1,721,007 research outputs found
Stability of equilibria and bifurcations for a fluid-solid interaction problem
No full textWe study certain significant properties of the equilibrium configurations of a rigid body subject to an undamped elastic restoring force, in the stream of a viscous liquid in an unbounded 3D domain. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity lambda. We show that if lambda is below a critical value, lambda(c )(say), there is a unique and stable time-independent configuration, where the body is in equilibrium and the flow is steady. We also prove that, if lambda < lambda(c) , no oscillatory flow may occur. Successively, we investigate possible loss of uniqueness by providing necessary and sufficient conditions for the occurrence of a steady bifurcation at some lambda(s) >= lambda (c) . (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/)
Steady-state flow of a shear-thinning liquid in an unbounded pipeline system
No full textWe show existence and uniqueness of steady-state solutions to the equations of generalized Newtonian liquids of shear-thinning type in an unbounded region that includes semi-infinite cylinders of constant cross-section (pipeline system). This result is established under the assumption that the datum (flow-rate) is sufficiently small. The main feature of our approach is the proof of a “global compactness” property of the sequence of approximating solutions
Flow-induced oscillations via Hopf bifurcation in a fluid–solid interaction problem
Get PDFWe furnish necessary and sufficient conditions for the occurrence of local Hopf bifurcation in a notably significant fluid-structure problem, where a Navier-Stokes liquid interacts with a rigid body that is subject to an undamped elastic restoring force. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity lambda>0. The study is particularly challenging since 0 is in the essential spectrum of the relevant linearized operator, for any value of lambda, which makes classical bifurcation theories inapplicable. To successfully address this situation, we build upon the method introduced by Galdi (Arch Ration Mech Anal 222:285-315, 2016) that overcomes the problem of the absence of a spectral gap. The most remarkable feature of our result is that no restriction is imposed on the frequency omega of the bifurcating solution, which may thus coincide with one of the natural structural frequencies omega(n) of the body. Therefore, resonance cannot occur as a result of this bifurcation. However, when omega ->omega(n), the amplitude of oscillations may become very large when the fluid density is negligible compared to the mass of the body. To our knowledge, our result is the first rigorous investigation of the existence of a Hopf bifurcation in a fluid-structure interaction problem
Womersley flow of generalized Newtonian liquid
Get PDFWe show that in an infinite straight pipe of arbitrary (sufficiently smooth) cross-section, a generalized non-Newtonian liquid admits one and only one fully developed time-periodic flow (Womersley flow) when either the flow rate (problem 1) or the axial pressure gradient (problem 2) is prescribed in analogous time-periodic fashion. In addition, we show that the relevant solution depends continuously upon the data in appropriate norms. As is well known from the Newtonian counterpart of the problem, the latter is pivotal for the analysis of flow in a general unbounded pipe system with cylindrical outlets (Leray's problem). It is also worth remarking that problem 1 possesses an intrinsic interest from both mathematical and physical viewpoints, in that it constitutes a (nonlinear) inverse problem with a significant bearing on several applications, including blood flow modelling in large arteries
Inertial Motions of a Rigid Body with a Cavity Filled with a Viscous Liquid
No full textWe study inertial motions of the coupled system, (Formula presented.) , constituted by a rigid body containing a cavity entirely filled with a viscous liquid. We show that for arbitrary initial data having only finite kinetic energy, every corresponding weak solution (à la Leray–Hopf) converges, as time goes to infinity, to a uniform rotation, unless two central moments of inertia of (Formula presented.) coincide and are strictly greater than the third one. This corroborates a famous “conjecture” of N.Ye. Zhukovskii in several physically relevant cases. Moreover, we show that, in a known range of initial data, this rotation may only occur along the central axis of inertia of (Formula presented.) with the larger moment of inertia. We also provide necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations, which improve and/or generalize results previously given by other authors under different types of approximation. Finally, we present results obtained by a targeted numerical simulation that, on the one hand, complement the analytical findings, whereas, on the other hand, point out new features that the analysis is yet not able to catch, and, as such, lay the foundation for interesting and challenging future investigation
Existence, Uniqueness and Regularity for the Second-Gradient Navier-Stokes Equations in Exterior Domains
No full textWe study the well-posedness of the problem
⎧
⎪
⎨
⎪
⎩
∂u
∂t
+ (Du)u + ∇p = νΔu − τΔΔu in ]0,+∞[×Ω,
divu = 0 in ]0,+∞[×Ω,
u(t,x) =
∂u
∂n (t,x) = 0
on ]0,+∞[×∂Ω,
u(0,x) = u 0 (x) in Ω,
where u :]0,+∞[×Ω → R n is the velocity field, p :]0,+∞[×Ω → R is the pressure,
ν is the kinematical viscosity, τ the so-called hyperviscosity and Ω is a general
domain as for existence and uniqueness of the solution, and an exterior domain as
for regularity results.
This problem has been physically well motivated in the recent years as the
simplest case of an isotropic second-order fluid, i.e. a fluid whose power expended
depends on second derivatives of the velocity field
Equilibrium configuration of a rectangular obstacle immersed in a channel flow
Get PDFFluid 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.SCOPUS: ar.jinfo:eu-repo/semantics/publishe
Inertial motions of a rigid body with a cavity filled with a viscous liquid
Get PDFIn this note we announce a number of analytical and numerical results related to the motion of a system S constituted by a rigid body with a cavity that is completely filled with a Navier-Stokes liquid, and that moves in absence of external forces (inertial motions). Our investigation shows, in particular, that the ultimate motion of S about its center of mass is a permanent rotation, thus proving a longstanding conjecture of N.Ye. Zhukovskii. We also present other interesting features of inertial motions that are emphasized by our numerical tests, but that still lack a rigorous mathematical proof. © 2013 Académie des sciences
Equilibrium Configurations of a Symmetric Body Immersed in a Stationary Navier–Stokes Flow in a Planar Channel
Get PDFWe study the equilibrium configurations for several fluid-structure interaction problems. The fluid is confined in a 2D unbounded channel that contains a body, free to move inside the channel with rigid motions (transversal translations and rotations). The motion of the fluid is generated by a Poiseuille inflow/outflow at infinity and governed by the stationary Navier-Stokes equations. For a model where the fluid is the air and the body represents the cross-section of a suspension bridge, therefore also subject to restoring elastic forces, we prove that for small inflows there exists a unique equilibrium position, while for large inflows we numerically show the appearance of additional equilibria. A similar uniqueness result is also obtained for a discretized 3D bridge, consisting in a finite number of cross-sections interacting with the adjacent ones. The very same model, but without restoring forces, is used to describe the mechanism of the Leonardo da Vinci ferry, which is able to cross a river without engines. We numerically determine the optimal orientation of the ferry that allows it to cross the river in minimal time.SCOPUS: ar.jinfo:eu-repo/semantics/publishe
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