Handbook of Numerical Analysis, Volume XVI, Special Volume: by R. Glowinski, Jinchao Xu, Philippe G. Ciarlet

By R. Glowinski, Jinchao Xu, Philippe G. Ciarlet

Non-Newtonian flows and their numerical simulations have generated an ample literature, in addition to many courses and references to which might be present in this volume's articles. This abundance of guides should be defined via the truth that non-Newtonian fluids ensue in lots of genuine lifestyles events: the foodstuff undefined, oil & gasoline undefined, chemical, civil and mechanical engineering, the bio-Sciences, to call quite a few. Mathematical and numerical research of non-Newtonian fluid stream versions offer difficult difficulties to partial differential equations experts and utilized computational mathematicians alike. This quantity deals investigations. effects and conclusions that would doubtless be precious to engineers and computational and utilized mathematicians who're excited about a number of points of non-Newtonian Fluid Mechanics. New overview of recognized computational tools for the simulation viscoelastic and viscoplastic types.; Discusses new numerical equipment that experience confirmed to be extra effective and extra exact than conventional methods.; Articles that debate the numerical simulation of particulate stream for viscoelastic fluids.;

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Extra resources for Handbook of Numerical Analysis, Volume XVI, Special Volume: Numerical Methods for Non-Newtonian Fluids

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F ; w/; 8r 2 L02 . 11) v; w/; 26 Chapter 1 V. Girault and F. Hecht The following proposition states that it is well posed. 4. Let be bounded, connected, and Lipschitz-continuous, > 0 and f 2 L2 . /2 . For any z in L2 . z// in V L02 . /. This solution satisfies the following bounds in H 1 . /2 L2 . z/jH 1 . z/kL2 . k f kL2 . / ; 1 / S2 k f kL2 . z/jH 1 . / kzkL2 . 3). 13) is independent of z. 19); but for existence, we only require the much simpler statement: 8u 2 W; 8z 2 H 1 . 16) Let fwi gi 1 be a basis of H 2 .

The case when the velocity is in W 1;1 will play an important part in the sequel. Again, we formulate it as follows: When does a right-hand side f in Lr . /d for some real number r > d imply that u belongs to W 2;r . /d ? The answer is given by Grisvard [1985] when d D 2 and by Dauge [1989], Kozlov, Maz’ya and Rossmann [2000] when d D 3. 9. (1) Let be a convex polygon in IR2 . Then there exists a real number r > 2 depending on the largest inner angle of @ such that for all r with 2 r r , f in Lr .

D k h k2L2 . curl uh . h /; k h k2L2 . / juh . h /jH 1 . k h k2L2 . / S2 k f kL2 . 0 for all h in Zh satisfying h/ D S2 k f kL2 . curl f ; h/ kcurl f kL2 . C kcurl f kL2 . / / k h kL2 . / k h kL 2 . / : kcurl f kL2 . 17). 16). 1). 1. 16), that there exists a subsequence of h (still denoted by h) and functions u 2 H01 . /2 , p 2 L2 . /, z 2 L2 . / such that lim uh D u weakly in H01 . 0 lim ph D p weakly in L2 . 18) lim zh D z weakly in L2 . 0 The compactness of the imbedding of H 1 . / into Lp .

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