Abstract
We consider equations describing the multidimensional motion of compressible viscous (non-Newtonian) Bingham-type fluids, i.e., fluids with multivalued function relating the stresses to the tensor of strain rates. We prove the global existence theorem in time and in the initial data for the first initial boundary-value problem corresponding to flows in a bounded domain in the class of “weak” generalized solutions. In this case, we admit an anisotropic relation between the stress and strain rate tensors and study admissible relations of this kind in detail.
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References
E. C. Bingham, Fluidity and Plasticity (McGrew-Hill Book Co., New York, 1922).
W. Prager, Introduction to Mechanics of Continua, in Introductions to Higher Mathematics (Ginn and Co., New York, 1961).
J. Serrin, Mathematical Principles of Classical Fluid Mechanics (Springer, Berlin-Gottingen-Heidelberg, 1959; Inostr. Lit., Moscow, 1963).
D. Prasad and H. K. Kytomaa, “Particle stress and viscous compaction during shear of dense suspensions,” Intern. J. of Multiphase Flow 21(5), 775–785 (1995).
L. E. Malvern, Introduction to the Mechanics of a Continuous Medium (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1969).
I. V. Basov and V. V. Shelukhin, “Generalized solutions to the equations of compressible Bingham flows,” ZAMM Z. Angew. Math. Mech. 79(3), 185–192 (1999).
V. V. Shelukhin, “Bingham viscoplastic as a limit of non-Newtonian fluids,” J. Math. Fluid Mech. 4(2), 109–127 (2002).
J. Málek, M. Růžička and V. V. Shelukhin, “Hershel-Bulkley fluids: existence and regularity of steady flows,” Math. Models Methods Appl. Sci. 15(12), 1845–1861 (2005).
A. E. Mamontov, “Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. I,” Sib. Mat. Zh. 40(2), 408–420 (1999) [Sib. Math. J. 40 (2), 351–362 (1999)].
A. E. Mamontov, “Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. II,” Sib. Mat. Zh. 40(3), 635–649 (1999) [Sib. Math. J. 40 (3), 541–555 (1999)].
M. A. Krasnosel’skii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, in Modern Problems of Mathematics (Fizmatgiz, Moscow, 1958) [in Russian].
A. E. Mamontov, “Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids,” Mat. Zametki 68(3), 360–376 (2000) [Math. Notes 68 (3–4), 312–325 (2000)].
A. V. Kazhikhov and A. E. Mamontov, “On a certain class of convex functions and the exact well-posedness classes of the Cauchy problem for the transport equation in Orlicz spaces,” Sib. Mat. Zh. 39(4), 831–850 (1998) [Sib. Math. J. 39 (4), 716–734 (1998)].
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod, Paris, 1969; Mir, Moscow, 1972).
R. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, 1970; Mir, Moscow, 1973).
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Original Russian Text © A. E. Mamontov, 2007, published in Matematicheskie Zametki, 2007, Vol. 82, No. 4, pp. 560–577.
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Mamontov, A.E. Existence of global solutions to multidimensional equations for Bingham fluids. Math Notes 82, 501–517 (2007). https://doi.org/10.1134/S000143460709026X
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DOI: https://doi.org/10.1134/S000143460709026X