Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Real Academia Ciencias Exactas Físicas Y Naturales
Google Scholar
Research Projects
Organizational Units
Journal Issue
We study the transient flow of an isothermal and incompressible Bingham fluid. Similar models arise in completely different contexts as, for instance, in material science, image processing and differential geometry. For the two-dimensional flow in a bounded domain we show the extinction in a finite time even under suitable nonzero external forces. We also consider the special case of a three-dimensional domain given as an infinitely long cylinder of bounded cross section. We give sufficient conditions leading to a scalar formulation on the cross section. We prove the stabilization of solutions, when t goes to infinity, to the solution u(infinity) of the associated stationary problem, once we assume a suitable convergence on the right hand forcing term. We give some sufficient conditions for the extinction in a finite time of solutions of the scalar problem. We show that, at least under radially symmetric conditions, when the stationary state is not trivial, u(infinity) not equal 0, there are cases in which the stabilization to the stationary solution needs an infinite time to take place. We end the paper with some numerical experiences on the scalar formulation. In particular, some of those experiences exhibit an instantaneous change of topology of the support of the solution: when the support of the initial datum is formed by two disjoint balls, but closed enough, then, instantaneously, for any t > 0, the support of the solution u(., t) becomes a connected set. Some other numerical experiences are devoted to the study of the "profile" of the solution and its extinction time.
ANDREU, F.; BALLESTER, C.; CASELLES, V. AND MAZÓN, J. M., (2001). Minimizing total variation flow. Differential Integral Equations, 14, 321–360. ANDREU, F.; BALLESTER, C.; CASELLES, V. AND MAZÓN, J. M., (2001). The Dirichlet problem for the total variation flow, J: Funct. Anal., 180, 2, 347–403. DOI: 10.1006/jfan.2000.3698 ANDREU, F.; CASELLES, V.; DÍAZ, J. I. AND MAZÓN, J. M., (2002). Some qualitative properties for the total variation flow, J: Funct. Anal., 188, 2, 516–547. DOI: 10.1006/jfan.2001.3829 ANZELLOTTI, G., (1983). Paring between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. (4), 135, 293–318. ANTONTSEV, S. N. AND DÍAZ, J. I., (1988). Applications of energy methods for localization of solutions of equations in Continuum Mechanics, Dokl. Akad. Nauk SSSR. (Mathematical-Physics), 303, 2, 320--325, (in russian). English trasslation Soviet Phys. Dokl., 33, 11, 813–816, (1989). ANTONTSEV, S. N. AND DÍAZ, J. I, (1989). Energy methods and localization of solutions for Continuum Mechanics Equations, J. Appl. Mech. Tech. Phys., 174, 2, 18–25, (in russian). DOI: 10.1007/BF00852162 ANTONTSEV, S. N.; DÍAZ, J. I. AND SHMAREV, S. I., (2002). Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications, 48, Birkhäuser, Boston. BATCHELOR, G. K., (1967). An introduction to fluid dynamics, Cambridge University Press, Cambridge. BEGIS, D. AND GLOWINSKI, R., (1982). Application de la méthode des éléments finis à l'approximation d'un problème de domaine optimal, R.A.I.R.O., Num. Anal., 16, 351–373. BELLETTINI, G.; CASELLES, V. AND NOVAGA, M., (2002). The Total Variation Flow in R N, J. Differential Equations, 184, 475–525. BINGHAM, E. C., (1916). An Investigation of the Laws of Plastic Flow, U.S. Bureau of Standards Bulletin, 13, 309–353. BREZIS, H., (1971). Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. In: E. Zarantonello (Ed.), Contributions to Nonlinear Functional Analysis, Academic Press, New York, 101–156. BREZIS, H., (1973). Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert North-Holland. Amsterdam. CEA, J. AND GLOWINSKI, R., (1972). Méthodes numériques pour l'écoulement lamminare d'un fluide rigide visco-plastique incompressible, Int. J. Comput. Math., Section B, 3, 225–255. DEAN, E. J.; GLOWINSKI, R. AND GUIDOBONI, G., (2007). On the numerical simulation of Bingham visco-plastic flow: Old and new results, J. non-Newtonian Fluid Mech., 142, 1–3, 36–62. DOI: 10.1016/j.jnnfm.2006.09.002 DÍAZ. J. I., (1992). Symmetrization of nonlinear elliptic and parabolic problems and applications: a particular overview. In C. Bandle et al. eds., Progress in partial differential equations. elliptic and parabolic problems, Pitman Research Notes in Mathematics Longman, Harlow, Essex, 1 16. DÍAZ, J. I. AND DE THELIN, F., (1994). On a nonlinear parabolic problems arising in some models related to turbulence flows, SIAM J. Math. Anal., 25, 4, 1085–1111. DOI: 10.1137/S0036141091217731 DUVAUT, G. AND LIONS, J. L., (1976). Inequalities in Mechanics an Physics, Springer, Berlin. DUVAUT, G. LIONS, J. L., (1972). Les Inéquations en Mécanique et Physique, Dunod, Paris. FRIEDMAN, A., (1964). Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, NJ. GLOWINSKI, R., (1984). Numerical Methods for Nonlinear Variational Problems, Springer, New-York, NY. GLOWINSKI, R., (2003). Finite element methods for incompressible viscous flow. In P. G. Ciarlet and J. L. Lions eds., Handbook of Numerical Analysis, IX, 3–1176, North-Holland, Amsterdam. GLOWINSKI, R. AND LE TALLEC, P., (1989). Augmented Lagrangian interpretation of the nonoverlapping Schwarz alternating method. In: T. F. Chan, R. Glowinski, J. Périaux, O. B. Widlund (Eds.), Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, 224–231, Houston, TX, SIAM, Philadelphia, USA, 1990. GLOWINSKI, R.; LIONS, J. L. AND TRÉMOLIÈRES, R., (1981). Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam. HARDT, M. AND ZHOU, X., (1998). An evolution problem for linear growth functionals, Interdisciplinary Applied Mathematics, 8, Springer, New York. KOBAYASHI, R. AND GIGA, Y., (1999). Equations with singular diffusivity, J. Statist. Phys., 95, 1187–1220. MOSSOLOV, P. P. AND MIASNIKOV, V. P., (1965). Variational methods in the theory of fluidity of a viscus-plastic medium, P. M. M., 29, 468–492. NAKAO, M., (1978). A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30, 4, 747–762. DOI: 10.2969/jmsj/03040747 RUDIN, L.; OSHER, S. AND FATEMI, E., (1992). Nonlinear total variation based noise removal algorithms, Phys. D, 60, 259–268. DOI: 10.1016/0167-2789(92)90242-F STRAUSS, M. J., Variation of Korn's and Sobolev's inequalities. In, Proceedings of Symposia in Pure Mathematics, 23, American Mathematics Society, Providence, RI., 1973, 207–214. TALENTI, G., (1976). Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4), 110, 1, 353–372. DOI: 10.1007/BF02418013 WAITE, L. AND FINE, J. M., (2007). Applied Biofluid Mechanics, McGraw-Hill Professional Publishing. YOSIDA, K., (1966). Functional Analysis, Springer-Verlag.