Global Existence and Uniqueness of the Weak Solution in Thixotropic Model

Article Sidebar

PDF
Cite as:
Amira Rahai, Amar Guesmia, Global Existence and Uniqueness of the Weak Solution in Thixotropic Model, Int. J. Anal. Appl., 19 (2) (2021), 193-204.

Main Article Content

Amira Rahai, Amar Guesmia

Abstract

In this paper, we study global existence, uniqueness and boundedness of the weak solution for the system (P) which is formulated by two subsystems (P1) and (P2), the first describes the thixotropic problem and the second describes the diffusion degradation of c, using Galerkin's method, Lax-Milgran's and maximum principle. Moreover we show that the unique solution is positive.

Article Details

References

  1. H.A. Barnes, Thixotropy a review, J. Non-Newtonian Fluid Mech. 70 (1997), 1-33.
  2. H. El-Gendy, M. Alcoutlabi, M. Jemmett, M. Deo, J. Magda, R. Venkatesan, A. Montesi, The propagation of pressure in a gelled waxy oil pipeline as studied by particle imaging velocimetry, AIChE J. 58 (2012), 302-311.
  3. H. Freundlich, Thixotropy, Paris, 1935.
  4. C.F. Goodeve, A general theory of thixotropy and viscosity, Trans. Faraday Soc. 35 (1939), 342-358.
  5. C.F. Goodeve, G.W. Whitfield, The measurement of thixotropy in absolute units. Trans. Faraday Soc. 34 (1938), 511-520.
  6. T. Hillen, K. Painter, Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding, Adv. Appl. Math. 26 (2001), 280-301.
  7. C.Mesikh, A. Guesmia, and S.Saadi, Global Existence and Uniqueness of the Weak Solution in Keller Segel Model. Glob. J. Sci. Front. Res. F, 14 (2014), 46-55.
  8. F. Moore, The rheology of cereamic slips and bodies. Trans. Br. Ceramic Soc. 58 (1959), 470-484
  9. A. Mujumdar, A.N. Beris, A.B. Metzner, Transient phenomena in thixotropic systems, J. Non-Newtonian Fluid Mech. 102 (2002), 157-178.
  10. Q.D. Nguyen, D.V. Boger, Thixotropic behaviour of concentrated bauxite residue suspensions. Rheol. Acta, 24(4) (1985), 427-437.
  11. J.O. Sibree, The viscosity of emulsions. Part I. Trans. Faraday Soc. 26 (1930), 26-36.
  12. J.O. Sibree, The viscosity of emulsions. Part II. Trans. Faraday Soc. 27 (1931), 161-176.
  13. P.R. de Souza Mendes, Modeling the thixotropic behavior of structured fluids, J. Non-Newtonian Fluid Mech. 164 (2009), 66-75.
  14. J. De Vicente, C. Berli, Aging, rejuvenation, and thixotropy in yielding magneto-rheological fluids. Rheol. Acta, 52 (2013), 467-483.
  15. X. Zhang, W. Li, X. Gong, Thixotropy of MR shear-thickening fluids. Smart materials and structures, 19(12) (2010), 125012.
  16. G. Allaire, Analyse num ´erique et optimisation. Ed. de l'Ecole Polytechnique, Paris, 2005.
  17. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence, RI, 2010.