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1. Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities

   https://doi.org/10.1093/imamat/hxaa006  

   Guo, Hongxia; Gui, Changfeng; Lin, Ping; Zhao, Mingfeng (March 2020, IMA Journal of Applied Mathematics)

   null (Ed.)

   Abstract The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. The numerical results suggest that there exist three solutions designated as type $$I$$, type $II$ and type $III$ for the asymmetric case, type $$I$$ solution exists for all non-negative Reynolds number and types $II$ and $III$ solutions appear simultaneously at a common Reynolds number that depends on the value of asymmetric parameter $$a$$ and with the increase of $$a$$ the common Reynolds numbers are decreasing. We also theoretically show that there exist three solutions. The corresponding asymptotic solution for each of the multiple solutions is constructed by the method of boundary layer correction or matched asymptotic expansion for the most difficult high Reynolds number case. These asymptotic solutions are all verified by their corresponding numerical solutions. 

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2. A model for the oscillatory flow in the cerebral aqueduct

   https://doi.org/10.1017/jfm.2020.463  

   Sincomb, S.; Coenen, W.; Sánchez, A. L.; Lasheras, J. C. (September 2020, Journal of Fluid Mechanics)

   null (Ed.)

   This paper addresses the pulsating motion of cerebrospinal fluid in the aqueduct of Sylvius, a slender canal connecting the third and fourth ventricles of the brain. Specific attention is given to the relation between the instantaneous values of the flow rate and the interventricular pressure difference, needed in clinical applications to enable indirect evaluations of the latter from direct magnetic resonance measurements of the former. An order of magnitude analysis accounting for the slenderness of the canal is used in simplifying the flow description. The boundary layer approximation is found to be applicable in the slender canal, where the oscillating flow is characterized by stroke lengths comparable to the canal length and periods comparable to the transverse diffusion time. By way of contrast, the flow in the non-slender opening regions connecting the aqueduct with the two ventricles is found to be inviscid and quasi-steady in the first approximation. The resulting simplified description is validated by comparison with results of direct numerical simulations. The model is used to investigate the relation between the interventricular pressure and the stroke length, in parametric ranges of interest in clinical applications. 

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3. Series Solutions for Clamped Peridynamic Beams Using Fourth-Order Eigenfunctions

   https://doi.org/10.1007/978-3-031-75626-9_34  

   Wang, Ziyu; Christov, Ivan C (June 2025, Springer Nature Switzerland)

   Altenbach, Holm; Eremeyev, Victor A (Ed.)

   We propose an analytical approach to solving nonlocal generalizations of the Euler–Bernoulli beam. Specifically, we consider a version of the governing equation recently derived under the theory of peridynamics. We focus on the clamped–clamped case, employing the natural eigenfunctions of the fourth derivative subject to these boundary conditions. Static solutions under different loading conditions are obtained as series in these eigenfunctions. To demonstrate the utility of our proposed approach, we contrast the series solution in terms of fourth-order eigenfunctions to the previously obtained Fourier sine series solution. Our findings reveal that the series in fourth-order eigenfunctions achieve a given error tolerance (with respect to a reference solution) with ten times fewer terms than the sine series. The high level of accuracy of the fourth-order eigenfunction expansion is due to the fact that its expansion coefficients decay rapidly with the number of terms in the series, one order faster than the Fourier series in our examples. 

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4. A High-Order Hybrid Approach Integrating Neural Networks and Fast Poisson Solvers for Elliptic Interface Problems

   https://doi.org/10.3390/computation13040083  

   Ren, Yiming; Zhao, Shan (April 2025, Computation)

   A new high-order hybrid method integrating neural networks and corrected finite differences is developed for solving elliptic equations with irregular interfaces and discontinuous solutions. Standard fourth-order finite difference discretization becomes invalid near such interfaces due to the discontinuities and requires corrections based on Cartesian derivative jumps. In traditional numerical methods, such as the augmented matched interface and boundary (AMIB) method, these derivative jumps can be reconstructed via additional approximations and are solved together with the unknown solution in an iterative procedure. Nontrivial developments have been carried out in the AMIB method in treating sharply curved interfaces, which, however, may not work for interfaces with geometric singularities. In this work, machine learning techniques are utilized to directly predict these Cartesian derivative jumps without involving the unknown solution. To this end, physics-informed neural networks (PINNs) are trained to satisfy the jump conditions for both closed and open interfaces with possible geometric singularities. The predicted Cartesian derivative jumps can then be integrated in the corrected finite differences. The resulting discrete Laplacian can be efficiently solved by fast Poisson solvers, such as fast Fourier transform (FFT) and geometric multigrid methods, over a rectangular domain with Dirichlet boundary conditions. This hybrid method is both easy to implement and efficient. Numerical experiments in two and three dimensions demonstrate that the method achieves fourth-order accuracy for the solution and its derivatives. 

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5. Equations of motion for weakly compressible point vortices

   https://doi.org/10.1098/rsta.2021.0052  

   Llewellyn Smith, Stefan G.; Chu, T.; Hu, Z. (June 2022, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Equations of motion for compressible point vortices in the plane are obtained in the limit of small Mach number, M , using a Rayleigh–Jansen expansion and the method of Matched Asymptotic Expansions. The solution in the region between vortices is matched to solutions around each vortex core. The motion of the vortices is modified over long time scales O ( M 2 log ⁡ M ) and O ( M 2 ) . Examples are given for co-rotating and co-propagating vortex pairs. The former show a correction to the rotation rate and, in general, to the centre and radius of rotation, while the latter recover the known result that the steady propagation velocity is unchanged. For unsteady configurations, the vortex solution matches to a far field in which acoustic waves are radiated. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’. 

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