# American Institute of Mathematical Sciences

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 and 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), 623-727. doi: 10.1002/cpa.3160120405. [2] G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech., 44 (1970), 419-440. doi: 10.1017/S002211207000191X. [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), 787-815. doi: 10.1017/S0022112075000614. [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), 513-554. doi: 10.1007/s00205-006-0015-7. [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, North-Holland, Amsterdam, 1 (2002), 653-791. [6] 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. [7] 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. [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), 642-646. doi: 10.1016/j.mechrescom.2010.09.001. [9] 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. [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 (, (). [11] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Martinus Nijhoff Publishers, The Hague, 1983. [12] R. E. Johnson, An improved slender-body theory for Stokes flow, J. Fluid Mech., 99 (1980), 411-431. doi: 10.1017/S0022112080000687. [13] J. B. Keller and S. I. Rubinow, Slender-body theory for slow viscous flow, J. Fluid Mech., 75 (1976), 705-714. doi: 10.1017/S0022112076000475. [14] J. Lighthill, Mathematical Biofluiddynamics, SIAM, Philadelphia, 1975. [15] ________, Flagellar hydrodynamics, SIAM Rev., 18 (1976), 161-230. doi: 10.1137/1018040. [16] J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, Paris, 1969. [17] A. Musesti, Isotropic linear constitutive relations for nonsimple fluids, Acta Mech., 204 (2009), 81-88. doi: 10.1007/s00707-008-0050-6. [18] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110. doi: 10.1007/BF03167757.

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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), 623-727. doi: 10.1002/cpa.3160120405. [2] G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech., 44 (1970), 419-440. doi: 10.1017/S002211207000191X. [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), 787-815. doi: 10.1017/S0022112075000614. [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), 513-554. doi: 10.1007/s00205-006-0015-7. [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, North-Holland, Amsterdam, 1 (2002), 653-791. [6] 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. [7] 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. [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), 642-646. doi: 10.1016/j.mechrescom.2010.09.001. [9] 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. [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 (, (). [11] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Martinus Nijhoff Publishers, The Hague, 1983. [12] R. E. Johnson, An improved slender-body theory for Stokes flow, J. Fluid Mech., 99 (1980), 411-431. doi: 10.1017/S0022112080000687. [13] J. B. Keller and S. I. Rubinow, Slender-body theory for slow viscous flow, J. Fluid Mech., 75 (1976), 705-714. doi: 10.1017/S0022112076000475. [14] J. Lighthill, Mathematical Biofluiddynamics, SIAM, Philadelphia, 1975. [15] ________, Flagellar hydrodynamics, SIAM Rev., 18 (1976), 161-230. doi: 10.1137/1018040. [16] J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, Paris, 1969. [17] A. Musesti, Isotropic linear constitutive relations for nonsimple fluids, Acta Mech., 204 (2009), 81-88. doi: 10.1007/s00707-008-0050-6. [18] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110. doi: 10.1007/BF03167757.
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