Eight(y) mathematical questions on fluids and structures

Bonheure, Denis; Gazzola, Filippo; Sperone, Gianmarco

doi:10.4171/rlm/870

Cited by 11 publications

(19 citation statements)

References 145 publications

(193 reference statements)

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“…In the second problem, the body B is immersed in the same channel R×(−L, L) but is only free to rotate around a pin located at its center of mass, see Figure 2. These two problems are inspired to some bridge models considered in [2,6]. The obstacle B represents the cross-section of the deck of a suspension bridge, that may display both vertical and torsional oscillations, see [5].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…There is an upwards restoring force due to the elastic action of both the hangers and the sustaining cables which, somehow, behave as linear springs which may slacken so that they have no downwards action. There is the weight of the deck which acts constantly downwards: this explains why there is no odd requirement on f in (2). Finally, there is a resistance to both bending and stretching of the whole deck for which B merely represents a cross-section: this force is superlinear and explains the infinite limit in (2), the deck is not allowed to go too far away from its equilibrium (horizontal) position due to the elastic resistance to deformations of the whole deck.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…There is the weight of the deck which acts constantly downwards: this explains why there is no odd requirement on f in (2). Finally, there is a resistance to both bending and stretching of the whole deck for which B merely represents a cross-section: this force is superlinear and explains the infinite limit in (2), the deck is not allowed to go too far away from its equilibrium (horizontal) position due to the elastic resistance to deformations of the whole deck. The torsional oscillations are symmetric, they are due to the possible different behaviors of the hangers and cables at the two endpoints of the cross-section, see Figure 2 and the right picture in Figure 3.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…by [3,Theorem II.3.3] this problem admits a solution z ∈ H 2 0 (Σ L ). Hence, if we extend z by zero outside Σ L we obtain that z ∈ H 2…”

Section: Lemma 2 There Exists a Constantmentioning

confidence: 99%

“…when R = 0, as u(h, 0), see (16), does not produce any lift whatever h is. In terms of the function f , defined in (2), this means that…”

Section: Lemma 3 For Any H ∈mentioning

confidence: 99%

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Equilibrium configuration of a rectangular obstacle immersed in a channel flow

Bonheure¹,

Galdi²,

Gazzola³

2020

Comptes Rendus. Mathématique

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Fluid flows around an obstacle generate vortices which, in turn, generate lift forces on the obstacle. Therefore, even in a perfectly symmetric framework equilibrium positions may be asymmetric. We show that this is not the case for a Poiseuille flow in an unbounded 2D channel, at least for small Reynolds number and flow rate. We consider both the cases of vertically moving obstacles and obstacles rotating around a fixed pin.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…by [3,Theorem II.3.3] this problem admits a solution z ∈ H 2 0 (Σ L ). Hence, if we extend z by zero outside Σ L we obtain that z ∈ H 2…”

Section: Lemma 2 There Exists a Constantmentioning

confidence: 99%

“…when R = 0, as u(h, 0), see (16), does not produce any lift whatever h is. In terms of the function f , defined in (2), this means that…”

Section: Lemma 3 For Any H ∈mentioning

confidence: 99%

See 3 more Smart Citations

Equilibrium configuration of a rectangular obstacle immersed in a channel flow

Bonheure¹,

Galdi²,

Gazzola³

2020

Comptes Rendus. Mathématique

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Uniqueness and Bifurcation Branches for Planar Steady Navier–Stokes Equations Under Navier Boundary Conditions

Arioli

Gazzola

Koch

2021

J. Math. Fluid Mech.

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The stationary Navier–Stokes equations under Navier boundary conditions are considered in a square. The uniqueness of solutions is studied in dependence of the Reynolds number and of the strength of the external force. For some particular forcing, it is shown that uniqueness persists on some continuous branch of solutions, when these quantities become arbitrarily large. On the other hand, for a different forcing, a branch of symmetric solutions is shown to bifurcate, giving rise to a secondary branch of nonsymmetric solutions. This proof is computer-assisted, based on a local representation of branches as analytic arcs.

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