Sur les formules de green généralisées qui se présentent dans l'hydrodynamique et sur quelquesunes de leurs applications

Oseen, C. W.

doi:10.1007/bf02393128

Cited by 25 publications

(17 citation statements)

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“…Since v is solution to the integral Equation (34), by the couple (v, π v ) one finds the desired solution to System (1) (cf. [14,15] or [18] Section 4.6). Concerning the initial condition v 0 , we first observe that the limit property (6) 4 trivially holds for v 0 (t, x), and then, via the integral Equation (9) and Lemma 9 for ∇E * (v ⊗ v), we get…”

Section: Proof Of the Main Results Stated In The Introductionmentioning

confidence: 99%

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Navier–Stokes Cauchy Problem with |v0(x)|2 Lying in the Kato Class K3

Crispo¹,

Maremonti²

2021

Mathematics

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Section: Proof Of the Main Results Stated In The Introductionmentioning

confidence: 99%

“…For the Oseen tensor the following estimates hold (cf. [14], estimates (VI) on page 215 and (VIII) on page 216, or [15]):…”

Section: Preliminariesmentioning

confidence: 99%

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Navier–Stokes Cauchy Problem with |v0(x)|2 Lying in the Kato Class K3

Crispo¹,

Maremonti²

2021

Mathematics

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“…The boundary-integral formulation for a three-dimensional Stokes flows has been pioneered by Oseen 34,35 and Odqvist 36 and later recalled by Ladyzhenskaya. 37 Nowadays, several textbooks adequately present this widely employed method and its practical implementation (see, among others, Kim and Karrila, 38 Pozrikidis, 39 Bonnet, 40 or Sellier 41 ).…”

Section: B Boundary Approach and Resulting Boundary-integral Equationmentioning

confidence: 99%

Slow viscous gravity-driven interaction between a bubble and a free surface with unequal surface tensions

2015

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The axisymmetric gravity-driven dynamics of a bubble rising toward a free surface is addressed for gas-liquid interfaces having unequal surface tensions. The liquid flow is governed by the Stokes equations which are here solved using a boundary element method in axisymmetric configuration. Within this framework, two dimensionless numbers arise: the Bond number Bo 1 based on the surface tension of the bubble interface and the surface tension ratioγ comparing the free surface and bubble surface tensions. Under a careful and discussed selection of the code key settings (number of boundary elements, initial bubble location, and distance beyond which the free surface is truncated), it has been possible to numerically and accurately track in time the bubble and free surface shapes for several values of (Bo 1 ,γ). The long-time shapes are found to deeply depend upon both Bo 1 andγ and also to compare well with the shapes predicted in Princen and Mason ["Shape of a fluid drop at a fluid-liquid interface. II. Theory for three-phase systems," J. Colloid. Sci. 20, 246-266 (1965)] using a hydrostatic model in which both surfaces are touching. Similarly, the drainage dynamics of the liquid film thickness between the bubble and the free surface depends on (Bo 1 ,γ). The long-time film thickness exponentially decays in time and a so-called thinning rate α for which the numerical behaviors and a simple model reveal two basic behaviors: (i) at small Bond number, α behaves as 1/Bo 1 and (ii) at large Bond number, α is nearly constant. In addition, it is found that in the entire range of the quantity χ = (1 +γ)Bo 1 /(2γ), the thinning rate α is well approximated by the function 1/(18 χ) + α ∞ with α ∞ ≈ 0.158. Such a result also permits one to estimate the typical drainage time versus the initial bubble radius a, the liquid density ρ and viscosity µ, the gravity and the free surface, and bubble surface tensions.

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“…Local existence of a smooth solution (A) was proved by Carl W. Oseen [Ose11,Ose27]. In 1933 in his thesis [Ler33], Jean Leray completely solved the twodimensional case, as Olga A. Ladyzhenskaya did put it, by a "happy circumstance [which] is valid only in the two-dimensional case and only for the Cauchy problem, for periodic boundary conditions, and for a certain special boundary condition" [Lad03,p.…”

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confidence: 99%

On the periodic Navier--Stokes equation: An elementary approach to existence and smoothness for all dimensions $n\geq 2$

di Dio

2020

Preprint

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In this paper we study the periodic Navier-Stokes equation. From the periodic Navier-Stokes equation and the linear equation ∂tu = ν∆u + P[v∇u] we derive the corresponding equations for the time dependent Fourier coefficients a k (t). We prove the existence of a smooth solution u of the linear equation by a Montel space version of Arzelà-Ascoli. We gain bounds on the a k 's of u depending on v. These bounds provide the small time existence of a unique smooth solution of the Navier-Stokes equation. They prove that if all first derivatives of the solution are bounded for all times t ∈ [0, T ], then the solution is smooth for all t ∈ [0, T ]. We prove that the Navier-Stokes equation with small initial data, e.g. whena unique smooth solution for all times t ≥ 0. All results hold for all dimensions n ≥ 2.

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Sur les formules de green généralisées qui se présentent dans l'hydrodynamique et sur quelquesunes de leurs applications

Cited by 25 publications

References 0 publications

Navier–Stokes Cauchy Problem with |v0(x)|2 Lying in the Kato Class K3

Navier–Stokes Cauchy Problem with |v0(x)|2 Lying in the Kato Class K3

Slow viscous gravity-driven interaction between a bubble and a free surface with unequal surface tensions

On the periodic Navier--Stokes equation: An elementary approach to existence and smoothness for all dimensions $n\geq 2$

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