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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

Giulio G. Giusteri 1, , Alfredo Marzocchi 1, and  Alessandro Musesti 1,

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.
Keywords: dimensional reduction., Slender-body theory, low-Reynolds-number flow, fluid-structure interaction, hyperviscosity.
Mathematics Subject Classification: Primary: 76D07; Secondary: 35Q35, 35J9.
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 and 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), 623-727. doi: 10.1002/cpa.3160120405.

[2]

H. Brenner, The Stokes resistance of an arbitrary particle-II: An extension, Chem. Eng. Sci., 19 (1964), 599-629. doi: 10.1016/0009-2509(64)85051-X.

[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), 513-554. doi: 10.1007/s00205-006-0015-7.

[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, Amsterdam, 1 (2002), 653-791. doi: 10.1016/S1874-5792(02)80014-3.

[5]

G. G. Giusteri, The multiple nature of concentrated interactions in second-gradient dissipative liquids, Z. Angew. Math. Phys., 64 (2013), 371-380. doi: 10.1007/s00033-012-0229-5.

[6]

G. G. Giusteri and E. Fried, Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization, Meccanica, 49 (2014), 2153-2167. doi: 10.1007/s11012-014-9890-4.

[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), 642-646. doi: 10.1016/j.mechrescom.2010.09.001.

[8]

G. G. Giusteri, A. Marzocchi, and A. Musesti, Nonsimple isotropic incompressible linear fluids surrounding one-dimensional structures, Acta Mech., 217 (2011), 191-204. doi: 10.1007/s00707-010-0387-5.

[9]

G. G. Giusteri, A. Marzocchi, and A. Musesti, Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids, DCDS-B, 19 (2014), 2145-2157. doi: 10.3934/dcdsb.2014.19.2145.

[10]

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Martinus Nijhoff Publishers, The Hague, 1983.

[11]

A. Musesti, Isotropic linear constitutive relations for nonsimple fluids, Acta Mech., 204 (2009), 81-88. doi: 10.1007/s00707-008-0050-6.

[12]

H. F. Weinberger, Variational properties of steady fall in Stokes flow, J. Fluid Mech., 52 (1972), 321-344. doi: 10.1017/S0022112072001442.

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), 623-727. doi: 10.1002/cpa.3160120405.

[2]

H. Brenner, The Stokes resistance of an arbitrary particle-II: An extension, Chem. Eng. Sci., 19 (1964), 599-629. doi: 10.1016/0009-2509(64)85051-X.

[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), 513-554. doi: 10.1007/s00205-006-0015-7.

[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, Amsterdam, 1 (2002), 653-791. doi: 10.1016/S1874-5792(02)80014-3.

[5]

G. G. Giusteri, The multiple nature of concentrated interactions in second-gradient dissipative liquids, Z. Angew. Math. Phys., 64 (2013), 371-380. doi: 10.1007/s00033-012-0229-5.

[6]

G. G. Giusteri and E. Fried, Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization, Meccanica, 49 (2014), 2153-2167. doi: 10.1007/s11012-014-9890-4.

[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), 642-646. doi: 10.1016/j.mechrescom.2010.09.001.

[8]

G. G. Giusteri, A. Marzocchi, and A. Musesti, Nonsimple isotropic incompressible linear fluids surrounding one-dimensional structures, Acta Mech., 217 (2011), 191-204. doi: 10.1007/s00707-010-0387-5.

[9]

G. G. Giusteri, A. Marzocchi, and A. Musesti, Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids, DCDS-B, 19 (2014), 2145-2157. doi: 10.3934/dcdsb.2014.19.2145.

[10]

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Martinus Nijhoff Publishers, The Hague, 1983.

[11]

A. Musesti, Isotropic linear constitutive relations for nonsimple fluids, Acta Mech., 204 (2009), 81-88. doi: 10.1007/s00707-008-0050-6.

[12]

H. F. Weinberger, Variational properties of steady fall in Stokes flow, J. Fluid Mech., 52 (1972), 321-344. doi: 10.1017/S0022112072001442.

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