The Eigenfunctions of the Stokes Operator in Special Domains. I

Rummler, B.

doi:10.1002/zamm.19970770817

Cited by 24 publications

(27 citation statements)

References 3 publications

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“…The eigenvalues of the Stokes operator can be found explicitly, indeed, a modification of the procedure used in [2,24] for the no-slip boundary condition gives, after a lengthy computation, the following general formula for the eigenvalues …”

Section: Influence Of the Change In The Spectrum Of The Stokes Operatormentioning

confidence: 99%

On the influence of boundary condition on stability of Hagen–Poiseuille flow

Prša

2009

Computers & Mathematics with Applications

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a b s t r a c tWe analyze the influence of choice of boundary condition (no-slip and Navier's slip boundary conditions) on linear stability of Hagen-Poiseuille flow. Several heuristic arguments based on detailed analysis of spectrum of the Stokes operator are given, and it is concluded that Navier's slip boundary condition should have a destabilizing effect on the flow. Finally the linear stability problem is solved by numerical means, and quantitative results confirming the heuristic prediction are obtained. It is shown that the destabilization is not strong enough to maintain an unstable disturbance, and that the significant destabilization effects of Navier's slip boundary condition are restricted to small values of the Reynolds number. As a byproduct we obtain explicit formulas for eigenfunctions of the Stokes operator subject to Navier's slip boundary condition.

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Section: Influence Of the Change In The Spectrum Of The Stokes Operatormentioning

confidence: 99%

On the influence of boundary condition on stability of Hagen–Poiseuille flow

Prša

2009

Computers & Mathematics with Applications

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“…More explicitly, for such a choice, the SLE problem reads

({, \begin{matrix} center \\ array Δ {\hat{bold-italicφ}}_{k}^{u} + {\hat{λ}}_{k} {\hat{bold-italicφ}}_{k}^{u} = \nabla {\hat{ϕ}}_{k}^{p} & array in \hat{γ}, \\ array Δ {\hat{ϕ}}_{k}^{p} = 0 & array in \hat{γ}, \\ array \nabla \cdot {\hat{bold-italicφ}}_{k}^{u} = 0 & array in \hat{γ}, \\ array {\hat{bold-italicφ}}_{k}^{u} = 0 & array on \partial \hat{γ}, \end{matrix})

being

{\overset{φ}{true}}_{k}^{u}

and

{\overset{ϕ}{true}}_{k}^{p}

the eigenfunctions for the velocity and the pressure, respectively. Following the procedure adopted for scalar problems, the eigenfunctions are computed as in Rummler, and they read as

\begin{matrix} right & left {\hat{ϕ}}_{k}^{p} (\hat{r}, \hat{ϑ}) = {\hat{ϕ}}_{j, n}^{p} (\hat{r}, \hat{ϑ}) = c_{1} {\hat{r}}^{n} \exp (i n \hat{ϑ}), & right \\ right & left {\hat{bold-italicφ}}_{k}^{u} (\hat{r}, \hat{ϑ}) = {\hat{bold-italicφ}}_{j, n}^{u} \end{matrix}

…”

Section: Himod In Cylindrical Domainsmentioning

confidence: 99%

“…beinĝu k and̂k the eigenfunctions for the velocity and the pressure, respectively. Following the procedure adopted for scalar problems, the eigenfunctions are computed as in Rummler, 32 and they read as k (r,̂) =̂, n (r,̂) = c 1r n exp(in̂), u k (r,̂) =̂u ,n (r,̂) = c 1 exp(in̂)…”

Section: The Top-down Approachmentioning

confidence: 99%

Hierarchical model reduction for incompressible fluids in pipes

Guzzetti

Perotto

Veneziani

2018

Numerical Meth Engineering

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Summary Hierarchical model reduction is intended to solve efficiently partial differential equations in domains with a geometrically dominant direction. In many engineering applications, these problems are often reduced to 1‐dimensional differential systems. This guarantees computational efficiency yet dumps local accuracy as nonaxial dynamics are dropped. Hierarchical model reduction recovers the secondary components of the dynamics of interest with a combination of different discretization techniques, following up a natural separation of variables. The dominant direction is generally solved by the finite element method or isogeometric analysis to guarantee flexibility, while the transverse components are solved by spectral methods, to guarantee a small number of degrees of freedom. By judiciously selecting the number of transverse modes, the method has been proven to improve significantly the accuracy of purely 1‐dimensional solvers, with great computational efficiency. A Cartesian framework has been used so far both in slab domains and cylindrical pipes (including arteries) mapped to Cartesian reference domains. In this paper, we investigate the alternative use of a polar coordinates system for the transverse dynamics in circular or elliptical pipes. This seems a natural choice for applications like computational hemodynamics. In spite of this, the selection of a basis function set for the transverse dynamics is troublesome. As pointed out in the literature—even for simple elliptical problems—there is no “best” basis available. In this paper, we perform an extensive investigation of hierarchical model reduction in polar coordinates to discuss different possible choices for the transverse basis, pointing out pros and cons of the polar coordinate system.

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“…We refer to and for formulas defining the complete sets of Laplace and Stokes eigenfunctions on the circular annuli

Ω_{σ}^{*}

and on the unit ball

Ω^{*}

. There the eigenvalues are given as (squares of) roots of certain transcendental equations.…”

Section: Theoretical Groundworkmentioning

confidence: 99%

“…If we are concerned with Poincaré constants, we are also concerned with the first eigenvalue of the Laplace or Stokes operator on the A -domains (with vanishing Dirichlet traces), respectively, because of the relations where λ 1,L (A) and λ 1,S (A) denote the first simple eigenvalue of the Laplace operator and the first simple eigenvalue of the Stokes operator, respectively. We refer to [7] for very detailed information with respect to our domain and to [12] for the open unit ball * . Our paper is organised as follows: The essential theoretical fundamentals are collected in Sect.…”

Section: Introductionmentioning

confidence: 99%

Exact Poincaré constants in two‐dimensional annuli

Rummler

Růžička

Thäter³

2016

Z Angew Math Mech

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We provide precise estimates of the Poincaré constants firstly for scalar functions and secondly for solenoidal (i.e. divergence free) vector fields (in both cases with vanishing Dirichlet traces on the boundary) on 2d‐annuli by the use of the first eigenvalues of the scalar Laplacian and the Stokes operator, respectively. In our non‐dimensional setting each annulus ΩA is defined via two concentrical circles with radii A/2 and A/2+1. Additionally, corresponding problems on domains Ωσ*, the 2d‐annuli from , are investigated ‐ for comparison but also to provide limits for A→0. In particular, the Green's function of the Laplacian on Ωσ* with vanishing Dirichlet traces on ∂Ωσ* is used to show that for σ→0 the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit circle. On the other hand, we take advantage of the so‐called small‐gap limit for A→∞.

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The Eigenfunctions of the Stokes Operator in Special Domains. I

Cited by 24 publications

References 3 publications

On the influence of boundary condition on stability of Hagen–Poiseuille flow

On the influence of boundary condition on stability of Hagen–Poiseuille flow

Hierarchical model reduction for incompressible fluids in pipes

Exact Poincaré constants in two‐dimensional annuli

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