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, New York, 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), 329-347. doi: 10.1007/BF01781075.  Google Scholar

[3]

H. Amann, Linear and Quasilinear Parabolic Problems, Vol. 1, Abstract linear theory, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

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M. Bause, Optimale Konvergenzraten F"ur voll Diskretisierte Navier-Stokes Approximationen H"oherer Ordnung in Gebieten mit Lipschitz-Rand, Dissertation, Paderborn, 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), 243-259. 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), Comm. Pure Appl. Math., 47 (1994), 1083-1115. doi: 10.1002/cpa.3160470805.  Google Scholar

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G. Benfatto and M. Pulvirenti, Generation of vorticity near the boundary in planar Navier-Stokes flows, Commun. Math. Phys., 96 (1984), 59-95. doi: 10.1007/BF01217348.  Google Scholar

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G. Benfatto and M. Pulvirenti, Convergence of Chorin-Marsden product formula in the half-plane, Commun. Math. Phys., 106 (1986), 427-458. 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), 308-340.  Google Scholar

[10]

A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57 (1973), 785-796. doi: 10.1017/S0022112073002016.  Google Scholar

[11]

A. J. Chorin, Vortex sheet approximation of boundary layers, J. Comput. Phys., 27 (1978), 428-442. 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), 205-256. 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), 111-148. 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), Pitman. Google Scholar

[15]

D. Fujiwara and H. Morimoto, An $L_r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Tokyo U., Sec 1A-Math., 24 (1977), 685-700.  Google Scholar

[16]

M.J. Lighthill, Introductions. Real and ideal fluids, II. Introduction. Boundary layer theory, in Laminar Boundary Layers, (ed. L.Rosenhead), Oxford University Press, (1963), 1-113. Google Scholar

[17]

J. Marsden, On product formulas for nonlinear semi-groups, J. Funct. Anal., 13 (1973), 51-72. doi: 10.1016/0022-1236(73)90066-9.  Google Scholar

[18]

J. Marsden, Applications of Global Analysis in Mathematical Physics, Publish/Perish, Boston, 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), 777-782. 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), 309-32. 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), 489-496. Google Scholar

[22]

L. Prandtl, The generation of vortices in fluids of small viscosity, Journal of the Royal Aeronautical Society, 31 (1927), 720-741. Google Scholar

[23]

R. Rautmann, Ein Vektorpotentialmodell für die Wirbelbildung am Rand umströmter Körper, Z. Angew. Math. Mech., 68 (1988), 383-387. doi: 10.1002/zamm.19880680823.  Google Scholar

[24]

R. Rautmann, Eine konvergente Produktformel für linearisierte Navier-Stokes Probleme, Z. Angew. Math. Mech., 69 (1989), T181-T83.  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, (eds. G. Butazzo, G. H. Galdi and L. Zanghirati), Ferrara 14-18, 1991, Plenum Press, New York, (1992), 135-156.  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), 55-108.  Google Scholar

[27]

H. Sohr, The Navier-Stokes Equations, Birkhäuser, Basel, 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, Collège de France Seminar III, (eds. H. Brezis and J. L. Lions), (1982), 340-423.  Google Scholar

[29]

L. Stupelis, Navier-Stokes Equations in Irregular Domains, Kluwer Academic Publishers, Dordrecht 1995.  Google Scholar

[30]

W. von Wahl, The equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg, Braunschweig 1985.  Google Scholar

[31]

L.-A. Ying, Viscous splitting method for the unbounded problem of the Navier-Stokes equations, Math. Comp., 55 (1990), 89-113. doi: 10.1090/S0025-5718-1990-1023053-0.  Google Scholar

[32]

L.-A. Ying and P. Zhang, Vortex Methods, Science Press, Beijing/New York, Kluwer Academic Publishers, Dordrecht/Boston/London 1997.  Google Scholar

[33]

Z. Yosida and Y. Giga, Remarks on spectra of operator rot, Math. Zeitschrift, 204 (1990), 235-245. doi: 10.1007/BF02570870.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 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), 329-347. doi: 10.1007/BF01781075.  Google Scholar

[3]

H. Amann, Linear and Quasilinear Parabolic Problems, Vol. 1, Abstract linear theory, Birkhäuser, Basel, 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, Paderborn, 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), 243-259. 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), Comm. Pure Appl. Math., 47 (1994), 1083-1115. 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), 59-95. 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), 427-458. 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), 308-340.  Google Scholar

[10]

A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57 (1973), 785-796. doi: 10.1017/S0022112073002016.  Google Scholar

[11]

A. J. Chorin, Vortex sheet approximation of boundary layers, J. Comput. Phys., 27 (1978), 428-442. 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), 205-256. 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), 111-148. 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), Pitman. Google Scholar

[15]

D. Fujiwara and H. Morimoto, An $L_r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Tokyo U., Sec 1A-Math., 24 (1977), 685-700.  Google Scholar

[16]

M.J. Lighthill, Introductions. Real and ideal fluids, II. Introduction. Boundary layer theory, in Laminar Boundary Layers, (ed. L.Rosenhead), Oxford University Press, (1963), 1-113. Google Scholar

[17]

J. Marsden, On product formulas for nonlinear semi-groups, J. Funct. Anal., 13 (1973), 51-72. doi: 10.1016/0022-1236(73)90066-9.  Google Scholar

[18]

J. Marsden, Applications of Global Analysis in Mathematical Physics, Publish/Perish, Boston, 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), 777-782. 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), 309-32. 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), 489-496. Google Scholar

[22]

L. Prandtl, The generation of vortices in fluids of small viscosity, Journal of the Royal Aeronautical Society, 31 (1927), 720-741. Google Scholar

[23]

R. Rautmann, Ein Vektorpotentialmodell für die Wirbelbildung am Rand umströmter Körper, Z. Angew. Math. Mech., 68 (1988), 383-387. doi: 10.1002/zamm.19880680823.  Google Scholar

[24]

R. Rautmann, Eine konvergente Produktformel für linearisierte Navier-Stokes Probleme, Z. Angew. Math. Mech., 69 (1989), T181-T83.  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, (eds. G. Butazzo, G. H. Galdi and L. Zanghirati), Ferrara 14-18, 1991, Plenum Press, New York, (1992), 135-156.  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), 55-108.  Google Scholar

[27]

H. Sohr, The Navier-Stokes Equations, Birkhäuser, Basel, 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, Collège de France Seminar III, (eds. H. Brezis and J. L. Lions), (1982), 340-423.  Google Scholar

[29]

L. Stupelis, Navier-Stokes Equations in Irregular Domains, Kluwer Academic Publishers, Dordrecht 1995.  Google Scholar

[30]

W. von Wahl, The equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg, Braunschweig 1985.  Google Scholar

[31]

L.-A. Ying, Viscous splitting method for the unbounded problem of the Navier-Stokes equations, Math. Comp., 55 (1990), 89-113. doi: 10.1090/S0025-5718-1990-1023053-0.  Google Scholar

[32]

L.-A. Ying and P. Zhang, Vortex Methods, Science Press, Beijing/New York, Kluwer Academic Publishers, Dordrecht/Boston/London 1997.  Google Scholar

[33]

Z. Yosida and Y. Giga, Remarks on spectra of operator rot, Math. Zeitschrift, 204 (1990), 235-245. doi: 10.1007/BF02570870.  Google Scholar

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