By Chérif Amrouche, Ulrich Razafison (auth.), Rolf Rannacher, Adélia Sequeira (eds.)

ISBN-10: 3642040675

ISBN-13: 9783642040672

ISBN-10: 3642040683

ISBN-13: 9783642040689

This ebook is a different choice of high-level papers dedicated to primary subject matters in mathematical fluid mechanics and their functions, typically in reference to the medical paintings of Giovanni Paolo Galdi. The contributions are customarily based at the examine of the elemental homes of the Navier-Stokes equations, together with life, distinctiveness, regularity, and balance of options. similar versions describing non-Newtonian flows, turbulence, and fluid-structure interactions also are addressed. the consequences are analytical, numerical and experimental in nature, making the ebook rather beautiful to an unlimited readership encompassing mathematicians, engineers and physicists. the range of the subjects, as well as the various methods, will offer readers a world and up to date review of either the newest findings at the topic and of the salient open questions.

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**Additional info for Advances in Mathematical Fluid Mechanics: Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday**

**Example text**

From (1) we get ⎧ ⎨ − Δ u + ∇ · G(x, ∇ u) + (u · ∇) u + ∇ π = f , ⎩ (5) ∇ · u = 0. Integrability of the Full Gradient for Generalized Power Law Models 39 In order to fix ideas we assume here the homogeneous Dirichlet boundary condition u |Γ = 0 . (6) However, many other boundary conditions fall within the above scheme. Actually, it is sufficient that an estimate like (21) holds for the usual Stokes linear system (20) under the desired boundary condition. We may also assume a non-homogeneous Dirichlet boundary condition u |Γ = a(x) , if a ∈ W 1, +∞ (Γ ) satisfy the necessary compatibility condition Γ a · n dΓ = 0 .

A Gronwall argument allows to conclude that ϕ ∈ L ∞ (0, ∞; H 2 (Ω)). From the a priori estimates follow weak convergences. For the convergence of the nonlinear term u · ∇ϕ, we estimate the time derivatives, which allows to conclude the Galerkin process. 1 The Numerical Scheme In order to simulate the behavior of a diphasic flow in thin film, we introduce a numerical scheme for the system (15), which consists in two steps. The first step is the computation of the pressure and the velocity by (13) and (14).

Taking u(x) = v(r, θ ), it is sufficient to prove θ∗ I = −θ ∗ ≤C ∞ 1 1 (1 + r )(α− 2 ) p (r − r cos θ )(β− 2 ) p r 2 |v| p dr dθ R θ∗ −θ ∗ ∞ αp βp 2− p (1 + r ) (r − r cos θ ) r R ∂v ∂θ (70) p dr dθ. Proceeding as for the case n ≥ 3 and the use of inequality (58) give us Inequality (70). Let R be a positive real number fixed large enough. In the sequel, we will need the following Hardy-type inequality (cf. Hardy-Littlewood-Polya [9] : we have ∀ f ∈ D(]R, ∞[), +∞ | f (r )| p r γ dr ≤ C R +∞ | f (r )| p r γ + p dr, with γ +1 = 0.

### Advances in Mathematical Fluid Mechanics: Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday by Chérif Amrouche, Ulrich Razafison (auth.), Rolf Rannacher, Adélia Sequeira (eds.)

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