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

  • Jesús Ildefonso Díaz
  • Roland Glowinski
  • Giovanna GuIDoboni
  • Taebeom Kim
Article

Abstract

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 threedimensional 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 \( u_\infty \) 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, \( u_\infty \ne 0 \) , 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.

Keywords

Bingham flows propagation of the support stabilzation finite extinction time numerical experiences 

Mathematics Subject Classifications

35K55 35R35 35K85 35B30 35B35 65N30 68U20 76A05 

Propiedades cualitativas y aproximación de las soluciones de problemas de fluIDos de Bingham: sobre la estabilización para tiempos grandes y la geometría del soporte de las soluciones

Resumen

ConsIDeramos el flujo transitorio de un fluIDo de Bingham isotérmico e incompresible. Modelos similares se plantean en contextos completamente diferentes como, por ejemplo, en ciencias de los materiales, tratamiento de imágenes y geometría diferencial. Para el flujo en un dominio bIDimensional mostramos la extinción en tiempo finito, incluso bajo adecuadas fuerzas externas no nulas. ConsIDeramos tambien el caso especial del dominio trIDimensional dado por un cilindro infinitamente largo de sección transversal acotada. Damos condiciones suficientes que conducen a una formulación escalar sobre el dominio transversal. Probamos la estabilización de las soluciones, cuando t tiende a infinito, a la solución \( u_\infty \) del problema estacionario asociado, una vez que se supone una cierta convergencia sobre los términos del lado derecho. Damos algunas condiciones suficientes para la extinción en tiempo finito de las soluciones del problema escalar. Se demuestra, asi mismo, que, al menos bajo condiciones de simetría radial, cuando el estado estacionario no es trivial, \( u_\infty \ne 0 \) , hay casos en los que la estabilización de la soluci ón estacionaria requiere un tiempo infinito. Para terminar, se ofrecen algunas experiencias numéricas para la formulación escalar. En particular, algunas de esas experiencias muestran un cambio instantáneo de la topología del soporte de la solución: cuando el soporte del dato inicial está formado por dos bolas disjuntas, pero suficiente cercanas, entonces, instantáneamente, para cualquier t > 0, el soporte de la solución u( ·, t) se convierte en un conjunto conexo. Algunas otras experiencias numéricas se dedican al estudio del «perfil» de la solución en su momento de extinción.

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Copyright information

© Springer 2010

Authors and Affiliations

  • Jesús Ildefonso Díaz
    • 1
    • 2
  • Roland Glowinski
    • 3
    • 5
  • Giovanna GuIDoboni
    • 3
    • 4
  • Taebeom Kim
    • 3
  1. 1.Departamento de Matemática Department of MathematicsUniversIDad Complutense de MadrIDMadrIDSpain
  2. 2.Real Academia de Ciencias Exactas, Físicas y NaturaleMadrIDSpain
  3. 3.Department of MathematicsUniversity of HoustonHoustonUSA
  4. 4.Department of Mathematical SciencesIndiana University and Purdue University at IndianapolisIndianapolisUSA
  5. 5.Laboratoire Jacques-Louis LionsUniversité P. et M. CurieParisFrance

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