We consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid-solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain. In particular, we study the problem of existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present. The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis. Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory.
We consider the well-posedness of a model for a flow-structure interaction. This model describes the dynamics of an elastic flexible plate with clamped boundary conditions immersed in a supersonic flow. A perturbed wave equation describes the flow potential. The plate's out-of-plane displacement can be modeled by various nonlinear plate equations (including von Karman and Berger). Supersonic regimes corresponding to the flow provide for new mathematical challenge that is related to the loss of ellipticity in a stationary dynamics. This difficulty is present also in the linear model. We show that the linearized model is well-posed on the state space (as given by finite energy considerations) and generates a strongly continuous semigroup. We make use of these results along with sharp regularity of Airy's stress function (obtained by compensated compactness method) to conclude global-in-time well-posedness for the fully nonlinear model. The proof of generation has two novel features, namely: (1) we introduce a new flow potential velocity-type variable which makes it possible to cover both subsonic and supersonic cases, and to split the dynamics generating operator into a skew-adjoint component and a perturbation acting outside of the state space. Performing semigroup analysis also requires a nontrivial approximation of the domain of the generator. The latter is due to the loss of ellipticity. And (2) we make critical use of hidden trace regularity for the flow component of the model (in the abstract setup for the semigroup problem) which allows us to develop a fixed point argument and eventually conclude well-posedness. This well-posedness result for supersonic flows (in the absence of regularizing rotational inertia) has been hereto open. The use of semigroup methods to obtain well-posedness opens this model to long-time behavior considerations.
This paper is devoted to a long time behavior analysis associated with flow structure interactions at subsonic and supersonic velocities. It turns out that an intrinsic component of that analysis is the study of attracting sets corresponding to von Karman plate equations with delayed terms and without rotational terms. The presence of delay terms in the dynamical system leads to the loss of gradient structure while the absence of rotational terms in von Karman plates leads to the loss of compactness of the orbits. Both these features make the analysis of long time behavior rather subtle rendering the established tools in the theory of PDE dynamical systems not applicable. It is our goal to develop methodology that is capable of handling this class of problems.Key terms: nonlinear plate, PDE with delay, long-time behavior of solutions, dynamical systems, global attractors, flow-structure interaction, MSC 2010: 35L20, 74F10, 35Q74. known from experiment (and also confirmed by numerics), that the potential flow (particularly at the supersonic speeds) has the ability of inducing a certain amount of stability in the moving structure. This is the case even when the structure itself does not possess mechanical damping mechanisms. If one writes down the equations for the interactive system, along with the standard energy balance, this dissipative effect is not exhibited at all; quantities are conserved and not dissipated. Thus, there must be some "hidden" mechanism which produces this dissipation. It is our goal to shed some light on this phenomenon. As it turns out, the decoupling technique introduced in [5, 6], which reduces the analysis of full flow-structure interaction to that of a certain delayed plate model, allows us to observe certain stabilizing effects of the flow. These occur in the form of non-conservative forces acting upon the structure as the "downwash" of the flow. This idea was already applied to Berger plate models [7,17] in the proof of existence of attractors corresponding to the associated reduced plate problem with a delayed term.In fact, well-posedness and long-time behavior analyses of nonlinear plate PDEs with delays have been treated in [8] (see also [14]): first, in the case of the von Karman model with rotational inertia, and secondly, in [7,17], in the case of the Berger model with a small intensity of delayed term (this corresponds to a large speed U of the flow of gas -hypersonic). These expositions flesh out the existence and properties of global attractors for the general plate with delay in the presence of a 'natural' form of interior damping, and then apply this general result to the specific delayed (aeroelastic) force given in the full flow-plate coupling.It should be noted that the presence of rotational inertia parameter, while drastically improving the topological properties of the model, is neither natural nor desirable in the context of flow-structure interaction. First, the original model for flow structure interaction describes the interaction between the mid-surface of the pl...
We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile U. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically [18]-the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a C0-semigroup e At t≥0 on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator A which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field U ∈ H 3 (O) has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.
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