<i>The Mathematical Theory of Viscous Incompressible Flow</i>

Ladyzhenskaya, O. A.; Silverman, Richard A.; Schwartz, Jacob T.; Romain, Jacques E.

doi:10.1063/1.3051412

Cited by 1,238 publications

(957 citation statements)

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“…where the second inequality represents the partial case of the Sobolev embedding inequality [10]. Hence, one can see that I → ∞ is equivalent to the divergence of the norm (2) for the Sobolev space H 2 (R 3 ).…”

Section: Discussionmentioning

confidence: 98%

Towards a sufficient criterion for collapse in 3D Euler equations

Кузнецов¹

2003

Physica D: Nonlinear Phenomena

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Section: Discussionmentioning

confidence: 98%

Towards a sufficient criterion for collapse in 3D Euler equations

Кузнецов¹

2003

Physica D: Nonlinear Phenomena

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“…Ladyzhenskaya (1963) showed that [2.14] where T ik and U ik are the traction and velocity kernels, u i is the velocity vector, and τ i is the traction vector τ = σ fluid •n. The axisymmetric form of the kernels used in this simulation may be found on pages 132−135 and appendix D in Becker (1992).…”

Section: Boundary Element Methods (Bem)mentioning

confidence: 99%

The pulsatile propagation of a finger of air within a fluid-occluded cylindrical tube

Smith

Gaver

2008

J. Fluid Mech.

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We computationally investigate the unsteady pulsatile propagation of a finger of air through a liquidfilled cylindrical rigid tube using a combined boundary element method and lubrication theory approach. The flow-field is governed by the dimensionless parameters Ca Q (t) = Ca M + Ca Ω sin (Ωt) = μQ*(t*)/πR 2 γ, Ω = μωR/γ and A = 2Ca Ω /Ω. Here, Ca Q (t) consists of both mean (Ca M ) and oscillatory (Ca Ω ) components. It is shown that the behavior of this system is appropriately described by steady-state responses until the onset of reverse flow, wherein the system operates in the unsteady regime (Ca Ω > Ca M ). When flows in this regime are considered, converging and diverging stagnation points move dynamically throughout the cycle and may temporarily separate from the interface at high Ω. We have also found that for Ca Ω < 10Ca M the bubble tip pressure drop ΔP tip may be estimated accurately from the pressure measured downstream of the bubble tip when corrections for the pressure drop due to Poiseuille flow are applied. The normal stress gradient at the tube wall (∂τ n /∂z) is discussed in detail, as this is believed to be the primary factor in airway epithelial cell damage (Bilek et al 2003). In the unsteady regime we find that local film-thinning produces high ∂τ n /∂z at low Ca Ω . Film thickening at moderate Ca Ω in the unsteady regime protects the tube wall from the large gradients near the bubble tip, therefore reducing ∂τ n /∂z. We find that the stress field is highly dynamic and exhibits intriguing spatial and temporal characteristics that may be of interest to our field of study, pulmonary airway reopening.

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“…The weak compactness follows directly from the energy estimate in (9). It remains to prove that if θ α −→θ weakly in L 2 (Ω) thenθ is a weak solution for the SQG equation in (1).…”

Section: Is a Weak Solution For The Sqg Equations (1)mentioning

confidence: 99%

An inviscid regularization for the surface quasi‐geostrophic equation

Khouider

Titi

2007

Comm Pure Appl Math

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Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter α tends to zero, for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite time blow up testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. Moreover, much like the original problem, the inviscid regularization admits a maximum principle.

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The Mathematical Theory of Viscous Incompressible Flow

Cited by 1,238 publications

References 0 publications

Towards a sufficient criterion for collapse in 3D Euler equations

Towards a sufficient criterion for collapse in 3D Euler equations

The pulsatile propagation of a finger of air within a fluid-occluded cylindrical tube

An inviscid regularization for the surface quasi‐geostrophic equation

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