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

This ebook is a distinct selection of high-level papers dedicated to basic themes in mathematical fluid mechanics and their purposes, regularly in reference to the medical paintings of Giovanni Paolo Galdi. The contributions are generally established at the learn of the fundamental homes of the Navier-Stokes equations, together with life, distinctiveness, regularity, and balance of recommendations. comparable versions describing non-Newtonian flows, turbulence, and fluid-structure interactions also are addressed. the implications are analytical, numerical and experimental in nature, making the publication relatively beautiful to an enormous readership encompassing mathematicians, engineers and physicists. the variety of the subjects, as well as the several ways, will supply readers an international and up to date evaluation of either the most recent findings at the topic and of the salient open questions.

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A The velocity u = (u, v) is determined from the pressure by: u= B− AB A ∂x p + 1 − A s A and y v=− ∂x u(x, z) dz. (14) 0 The whole system (Reynolds and Cahn-Hilliard equations) reads, in the case where the capillarity coefficient κ is of order ε: A New Model of Diphasic Fluids in Thin Films 31 ⎧ ∂x ( D(ϕ) ∂x p) = s ∂x E(ϕ) ⎪ ⎪ ⎪ ⎪ ⎪ A AB ⎪ ⎪ ⎪ u(x, y) = B − ∂x p + s 1 − ⎪ ⎪ A A ⎪ ⎪ ⎨ y v(x, y) = − ∂x u(x, z)dz ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ div(B(ϕ)∇μ) = 0 ⎪ ∂t ϕ + u ∂ x ϕ + v ∂ y ϕ − ⎪ Pe ⎪ ⎪ ⎩ μ = −α 2 Δϕ + F (ϕ).

In: International Conference on Differential Equations, Vols. 1, 2 (Berlin, 1999), 488–490. World Science Publication, River Edge, NJ (2000) 8. : Nonhomogeneous Cahn-Hilliard fluids. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 18(2), 225–259 (2001) A New Model of Diphasic Fluids in Thin Films 35 9. : Numerical study of viscoelastic mixtures through a CahnHilliard flow model. Eur. J. Mech. B Fluids 23(5), 759–780 (2004) 10. : Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation.

The boundary conditions read ϕ|Γl = ϕl , ∂ϕ ∂∇ Γ \Γl = 0, μ|Γl = 0, ∂μ ∂∇ Γ = 0. (10) In order to take into account the surface tension effects, we add to the external forces F in (1) an additional term κμ∇ϕ, where κ is the capillarity coefficient (related to the surface tension). The Navier-Stokes equation becomes: (∂t u + u · ∇u) − div (ηD(u)) + ∇ p = κμ∇ϕ, Fig. 1 Domain Ω of boundary Γ and notations for the boundary conditions on ϕ and on u The system (11)–(9) has been studied in [6, 10]. div u = 0.