Self-propelled motion of a rigid body inside a density dependent incompressible fluid

Š. Nečasová¹ ; M. Ramaswamy² ; A. Roy¹ ; A. Schlömerkemper³

¹ Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic.
² Chennai Mathematical Institute, H1, SITCOT IT Park, Siruseri 603103, India.
³ Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 9.

Résumé

This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole ℝ3. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution.

Détail

DOI : 10.1051/mmnp/2020052

Bibliographie

[1] H. Al Baba, N.V. Chemetov, Š. Nečasová, B. Muha Strong solutions in L2 framework for fluid-rigid body interaction problem. Mixed case Topol. Methods Nonlinear Anal. 2018 337 350

[2] F. Alouges, A. Desimone, A. Lefebvre Optimal strokes for low Reynolds number swimmers : an example J. Nonlinear Sci 2008 277 302

[3] T. Bodnár, G.P. Galdi and Š. Nečasová, Fluid-Structure Interaction and Biomedical Applications. Birkhäuser/Springer, Basel (2014).

[4] T. Bodnár, G.P. Galdi and Š. Nečasová, Particles in flows. Advances in Mathematical Fluid Mechanics. Birkhäuser/Springer, Cham (2017).

[5] F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models. Vol. 183 of Applied Mathematical Sciences. Springer, New York (2013).

[6] A. Bressan Impulsive control of Lagrangian systems and locomotion in fluids Discrete Contin. Dyn. Syst 2008 1 35

[7] D. Bucur, E. Feireisl, Š. Nečasová Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions Arch. Ration. Mech. Anal 2010 117 138

[8] T. Chambrion, A. Munnier locomotion and control of a self-propelled shape-changing body in a fluid J. Nonlinear Sci 2011 325 385

[9] N.V. Chemetov, Š. Nečasová The motion of the rigid body in the viscous fluid including collisions. Global solvability result Nonlinear Anal. Real World Appl. 2017 416 445

[10] C. Conca, M. Tucsnak Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid Commun. Partial Differ. Equ 2000 1019 1042

[11] P. Cumsille, T. Takahashi Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid Czechoslovak Math. J 2008 961 992

[12] B. Desjardins Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space Differ. Integral Equ 1997 587 598

[13] B. Desjardins, M.J. Esteban Existence of weak solutions for the motion of rigid bodies in a viscous fluid Arch. Ration. Mech. Anal 1999 59 71

[14] B. Desjardins, M.J. Esteban On weak solutions for fluid-rigid structure interaction : compressible and incompressible models Commun. Partial Differ. Equ 2000 1399 1413

[15] R.J. Diperna, P.L. Lions Ordinary differential equations, transport theory and Sobolev spaces Invent. Math 1989 511 547

[16] G. Galdi On the steady self-propelled motion of a body in a viscous incompressible fluid Arch. Rat. Mech. Anal 1999 53 88

[17] G.P. Galdi, On the motion of a rigid body in a viscous liquid : a mathematical analysis with applications, in Vol. I of Handbook of mathematical fluid dynamics. North-, Amsterdam (2002) 653–791.

[18] M. Geissert, K. Götze, M. Hieber Lp-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids Trans. Am. Math. Soc 2013 1393 1439

[19] D. Gérard-Varet, M. Hillairet Existence of weak solutions up to collision for viscous fluid-solid systems with slip Commun. Pure Appl. Math 2014 2022 2075

[20] D. Gérard-Varet, M. Hillairet, C. Wang The influence of boundary conditions on the contact problem in a 3d Navier-Stokes flow J. Math. Pures Appl 2015 1 38

[21] D. Gérard-Varet, N. Masmoudi Relevance of the slip condition for fluid flows near an irregular boundary Commun. Math. Phys 2010 99 137

[22] M.D. Gunzburger, H.-C. Lee, G.A. Seregin Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions J. Math. Fluid Mech 2000 219 266

[23] W. Jäger, A. Mikelić On the roughness-induced effective boundary conditions for an incompressible viscous flow J. Differ. Equ 2001 96 122

[24] P.-L. Lions, Mathematical topics in fluid mechanics. Incompressible models. Vols. 1 and vol. 3 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1996).

[25] D. Maity and M. Tucsnak, Lp-Lq maximal regularity for some operators associated with linearized incompressible fluid-rigid body problems. Mathematical analysis in fluid mechanics–selected recent results. In Vol. 710 of Contemporary Mathematics. AMS Providence (2018) 175–201.

[26] J.A.S. Martín, V. Starovoitov, M. Tucsnak Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid Arch. Ration. Mech. Anal 2002 113 147

[27] M. Padula On the existence and uniqueness of nonhomogeneous motions in exterior domains Math. Z 1990 581 604

[28] G. Planas, F. Sueur On the “viscous incompressible fluid + rigid body” system with Navier conditions Ann. Inst. Henri Poincaré Anal. Non Linéaire 2014 55 80

[29] J. San Martín, T. Takahashi, M. Tucsnak A control theoretic approach to the swimming of microscopic organisms Quart. Appl. Math 2007 405 424

[30] D. Serre Chute libre d’un solide dans un fluide visqueux incompressible Jpn. J. Appl. Math 1987 99 110

[31] M. Sigalotti, J.-C. Vivalda Controllability properties of a class of systems modeling swimming microscopic organisms ESAIM: COCV 2010 1053 1076

[32] A.L. Silvestre On the self-propelled motion of a rigid body in a viscous liquid and on the attainability of steady symmetric self-propelled motions J. Math. Fluid Mech 2002 285 326

[33] A.L. Silvestre On the slow motion of a self-propelled rigid body in a viscous incompressible fluid J. Math. Anal. Appl 2002 203 227

[34] J. Simon Compact sets in the space Lp(0, T Ann. Mat. Pura Appl 1987 65 96

[35] J. Simon Nonhomogeneous viscous incompressible fluids : existence of velocity, density, and pressure SIAM J. Math. Anal 1990 1093 1117

[36] V.N. Starovoitov Solvability of the problem of the self-propelled motion of several rigid bodies in a viscous incompressible fluid Comput. Math. Appl 2007 413 435

[37] T. Takahashi Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain Adv. Differ. Equ 2003 1499 1532

[38] C. Wang Strong solutions for the fluid-solid systems in a 2-d domain Asymptot. Anal 2014 263 306