September  2014, 3(3): 429-445. doi: 10.3934/eect.2014.3.429

Steady free fall of one-dimensional bodies in a hyperviscous fluid at low Reynolds number

1. 

Dipartimento di Matematica e Fisica "N. Tartaglia", Università Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia

Received  April 2013 Revised  January 2014 Published  August 2014

The paper is devoted to the study of the motion of one-dimensional rigid bodies during a free fall in a quasi-Newtonian hyperviscous fluid at low Reynolds number. We show the existence of a steady solution and furnish sufficient conditions on the geometry of the body in order to get purely translational motions. Such conditions are based on a generalized version of the so-called Reciprocal Theorem for fluids.
Citation: Giulio G. Giusteri, Alfredo Marzocchi, Alessandro Musesti. Steady free fall of one-dimensional bodies in a hyperviscous fluid at low Reynolds number. Evolution Equations & Control Theory, 2014, 3 (3) : 429-445. doi: 10.3934/eect.2014.3.429
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,, Comm. Pure Appl. Math., 12 (1959), 623.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

H. Brenner, The Stokes resistance of an arbitrary particle-II: An extension,, Chem. Eng. Sci., 19 (1964), 599.  doi: 10.1016/0009-2509(64)85051-X.  Google Scholar

[3]

E. Fried and M. E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales,, Arch. Ration. Mech. Anal., 182 (2006), 513.  doi: 10.1007/s00205-006-0015-7.  Google Scholar

[4]

G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in Handbook of mathematical fluid dynamics. North-Holland, 1 (2002), 653.  doi: 10.1016/S1874-5792(02)80014-3.  Google Scholar

[5]

G. G. Giusteri, The multiple nature of concentrated interactions in second-gradient dissipative liquids,, Z. Angew. Math. Phys., 64 (2013), 371.  doi: 10.1007/s00033-012-0229-5.  Google Scholar

[6]

G. G. Giusteri and E. Fried, Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization,, Meccanica, 49 (2014), 2153.  doi: 10.1007/s11012-014-9890-4.  Google Scholar

[7]

G. G. Giusteri, A. Marzocchi and A. Musesti, Three-dimensional nonsimple viscous liquids dragged by one-dimensional immersed bodies,, Mech. Res. Commun., 37 (2010), 642.  doi: 10.1016/j.mechrescom.2010.09.001.  Google Scholar

[8]

G. G. Giusteri, A. Marzocchi, and A. Musesti, Nonsimple isotropic incompressible linear fluids surrounding one-dimensional structures,, Acta Mech., 217 (2011), 191.  doi: 10.1007/s00707-010-0387-5.  Google Scholar

[9]

G. G. Giusteri, A. Marzocchi, and A. Musesti, Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids,, DCDS-B, 19 (2014), 2145.  doi: 10.3934/dcdsb.2014.19.2145.  Google Scholar

[10]

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media,, Martinus Nijhoff Publishers, (1983).   Google Scholar

[11]

A. Musesti, Isotropic linear constitutive relations for nonsimple fluids,, Acta Mech., 204 (2009), 81.  doi: 10.1007/s00707-008-0050-6.  Google Scholar

[12]

H. F. Weinberger, Variational properties of steady fall in Stokes flow,, J. Fluid Mech., 52 (1972), 321.  doi: 10.1017/S0022112072001442.  Google Scholar

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,, Comm. Pure Appl. Math., 12 (1959), 623.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

H. Brenner, The Stokes resistance of an arbitrary particle-II: An extension,, Chem. Eng. Sci., 19 (1964), 599.  doi: 10.1016/0009-2509(64)85051-X.  Google Scholar

[3]

E. Fried and M. E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales,, Arch. Ration. Mech. Anal., 182 (2006), 513.  doi: 10.1007/s00205-006-0015-7.  Google Scholar

[4]

G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in Handbook of mathematical fluid dynamics. North-Holland, 1 (2002), 653.  doi: 10.1016/S1874-5792(02)80014-3.  Google Scholar

[5]

G. G. Giusteri, The multiple nature of concentrated interactions in second-gradient dissipative liquids,, Z. Angew. Math. Phys., 64 (2013), 371.  doi: 10.1007/s00033-012-0229-5.  Google Scholar

[6]

G. G. Giusteri and E. Fried, Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization,, Meccanica, 49 (2014), 2153.  doi: 10.1007/s11012-014-9890-4.  Google Scholar

[7]

G. G. Giusteri, A. Marzocchi and A. Musesti, Three-dimensional nonsimple viscous liquids dragged by one-dimensional immersed bodies,, Mech. Res. Commun., 37 (2010), 642.  doi: 10.1016/j.mechrescom.2010.09.001.  Google Scholar

[8]

G. G. Giusteri, A. Marzocchi, and A. Musesti, Nonsimple isotropic incompressible linear fluids surrounding one-dimensional structures,, Acta Mech., 217 (2011), 191.  doi: 10.1007/s00707-010-0387-5.  Google Scholar

[9]

G. G. Giusteri, A. Marzocchi, and A. Musesti, Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids,, DCDS-B, 19 (2014), 2145.  doi: 10.3934/dcdsb.2014.19.2145.  Google Scholar

[10]

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media,, Martinus Nijhoff Publishers, (1983).   Google Scholar

[11]

A. Musesti, Isotropic linear constitutive relations for nonsimple fluids,, Acta Mech., 204 (2009), 81.  doi: 10.1007/s00707-008-0050-6.  Google Scholar

[12]

H. F. Weinberger, Variational properties of steady fall in Stokes flow,, J. Fluid Mech., 52 (1972), 321.  doi: 10.1017/S0022112072001442.  Google Scholar

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