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

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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

Abstract
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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 & 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

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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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