Equilibrium configuration of a rectangular obstacle immersed in a channel flow. (English) Zbl 1455.35168
This is an interesting paper in which symmetry breaking of an equilibrum configuration or a rectangular body emmersed in a viscous channel flow (Poiseuille flow at infinity) is ruled out at low Reynolds number. The body can either be pinned so that it can only be displaced vertically in the channel, or else pinned so that it can only rotate. The situation is supposed to model a suspended bridge. The equilibrium position is found to be independent of the driving parameters (Reynolds and Poiseuille amplitude) in its regime of uniqueness.
Reviewer: Theodore D. Drivas (Princeton)
MSC:
| 35Q30 | Navier-Stokes equations |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |
| 76D17 | Viscous vortex flows |
| 76U05 | General theory of rotating fluids |
References:
| [1] | Bello, Juan Antonio; Fernández-Cara, Enrique; Lemoine, Jérôme; Simon, Jacques, The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow, SIAM J. Control Optimization, 35, 2, 626-640 (1997) · Zbl 0873.76019 · doi:10.1137/S0363012994278213 |
| [2] | Bonheure, Denis; Gazzola, Filippo; Sperone, Gianmarco, Eight(y) mathematical questions on fluids and structures, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., 30, 4, 759-815 (2019) · Zbl 1447.76016 · doi:10.4171/RLM/870 |
| [3] | Galdi, Giovanni P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems (2011), Springer · Zbl 1245.35002 |
| [4] | Galdi, Giovanni P.; Heuveline, Vincent, Free and moving boundaries, 252, Lift and sedimentation of particles in the flow of a viscoelastic liquid in a channel (2007), Chapman & Hall/CRC · Zbl 1403.76187 |
| [5] | Gazzola, Filippo, Mathematical models for suspension bridges. Nonlinear structural instability, 15 (2015), Springer · Zbl 1325.00032 |
| [6] | Gazzola, Filippo; Sperone, Gianmarco, Steady Navier-Stokes equations in planar domains with obstacle and explicit bounds for unique solvability, Arch. Ration. Mech. Anal., 238, 3, 1283-1347 (2020) · Zbl 1451.35107 · doi:10.1007/s00205-020-01565-9 |
| [7] | Henrot, Antoine; Pierre, Michel, Shape Variation and Optimization: A Geometrical Analysis, 28 (2018), European Mathematical Society · Zbl 1392.49001 |
| [8] | Ho, B. P.; Leal, L. Gary, Inertial migration of rigid spheres in two-dimensional unidirectional flows, J. Fluid Mech., 65, 365-400 (1974) · Zbl 0284.76076 |
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