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 and 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.

[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.

[3]

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

[4]

M. Bause, Optimale Konvergenzraten F"ur voll Diskretisierte Navier-Stokes Approximationen H"oherer Ordnung in Gebieten mit Lipschitz-Rand, Dissertation, Paderborn, 1997.

[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.

[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.

[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.

[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.

[9]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova, 31 (1961), 308-340.

[10]

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

[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.

[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.

[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.

[14]

A. Douglis and E. Fabes, A layering method for viscous, incompressible $L_p$ flows occupying $\mathbb{R}^{N}$, Research Notes in Math., 108 (1984), Pitman.

[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.

[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.

[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.

[18]

J. Marsden, Applications of Global Analysis in Mathematical Physics, Publish/Perish, Boston, 1974.

[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.

[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.

[21]

L. Prandtl, Die Entstehung von Wirbeln in einer Flüssigkeit mit kleiner Reibung, Zeitschrift für Flugtechnik und Motorluftschiffahrt, 18 (1927), 489-496.

[22]

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

[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.

[24]

R. Rautmann, Eine konvergente Produktformel für linearisierte Navier-Stokes Probleme, Z. Angew. Math. Mech., 69 (1989), T181-T83.

[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.

[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.

[27]

H. Sohr, The Navier-Stokes Equations, Birkhäuser, Basel, 2001.

[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.

[29]

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

[30]

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

[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.

[32]

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

[33]

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

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[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.

[3]

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

[4]

M. Bause, Optimale Konvergenzraten F"ur voll Diskretisierte Navier-Stokes Approximationen H"oherer Ordnung in Gebieten mit Lipschitz-Rand, Dissertation, Paderborn, 1997.

[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.

[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.

[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.

[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.

[9]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova, 31 (1961), 308-340.

[10]

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

[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.

[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.

[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.

[14]

A. Douglis and E. Fabes, A layering method for viscous, incompressible $L_p$ flows occupying $\mathbb{R}^{N}$, Research Notes in Math., 108 (1984), Pitman.

[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.

[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.

[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.

[18]

J. Marsden, Applications of Global Analysis in Mathematical Physics, Publish/Perish, Boston, 1974.

[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.

[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.

[21]

L. Prandtl, Die Entstehung von Wirbeln in einer Flüssigkeit mit kleiner Reibung, Zeitschrift für Flugtechnik und Motorluftschiffahrt, 18 (1927), 489-496.

[22]

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

[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.

[24]

R. Rautmann, Eine konvergente Produktformel für linearisierte Navier-Stokes Probleme, Z. Angew. Math. Mech., 69 (1989), T181-T83.

[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.

[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.

[27]

H. Sohr, The Navier-Stokes Equations, Birkhäuser, Basel, 2001.

[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.

[29]

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

[30]

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

[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.

[32]

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

[33]

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

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