By Chérif Amrouche, Ulrich Razafison (auth.), Rolf Rannacher, Adélia Sequeira (eds.)
This e-book is a different selection of high-level papers dedicated to primary themes in mathematical fluid mechanics and their functions, more often than not in reference to the clinical paintings of Giovanni Paolo Galdi. The contributions are regularly based at the learn of the fundamental houses of the Navier-Stokes equations, together with lifestyles, strong point, regularity, and balance of suggestions. similar types describing non-Newtonian flows, turbulence, and fluid-structure interactions also are addressed. the implications are analytical, numerical and experimental in nature, making the ebook really beautiful to an enormous readership encompassing mathematicians, engineers and physicists. the range of the subjects, as well as the several ways, will offer readers an international and updated review of either the newest findings at the topic and of the salient open questions.
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Additional resources for Advances in Mathematical Fluid Mechanics: Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday
These models are not only based on mechanical considerations but also on chemical properties at the interface between the two fluids, which enable an exchange between the two phases. In this chapter, we use the Cahn-Hilliard equation, which involves an interaction potential. To this end, we introduce an order parameter ϕ, for example the volumic fraction of one phase in the mixture. This kind of model has already been studied for the complete Navier-Stokes equations in [6, 10]. In this chapter, we describe the governing equations (in Sect.
598). Weighted Sobolev Spaces for the Oseen Equation 21 Remark 9 Note that the particular case n = 3, p = 2, β > 0 and α + β + 12 > 0 of previous theorem for was proved by Farwig (see ). 3) for the case n = 3, p = 2, β > 0, α ≥ 0 and α + β < 32 . Lemma 5 Let α, β be two reals such that β ≤ 0 and α + n/ p − 1 < 0 or α + β + n/ p − 1 > 0. Then, for any large enough positive real number R, we have ∀u ∈ D(B R ), u p L α−1,β (B R ) ≤ C ∇u L α,β (B R ) . (81) p Proof Let u ∈ D(B R ). We first prove (81) for n ≥ 3.
The boundary conditions on p are deduced from the ones on u. Indeed, the choice of q corresponds to a Neumann condition on p at x = 0: it follows from (6) that h(0) q= u(0, y)dy = −∂x p(0) 0 sh(0) h(0)3 + . 12η 2 This expression determines ∂x p(0) as a function of q. Moreover, since the pressure p is defined up to a constant, we have to impose another condition. Finally, the boundary conditions on p read: ∂x p(0) = 12η h(0)3 sh(0) −q , 2 p(L) = 0. 1 Modelling a Mixture and Taking the Surface Tension into Account In order to describe the mixture of two miscible fluids, we introduce an order parameter ϕ ∈ [−1, 1] (corresponding to the volumic fraction of one fluid in the flow).