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1. A time domain approach for the exponential stability of a linearized compressible flow‐structure PDE system

   https://doi.org/10.1002/mma.6833  

   Guven Geredeli, Pelin (September 2020, Mathematical Methods in the Applied Sciences)

   This work is motivated by a longstanding interest in the long time behavior of flow‐structure interaction (FSI) PDE dynamics. We consider a linearized compressible flow structure interaction (FSI) PDE model with a view of analyzing the stability properties of both the compressible flow and plate solution components. In our earlier work, we gave an answer in the affirmative to question of uniform stability for finite energy solutions of said compressible flow‐structure system, by means of a “frequency domain” approach. However, the frequency domain method of proof in that work is not “robust” (insofar as we can see), when one wishes to study longtime behavior of solutions of compressible flow‐structure PDE models, which track the appearance of the ambient state onto the boundary interface. Nor is a frequency domain approach in this earlier work availing when one wishes to consider the dynamics, in long time, of solutions to physically relevant nonlinear versions of the compressible flow‐structure PDE system under present consideration (e.g., the Navier–Stokes nonlinearity in the PDE flow component or a nonlinearity of Berger/Von Karman type in the plate equation). Accordingly, in the present work, we operate in the time domain by way of obtaining the necessary energy estimates, which culminate in an alternative proof for the uniform stability of finite energy compressible flow‐structure solutions. Since there is a need to avoid steady states in our stability analysis, as a prerequisite result, we also show here that zero is an eigenvalue for the generators of flow‐structure systems, whether the material derivative term be absent or present. Moreover, we provide a clean characterization of the (one dimensional) zero eigenspace, with or without material derivative, under an appropriate assumption on the underlying ambient vector field. 

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2. Analysis of a 3D nonlinear moving boundary problem describing fluid-mesh-sell interaction

   Suncica Canic, Marija Galic (January 2020, Transactions of the American Mathematical Society)

   Abstract. We consider a nonlinear, moving boundary, fluid-structure interaction problem between a time dependent incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh of elastic rods. The fluid flow is modeled by the time-dependent Navier- Stokes equations in a three-dimensional cylindrical domain, while the lateral wall of the cylinder is modeled by the two-dimensional linearly elastic Koiter shell equations coupled to a one-dimensional system of conservation laws defined on a graph domain, describing a mesh of curved rods. The mesh supported shell allows displacements in all three spatial directions. Two-way coupling based on kinematic and dynamic coupling conditions is assumed between the fluid and composite structure, and between the mesh of curved rods and Koiter shell. Problems of this type arise in many ap- plications, including blood flow through arteries treated with vascular prostheses called stents. We prove the existence of a weak solution to this nonlinear, moving boundary problem by using the time discretization via Lie operator splitting method combined with an Arbitrary Lagrangian-Eulerian approach, and a non-trivial extension of the Aubin-Lions-Simon compactness result to problems on moving domains. 

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3. A Bilevel Network Interdiction Problem to Minimize the Number of Active Special Arcs in the Maximum Flow

   https://doi.org/10.1287/ijoc.2023.0423  

   Lopes_da_Silva, Daniel B; Sharkey, Thomas C; Song, Yongjia (April 2025, INFORMS Journal on Computing)

   We consider a bilevel network interdiction problem where the follower aims to maximize the amount of flow from the source node to the sink node, and the leader aims to minimize the number of arcs from a critical set that have positive flow on them, that is, active arcs, in the maximum flow solution obtained by the follower. This problem is motivated by an application in human trafficking disruption. We consider both the optimistic and pessimistic variants of this bilevel optimization problem and develop their respective single-level reformulations. We present a tailored solution method to the pessimistic problem, which solves the problem to optimality for one practically important class of networks. Through computational experiments on randomly generated layered network instances, we show the effectiveness of the proposed methods and demonstrate that the tailored method is orders of magnitude faster than existing approaches in the literature. We also conduct computational experiments on randomly generated test instances inspired by domestic human trafficking networks and draw domain-specific insights. 

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4. Stability of a Two-Dimensional Poiseuille-Type Flow for a Viscoelastic Fluid

   Masakazu Endo, Yoshikazu Giga (March 2017, Journal of mathematical fluid mechanics)

   A viscoelastic flow in a two-dimensional layer domain is considered. An L 2-stability of the Poiseuille-type flow is established provided that both Poiseuille flow and perturbation is sufficiently small. Our analysis is based on a stream function formulation introduced by Lin et al. 

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5. Volumetric Flow Field inside a Gas Stirred Cylindrical Water Tank

   https://doi.org/10.3390/fluids9010011  

   Jojo-Cunningham, Yasmeen; Guo, Xipeng; Zhou, Chenn; Liu, Yun (January 2024, Fluids)

   Ladle metallurgy serves as a crucial component of the steelmaking industry, where it plays a pivotal role in manipulating the molten steel to exercise precise control over its composition and properties. Turbulence in ladle metallurgy influences various important aspects of the steelmaking process, including mixing and distribution of additives, alongside the transport and removal of inclusions within the ladle. Consequently, gaining a clear understanding of the stirred flow field holds the potential of optimizing ladle design, improving control strategies, and enhancing the overall efficiency and steel quality. In this project, an advanced Particle-Tracking-Velocimetry system known as “Shake-the-Box” is implemented on a cylindrical water ladle model while compressed air injections through two circular plugs positioned at the bottom of the model are employed to actively stir the flow. To mitigate the particle images distortion caused by the cylindrical plexi-glass walls, the method of refractive matching is utilized with an outer polygon tank filled with a sodium iodide solution. The volumetric flow measurement is achieved on a 6 × 6 × 2 cm domain between the two plugs inside the cylindrical container while the flow rate of gas injection is set from 0.1 to 0.4 L per minute. The volumetric flow field result suggests double gas injection at low flow rate (0.1 L per minute) produce the least disturbed flow while highly disturbed and turbulent flow can be created at higher flow rate of gas injection. 

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