By Joaquim P. Marques de Sá
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Extra info for Applied Partial Differential Equations A Visual Approach
Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows (2nd edition), Gordon and Breach, 1969  J. Smoller, Shock Waves and Reaction-Diffusion Equations, (second edition), Springer-Verlag, Vol. 258, Grundlehren Series, 1994  R. DiPerna, Convergence of the Viscosity Method for Isentropic Gas Dynamics, Comm. Math. , Vol. 91, Nr. 1, 1983 37 3. Granular Material Flows Peter A. Markowich and Giuseppe Toscani1 We cite from the webpage of the granular ﬂows research group of the California Institute of Technology2 : A granular material ﬂow is a form of two-phase ﬂow consisting of particulates and an interstitial ﬂuid.
Math. , Vol. 91, Nr. 1, 1983 37 3. Granular Material Flows Peter A. Markowich and Giuseppe Toscani1 We cite from the webpage of the granular ﬂows research group of the California Institute of Technology2 : A granular material ﬂow is a form of two-phase ﬂow consisting of particulates and an interstitial ﬂuid. When sheared the particulates may either ﬂow in a manner similar to a ﬂuid, or resist the shearing like a solid. The dual nature of these types of ﬂows makes them very difﬁcult to analyze. Granular materials are all around us – examples include food products such as rice, corn, and breakfast cereal ﬂakes, building materials such as sand, gravel and soil, chemicals such as plastics, and pharmaceutical pills.
F = f (x, t) is the (given) external force ﬁeld (again two and, resp. three-dimensional) acting on the ﬂuid and ν > 0 the kinematic viscosity parameter. The functions u and p are the solutions of the PDE system, the ﬂuid density is assumed to be constant (say, 1) here as consistent with the incompressibility assumption. The nonlinear Navier–Stokes system has to be supplemented by an initial condition for the velocity ﬁeld and by boundary conditions if spatially conﬁned ﬂuid ﬂows are considered (or by decay conditions on whole space).
Applied Partial Differential Equations A Visual Approach by Joaquim P. Marques de Sá