell für die Wirbelbildung am Rand umströmter Körper,, Z. Angew. Math. Mech., 68 (1988), 383. doi: 10.1002/zamm.19880680823. Google Scholar [24] R. Rautmann, Eine konvergente Produktformel für linearisierte Navier-Stokes Probleme,, Z. Angew. Math. Mech., 69 (1989). Google Scholar [25] R. Rautmann, $H^2$-Convergent Linearizations to the Navier-Stokes Initial Value Problem,, in Proc. Intern Conf. on New Developments in Partial Differential Equations and Applications to Mathematical Physics, (1992), 14. Google Scholar [26] R. Rautmann and K. Masuda, $H^2$-Convergent Approximation Schemes to the Navier-Stokes Equations,, Comm. Math. Univ. Sancti Pauli, 43 (1994), 55. Google Scholar [27] H. Sohr, The Navier-Stokes Equations,, Birkhäuser, (2001). Google Scholar [28] V. A. Solonnikow, On the Stokes equations in domains with non-smooth boundaries and on viscous incompressible flow with a free surface,, in Nonlinear partial differential equations and their applications, (1982), 340. Google Scholar [29] L. Stupelis, Navier-Stokes Equations in Irregular Domains,, Kluwer Academic Publishers, (1995). Google Scholar [30] W. von Wahl, The equations of Navier-Stokes and Abstract Parabolic Equations,, Vieweg, (1985). Google Scholar [31] L.-A. Ying, Viscous splitting method for the unbounded problem of the Navier-Stokes equations,, Math. Comp., 55 (1990), 89. doi: 10.1090/S0025-5718-1990-1023053-0. Google Scholar [32] L.-A. Ying and P. Zhang, Vortex Methods,, Science Press, (1997). Google Scholar [33] Z. Yosida and Y. Giga, Remarks on spectra of operator rot,, Math. Zeitschrift, 204 (1990), 235. doi: 10.1007/BF02570870. Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [2] G. Alessandrini, A. Douglis and E. Fabes, An approximate layering method for the Navier-Stokes equations in bounded cylinders,, Ann. Mat. Pura Appl., 135 (1983), 329. doi: 10.1007/BF01781075. Google Scholar [3] H. Amann, Linear and Quasilinear Parabolic Problems,, Vol. 1, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar [4] M. Bause, Optimale Konvergenzraten F"ur voll Diskretisierte Navier-Stokes Approximationen H"oherer Ordnung in Gebieten mit Lipschitz-Rand,, Dissertation, (1997). Google Scholar [5] J. T. Beale and A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations,, Math. Comp., 37 (1981), 243. doi: 10.1090/S0025-5718-1981-0628693-0. Google Scholar [6] J. T. Beale and C. Greengard, Convergence of Euler-Stokes splitting of the Navier-Stokes equations,, IBM Research Report RC 18072 (1992), 47 (1992), 1083. doi: 10.1002/cpa.3160470805. Google Scholar [7] G. Benfatto and M. Pulvirenti, Generation of vorticity near the boundary in planar Navier-Stokes flows,, Commun. Math. Phys., 96 (1984), 59. doi: 10.1007/BF01217348. Google Scholar [8] G. Benfatto and M. Pulvirenti, Convergence of Chorin-Marsden product formula in the half-plane,, Commun. Math. Phys., 106 (1986), 427. doi: 10.1007/BF01207255. Google Scholar [9] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, Rend. Mat. Sem. Univ. Padova, 31 (1961), 308. Google Scholar [10] A. J. Chorin, Numerical study of slightly viscous flow,, J. Fluid Mech., 57 (1973), 785. doi: 10.1017/S0022112073002016. Google Scholar [11] A. J. Chorin, Vortex sheet approximation of boundary layers,, J. Comput. Phys., 27 (1978), 428. doi: 10.1016/0021-9991(78)90019-0. Google Scholar [12] A. J. Chorin, T. J. R. Hughes, M. F. McCracken and J. E. Marsden, Product formulas and numerical algorithms,, Comm. Pure Appl. Math., 31 (1978), 205. doi: 10.1002/cpa.3160310205. Google Scholar [13] P. Deuring and W. von Wahl, Strong solutions of the Navier-Stokes system in Lipschitz bounded domains,, Math. Nachr., 171 (1995), 111. doi: 10.1002/mana.19951710108. Google Scholar [14] A. Douglis and E. Fabes, A layering method for viscous, incompressible $L_p$ flows occupying $\mathbbR^n$,, Research Notes in Math., 108 (1984). Google Scholar [15] D. Fujiwara and H. Morimoto, An $L_r$-theorem of the Helmholtz decomposition of vector fields,, J. Fac. Sci. Tokyo U., 24 (1977), 685. Google Scholar [16] M.J. Lighthill, Introductions. Real and ideal fluids, II. Introduction. Boundary layer theory,, in Laminar Boundary Layers, (1963), 1. Google Scholar [17] J. Marsden, On product formulas for nonlinear semi-groups,, J. Funct. Anal., 13 (1973), 51. doi: 10.1016/0022-1236(73)90066-9. Google Scholar [18] J. Marsden, Applications of Global Analysis in Mathematical Physics,, Publish/Perish, (1974). Google Scholar [19] V. G. Maz'ja and B. A. Plamenevskii, First boundary value problem for the equation of hydrodynamic in a domain with a piecewise-smooth boundary,, J. Soviet. Math., (1983), 777. doi: 10.1007/BF01094440. Google Scholar [20] O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equation,, Numer. Math., 38 (1982), 309. doi: 10.1007/BF01396435. Google Scholar [21] L. Prandtl, Die Entstehung von Wirbeln in einer Flüssigkeit mit kleiner Reibung,, Zeitschrift für Flugtechnik und Motorluftschiffahrt, 18 (1927), 489. Google Scholar [22] L. Prandtl, The generation of vortices in fluids of small viscosity,, Journal of the Royal Aeronautical Society, 31 (1927), 720. Google Scholar [23] R. Rautmann, Ein Vektorpotentialmodell für die Wirbelbildung am Rand umströmter Körper,, Z. Angew. Math. Mech., 68 (1988), 383. doi: 10.1002/zamm.19880680823. Google Scholar [24] R. Rautmann, Eine konvergente Produktformel für linearisierte Navier-Stokes Probleme,, Z. Angew. Math. Mech., 69 (1989). Google Scholar [25] R. Rautmann, $H^2$-Convergent Linearizations to the Navier-Stokes Initial Value Problem,, in Proc. Intern Conf. on New Developments in Partial Differential Equations and Applications to Mathematical Physics, (1992), 14. Google Scholar [26] R. Rautmann and K. Masuda, $H^2$-Convergent Approximation Schemes to the Navier-Stokes Equations,, Comm. Math. Univ. Sancti Pauli, 43 (1994), 55. Google Scholar [27] H. Sohr, The Navier-Stokes Equations,, Birkhäuser, (2001). Google Scholar [28] V. A. Solonnikow, On the Stokes equations in domains with non-smooth boundaries and on viscous incompressible flow with a free surface,, in Nonlinear partial differential equations and their applications, (1982), 340. Google Scholar [29] L. Stupelis, Navier-Stokes Equations in Irregular Domains,, Kluwer Academic Publishers, (1995). Google Scholar [30] W. von Wahl, The equations of Navier-Stokes and Abstract Parabolic Equations,, Vieweg, (1985). Google Scholar [31] L.-A. Ying, Viscous splitting method for the unbounded problem of the Navier-Stokes equations,, Math. Comp., 55 (1990), 89. doi: 10.1090/S0025-5718-1990-1023053-0. Google Scholar [32] L.-A. Ying and P. Zhang, Vortex Methods,, Science Press, (1997). Google Scholar [33] Z. Yosida and Y. Giga, Remarks on spectra of operator rot,, Math. Zeitschrift, 204 (1990), 235. doi: 10.1007/BF02570870. Google Scholar
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