September  2014, 19(7): 2145-2157. doi: 10.3934/dcdsb.2014.19.2145

Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids

1. 

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

Received  May 2013 Revised  March 2014 Published  August 2014

We consider the free fall of slender rigid bodies in a viscous incompressible fluid. We show that the dimensional reduction (DR), performed by substituting the slender bodies with one-dimensional rigid objects, together with a hyperviscous regularization (HR) of the Navier--Stokes equation for the three-dimensional fluid lead to a well-posed fluid-structure interaction problem. In contrast to what can be achieved within a classical framework, the hyperviscous term permits a sound definition of the viscous force acting on the one-dimensional immersed body. Those results show that the DR/HR procedure can be effectively employed for the mathematical modeling of the free fall problem in the slender-body limit.
Citation: Giulio G. Giusteri, Alfredo Marzocchi, Alessandro Musesti. Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2145-2157. doi: 10.3934/dcdsb.2014.19.2145
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]

G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow,, J. Fluid Mech., 44 (1970), 419.  doi: 10.1017/S002211207000191X.  Google Scholar

[3]

A. T. Chwang and T. Y.-T. Wu, Hydromechanics of low-Reynolds-number flow. II. Singularity method for Stokes flows,, J. Fluid Mech., 67 (1975), 787.  doi: 10.1017/S0022112075000614.  Google Scholar

[4]

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

[5]

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, 1 (2002), 653.   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

G. G. Giusteri, A. Marzocchi and A. Musesti, Steady free fall of one-dimensional bodies in a hyperviscous fluid at low Reynolds number,, to appear in Evol. Equ. Control Theory (, ().   Google Scholar

[11]

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

[12]

R. E. Johnson, An improved slender-body theory for Stokes flow,, J. Fluid Mech., 99 (1980), 411.  doi: 10.1017/S0022112080000687.  Google Scholar

[13]

J. B. Keller and S. I. Rubinow, Slender-body theory for slow viscous flow,, J. Fluid Mech., 75 (1976), 705.  doi: 10.1017/S0022112076000475.  Google Scholar

[14]

J. Lighthill, Mathematical Biofluiddynamics,, SIAM, (1975).   Google Scholar

[15]

________, Flagellar hydrodynamics,, SIAM Rev., 18 (1976), 161.  doi: 10.1137/1018040.  Google Scholar

[16]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires,, Dunod, (1969).   Google Scholar

[17]

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

[18]

D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence,, Japan J. Appl. Math., 4 (1987), 99.  doi: 10.1007/BF03167757.  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]

G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow,, J. Fluid Mech., 44 (1970), 419.  doi: 10.1017/S002211207000191X.  Google Scholar

[3]

A. T. Chwang and T. Y.-T. Wu, Hydromechanics of low-Reynolds-number flow. II. Singularity method for Stokes flows,, J. Fluid Mech., 67 (1975), 787.  doi: 10.1017/S0022112075000614.  Google Scholar

[4]

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

[5]

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, 1 (2002), 653.   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

G. G. Giusteri, A. Marzocchi and A. Musesti, Steady free fall of one-dimensional bodies in a hyperviscous fluid at low Reynolds number,, to appear in Evol. Equ. Control Theory (, ().   Google Scholar

[11]

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

[12]

R. E. Johnson, An improved slender-body theory for Stokes flow,, J. Fluid Mech., 99 (1980), 411.  doi: 10.1017/S0022112080000687.  Google Scholar

[13]

J. B. Keller and S. I. Rubinow, Slender-body theory for slow viscous flow,, J. Fluid Mech., 75 (1976), 705.  doi: 10.1017/S0022112076000475.  Google Scholar

[14]

J. Lighthill, Mathematical Biofluiddynamics,, SIAM, (1975).   Google Scholar

[15]

________, Flagellar hydrodynamics,, SIAM Rev., 18 (1976), 161.  doi: 10.1137/1018040.  Google Scholar

[16]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires,, Dunod, (1969).   Google Scholar

[17]

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

[18]

D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence,, Japan J. Appl. Math., 4 (1987), 99.  doi: 10.1007/BF03167757.  Google Scholar

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