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Academic literature on the topic 'PDEs in fluid mechanics'

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Published: 1 March 2025

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Journal articles on the topic "PDEs in fluid mechanics"

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BREIT, D., L. DIENING, and S. SCHWARZACHER. "SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs." Mathematical Models and Methods in Applied Sciences 23, no. 14 (2013): 2671–700. http://dx.doi.org/10.1142/s0218202513500437.

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We consider functions u ∈ L∞(L2)∩Lp(W1, p) with 1 < p < ∞ on a time–space domain. Solutions to nonlinear evolutionary PDEs typically belong to these spaces. Many applications require a Lipschitz approximation uλ of u which coincides with u on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids of Diening, Ruzicka and Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010) 1–46. Since div uλ = 0, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of Breit, Diening and Fuchs, Solenoidal Lipschitz truncation and applications in fluid mechanics, J. Differential Equations253 (2012) 1910–1942.

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Bilige, Sudao, and Yanqing Han. "Symmetry reduction and numerical solution of a nonlinear boundary value problem in fluid mechanics." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 3 (2018): 518–31. http://dx.doi.org/10.1108/hff-08-2016-0304.

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Purpose The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics. Design/methodology/approach The authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods. Findings First, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions. Originality/value The application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.

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Swapna, Y. "Applications of Partial Differential Equations in Fluid Physics." Communications on Applied Nonlinear Analysis 31, no. 1 (2024): 207–20. http://dx.doi.org/10.52783/cana.v31.396.

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Partial differential equations, or PDEs, assume a critical part in grasping and outlining different fluid physics peculiarities. They have an expansive scope of utilizations, from expecting weather patterns to consolidating ocean streams, fire cycles, and fluid streams into system plan. These equations oversee the way of behaving of fluid amounts like as speed, stress, temperature, and consistency. They portray complex collaborations like changes in precipitation, scattering, and fluid-solid associations. Partial differential equations are utilized to apply the developing methodology. The arrangement is equivalent to for the recently concentrated on examples of typical differential equations. There are two kinds of partial differential equations: nonlinear and straight. Some certifiable equations, for example, those in electrostatics, heat conduction, transmission lines, quantum mechanics, and wave hypothesis, feature the significance of partial differential equations (PDEs). To make sense of something other than one, two, or three pieces of the partial differential equations, we will check out at the speculative piece of those applications that utilization PDEs in this examination. In all parts of science and development, partial differential equations, or PDEs, are generally utilized. Partial differential equations handle most of genuine frameworks. A condition communicating a connection between a piece of no less than two free factors and the partial helpers of this cutoff concerning these free factors is known as a partial differential condition, or PDE.

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Cao, Ruohan, Jin Su, Jinqian Feng, and Qin Guo. "PhyICNet: Physics-informed interactive learning convolutional recurrent network for spatiotemporal dynamics." Electronic Research Archive 32, no. 12 (2024): 6641–59. https://doi.org/10.3934/era.2024310.

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<p>The numerical solution of spatiotemporal partial differential equations (PDEs) using the deep learning method has attracted considerable attention in quantum mechanics, fluid mechanics, and many other natural sciences. In this paper, we propose an interactive temporal physics-informed neural network architecture based on ConvLSTM for solving spatiotemporal PDEs, in which the information feedback mechanism in learning is introduced between the current input and the previous state of network. Numerical experiments on four kinds of classical spatiotemporal PDEs tasks show that the extended models have superiority in accuracy, long-range learning ability, and robustness. Our key takeaway is that the proposed network architecture is capable of learning information correlation of the PDEs model with spatiotemporal data through the input state interaction process. Furthermore, our method also has a natural advantage in carrying out physical information and boundary conditions, which could improve interpretability and reduce the bias of numerical solutions.</p>

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Boyaval, Sébastien. "A class of symmetric-hyperbolic PDEs modelling fluid and solid continua." ESAIM: Proceedings and Surveys 76 (2024): 2–19. http://dx.doi.org/10.1051/proc/202476002.

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We generalize a new symmetric-hyperbolic system of PDEs proposed in [ESAIM:M2AN 55 (2021) 807-831] for Maxwell fluids to a class of systems that define unequivocally multi-dimensional visco-elastic flows. Precisely, within a general setting for continuum mechanics, we specify constitutive assumptions i) that ensure the unequivocal definition of motions satisfying widely-admitted physical principles, and ii) that contain [ESAIM:M2AN 55 (2021) 807-831] as one particular realization of those assumptions. The new class can capture the mechanics of various materials, from solids to viscous fluids, possibly with temperature dependence and heat conduction.

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Dalir, Nemat. "Modified Decomposition Method with New Inverse Differential Operators for Solving Singular Nonlinear IVPs in First- and Second-Order PDEs Arising in Fluid Mechanics." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/793685.

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Singular nonlinear initial-value problems (IVPs) in first-order and second-order partial differential equations (PDEs) arising in fluid mechanics are semianalytically solved. To achieve this, the modified decomposition method (MDM) is used in conjunction with some new inverse differential operators. In other words, new inverse differential operators are developed for the MDM and used with the MDM to solve first- and second-order singular nonlinear PDEs. The results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. The comparisons show excellent agreement.

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Moaddy, K., S. Momani, and I. Hashim. "The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics." Computers & Mathematics with Applications 61, no. 4 (2011): 1209–16. http://dx.doi.org/10.1016/j.camwa.2010.12.072.

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Da Prato, Giuseppe, and Vicenţiu D. Rădulescu. "Special issue on stochastic PDEs in fluid dynamics, particle physics and statistical mechanics." Journal of Mathematical Analysis and Applications 384, no. 1 (2011): 1. http://dx.doi.org/10.1016/j.jmaa.2011.06.058.

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Sharma, Nishchal. "Deep Learning for Solving Partial Differential Equations: A Review of Literature." International Journal for Research in Applied Science and Engineering Technology 12, no. 10 (2024): 588–91. http://dx.doi.org/10.22214/ijraset.2024.64623.

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Partial Differential Equations (PDEs) are fundamental in modeling various phenomena in physics, engineering, and finance. Traditional numerical methods for solving PDEs, such as finite element and finite difference methods, often face limitations when applied to high-dimensional and complex systems. In recent years, deep learning has emerged as a promising alternative for approximating solutions to PDEs, offering potential improvements in both efficiency and scalability. This paper provides a comprehensive review of the literature on deep learning-based methods for solving PDEs, focusing on key approaches such as Physics-Informed Neural Networks (PINNs), deep Galerkin methods, and neural operators. These methods leverage the expressiveness of neural networks to capture underlying physics while avoiding the curse of dimensionality associated with classical techniques. We explore the theoretical foundations, advantages, and limitations of these deep learning models, along with their applications in diverse fields like fluid dynamics, quantum mechanics, and financial modeling. Additionally, this review examines recent advancements in hybrid models that combine traditional numerical methods with deep learning approaches to enhance accuracy and stability. Through this review, we highlight key trends and open challenges in the field, paving the way for future research at the intersection of deep learning and computational mathematics.

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Naowarat, Surapol, Sayed Saifullah, Shabir Ahmad, and Manuel De la Sen. "Periodic, Singular and Dark Solitons of a Generalized Geophysical KdV Equation by Using the Tanh-Coth Method." Symmetry 15, no. 1 (2023): 135. http://dx.doi.org/10.3390/sym15010135.

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KdV equations have a lot of applications of in fluid mechanics. The exact solutions of the KdV equations play a vital role in the wave dynamics of fluids. In this paper, some new exact solutions of a generalized geophysical KdV equation are computed with the aid of tanh-coth method. To implement the tanh-coth procedure, we first convert the PDEs to ODEs with the help of wave transformation. Then, using a system of algebraic equations, we obtain several soliton solutions. To verify and clearly illustrate the exact solutions, several graphic presentations are developed by giving the parameter values, which are then thoroughly discussed in the relevant components.

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Dissertations / Theses on the topic "PDEs in fluid mechanics"

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Li, Siran. "Analysis of several non-linear PDEs in fluid mechanics and differential geometry." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:20866cbb-e5ab-4a6b-b9dc-88a247d15572.

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In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L<sup> p</sup> continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H<sup> r+1</sup> (r &GT; 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H<sup> r</sup>, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.

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Bocchi, Edoardo. "Compressible-incompressible transitions in fluid mechanics : waves-structures interaction and rotating fluids." Thesis, Bordeaux, 2019. http://www.theses.fr/2019BORD0279/document.

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Ce manuscrit porte sur les transitions compressible-incompressible dans les équations aux dérivées partielles de la mécanique des fluides. On s'intéresse à deux problèmes : les structures flottantes et les fluides en rotation. Dans le premier problème, l'introduction d'un objet flottant dans les vagues induit une contrainte sur le fluide et les équations gouvernant le mouvement acquièrent une structure compressible-incompressible. Dans le deuxième problème, le mouvement de fluides géophysiques compressibles est influencé par la rotation de la Terre. L'étude de la limite à rotation rapide montre que le champ vectoriel de vitesse tend vers une configuration horizontale et incompressible.Les structures flottantes constituent un exemple particulier d'interaction fluide-structure, où un solide partiellement immergé flotte à la surface du fluide. Ce problème mathématique modélise le mouvement de convertisseurs d'énergie marine. En particulier, on s'intéresse aux bouées pilonnantes, installées proche de la côte où les modèles asymptotiques en eaux peu profondes sont valables. On étudie les équations de Saint-Venant axisymétriques en dimension deux avec un objet flottant à murs verticaux se déplaçant seulement verticalement. Les hypothèses sur le solide permettent de supprimer le problème à bord libre associé avec la ligne de contact entre l'air, le fluide et le solide. Les équations pour le fluide dans le domaine extérieur au solide sont donc écrites comme un problème au bord quasi-linéaire hyperbolique. Celui-ci est couplé avec une EDO non-linéaire du second ordre qui est dérivée de l'équation de Newton pour le mouvement libre du solide. On montre le caractère bien posé localement en temps du système couplé lorsque que les données initiales satisfont des conditions de compatibilité afin de générer des solutions régulières.Ensuite on considère une configuration particulière: le retour à l'équilibre. Il s'agit de considérer un solide partiellement immergé dans un fluide initialement au repos et de le laisser retourner à sa position d'équilibre. Pour cela, on utilise un modèle hydrodynamique différent, où les équations sont linearisées dans le domaine extérieur, tandis que les effets non-linéaires sont considérés en dessous du solide. Le mouvement du solide est décrit par une équation intégro-différentielle non-linéaire du second ordre qui justifie rigoureusement l'équation de Cummins, utilisée par les ingénieurs pour les mouvements des objets flottants. L'équation que l'on dérive améliore l'approche linéaire de Cummins en tenant compte des effets non-linéaires. On montre l'existence et l'unicité globale de la solution pour des données petites en utilisant la conservation de l'énergie du système fluide-structure.Dans la deuxième partie du manuscrit, on étudie les fluides en rotation rapide. Ce problème mathématique modélise le mouvement des flots géophysiques à grandes échelles influencés par la rotation de la Terre. Le mouvement est aussi affecté par la gravité, ce qui donne lieu à une stratification de la densité dans les fluides compressibles. La rotation génère de l'anisotropie dans les flots visqueux et la viscosité turbulente verticale tend vers zéro dans la limite à rotation rapide. Notre interêt porte sur ce problème de limite singulière en tenant compte des effets gravitationnels et compressibles. On étudie les équations de Navier-Stokes-Coriolis anisotropes compressibles avec force gravitationnelle dans la bande infinie horizontale avec une condition au bord de non glissement. Celle-ci et la force de Coriolis donnent lieu à l'apparition des couches d'Ekman proche du bord. Dans ce travail on considère des données initiales bien préparées. On montre un résultat de stabilité des solutions faibles globales pour des lois de pression particulières. La dynamique limite est décrite par une équation quasi-géostrophique visqueuse en dimension deux avec un terme d'amortissement qui tient compte des couches limites<br>This manuscript deals with compressible-incompressible transitions arising in partial differential equations of fluid mechanics. We investigate two problems: floating structures and rotating fluids. In the first problem, the introduction of a floating object into water waves enforces a constraint on the fluid and the governing equations turn out to have a compressible-incompressible structure. In the second problem, the motion of geophysical compressible fluids is affected by the Earth's rotation and the study of the high rotation limit shows that the velocity vector field tends to be horizontal and with an incompressibility constraint.Floating structures are a particular example of fluid-structure interaction, in which a partially immersed solid is floating at the fluid surface. This mathematical problem models the motion of wave energy converters in sea water. In particular, we focus on heaving buoys, usually implemented in the near-shore zone, where the shallow water asymptotic models describe accurately the motion of waves. We study the two-dimensional nonlinear shallow water equations in the axisymmetric configuration in the presence of a floating object with vertical side-walls moving only vertically. The assumptions on the solid permit to avoid the free boundary problem associated with the moving contact line between the air, the water and the solid. Hence, in the domain exterior to the solid the fluid equations can be written as an hyperbolic quasilinear initial boundary value problem. This couples with a nonlinear second order ODE derived from Newton's law for the free solid motion. Local in time well-posedness of the coupled system is shown provided some compatibility conditions are satisfied by the initial data in order to generate smooth solutions.Afterwards, we address a particular configuration of this fluid-structure interaction: the return to equilibrium. It consists in releasing a partially immersed solid body into a fluid initially at rest and letting it evolve towards its equilibrium position. A different hydrodynamical model is used. In the exterior domain the equations are linearized but the nonlinear effects are taken into account under the solid. The equation for the solid motion becomes a nonlinear second order integro-differential equation which rigorously justifies the Cummins equation, assumed by engineers to govern the motion of floating objects. Moreover, the equation derived improves the linear approach of Cummins by taking into account the nonlinear effects. The global existence and uniqueness of the solution is shown for small data using the conservation of the energy of the fluid-structure system.In the second part of the manuscript, highly rotating fluids are studied. This mathematical problem models the motion of geophysical flows at large scales affected by the Earth's rotation, such as massive oceanic and atmospheric currents. The motion is also influenced by the gravity, which causes a stratification of the density in compressible fluids. The rotation generates anisotropy in viscous flows and the vertical turbulent viscosity tends to zero in the high rotation limit. Our interest lies in this singular limit problem taking into account gravitational and compressible effects. We study the compressible anisotropic Navier-Stokes-Coriolis equations with gravitational force in the horizontal infinite slab with no-slip boundary condition. Both this condition and the Coriolis force cause the apparition of Ekman layers near the boundary. They are taken into account in the analysis by adding corrector terms which decay in the interior of the domain. In this work well-prepared initial data are considered. A stability result of global weak solutions is shown for power-type pressure laws. The limit dynamics is described by a two-dimensional viscous quasi-geostrophic equation with a damping term that accounts for the boundary layers

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Barker, Tobias. "Uniqueness results for viscous incompressible fluids." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:db1b3bb9-a764-406d-a186-5482827d64e8.

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First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calderón. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L<sub>∞</sub>(-1; 0; L<sup>3, β</sup>(B(1) &xcap; &Ropf;<sup>3</sup> <sub>+</sub>)) with 3 ≤ β < ∞. What enables us to build upon the work of Escauriaza, Seregin and &Scaron;verák [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new ϵ-regularity criterion. Third, we show that if a weak Leray-Hopf solution in &Ropf;<sup>3</sup> <sub>+</sub>×]0,∞[ has a finite blow-up time T, then necessarily lim<sub>t↑T</sub>||v(·, t)||<sub>L<sup>3,β</sup>(&Ropf;<sup>3</sup> <sub>+</sub>)</sub> = ∞ with 3 < β < ∞. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and &Scaron;verák [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in &Ropf;<sup>3</sup>, with solenoidal initial data in the critical Besov space ?<sup>-1/4</sup><sub>4,∞</sub>(&Ropf;<sup>3</sup>), which has certain continuity properties with respect to weak&ast; convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in &Ropf;<sup>3</sup>×]0,∞[ has a finite blow-up time T, then necessarily lim<sub>t↑T</sub> ||v(·, t)||<sub>L<sub>3</sub>(&Ropf;<sup>3</sup>)</sub> = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.

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Kolumban, Jozsef. "Control issues for some fluid-solid models." Thesis, Paris Sciences et Lettres (ComUE), 2018. http://www.theses.fr/2018PSLED012/document.

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L'analyse du comportement d'un solide ou de plusieurs solides à l'intérieur d'un fluide est un problème de longue date, que l'on peut voir décrit dans de nombreux manuels classiques d'hydrodynamique. Son étude d'un point de vue mathématique a suscité une attention croissante, en particulier au cours des 15 dernières années. Ce projet de recherche vise à mettre l'accent sur plusieurs aspects de cette analyse mathématique, en particulier sur le contrôle et les problèmes asymptotiques. Un modèle simple d'évolution fluide-solide est celui d'un seul corps rigide entouré d'un fluide incompressible parfait. Le fluide est modelé par les équations d'Euler, tandis que le solide évolue selon la loi de Newton et est influencé par la pression du fluide sur la limite. L'objectif de cette thèse de doctorat consisterait en diverses études dans cette branche et, en particulier, étudierait les questions de contrôlabilité de ce système, ainsi que des modèles de limite pour les solides minces qui convergent vers une courbe. Nous souhaitons également étudier le système de contrôle Navier-Stokes / solid d'une manière similaire au problème de contrôlabilité du système Euler / solid. Une autre direction pour ce projet de doctorat est d'obtenir une limite lorsque le solide se concentre dans une courbe. Est-il possible d'obtenir un modèle simplifié d'un objet mince évoluant dans un fluide parfait, de la même manière que des modèles simplifiés ont été obtenus pour des objets qui sont petits dans toutes les directions? Cela pourrait ouvrir la voie à des recherches futures sur la dérivation des flux de cristaux liquides comme limite du système décrivant l'interaction entre le fluide et un filet de tubes solides lorsque le diamètre des tubes converge à zéro<br>The analysis of the behavior of a solid or several solids inside a fluid is a long-standing problem, that one can see described in many classical textbooks of hydrodynamics. Its study from a mathematical viewpoint has attracted a growing attention, in particular in the last 15 years. This research project aims at focusing on several aspect of this mathematical analysis, in particular on control and asymptotic issues. A simple model of fluid-solid evolution is that of a single rigid body surrounded by a perfect incompressible fluid. The fluid is modeled by the Euler equations, while the solid evolves according to Newton’s law, and is influenced by the fluid’s pressure on the boundary. The goal of this PhD thesis would consist in various studies in this branch, and in particular would investigate questions of controllability of this system, as well as limit models for thin solids converging to a curve. We would also like to study the Navier-Stokes/solid control system in a similar manner to the previously discussed controllability problem for the Euler/solid system. Another direction for this PhD project is to obtain a limit when the solid concentrates into a curve. Is it possible to obtain a simplified model of a thin object evolving in a perfect fluid, in the same way as simplified models were obtained for objects that are small in all directions? This could open the way to future investigations on derivation of liquid crystal flows as the limit of the system describing the interaction between the fluid and a net of solid tubes when the diameter of the tubes is converging to zero

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Helluy, Philippe. "Simulation numérique des écoulements multiphasiques: de la théorie aux applications." Habilitation à diriger des recherches, Université du Sud Toulon Var, 2005. http://tel.archives-ouvertes.fr/tel-00657839.

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Perrin, Charlotte. "Modèles hétérogènes en mécanique des fluides : phénomènes de congestion, écoulements granulaires et mouvement collectif." Thesis, Université Grenoble Alpes (ComUE), 2016. http://www.theses.fr/2016GREAM023/document.

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Cette thèse est dédiée à la description et à l'analyse mathématique de phénomènes d'hétérogénéités et de congestion dans les modèles de la mécanique des fluides.On montre un lien rigoureux entre des modèles de congestion douce de type Navier-Stokes compressible qui intègrent des forces de répulsion à très courte portée entre composants élémentaires; et des modèles de congestion dure de type compressible/incompressible décrivant les transitions entre zones libres et zones congestionnées.On s'intéresse ensuite à la modélisation macroscopique de mélanges formés par des particules solides immergées dans un fluide.On apporte dans ce cadre une première réponse mathématique à la question de la transition entre les régimes de suspensions dictés par les interactions hydrodynamiques et les régimes granulaires dictés par les contacts entre les particules solides.On met par cette démarche en évidence le rôle crucial joué par les effets de mémoire dans le régime granulaire.Cette approche permet également un nouveau point de vue pour l'étude mathématique des fluides avec viscosité dépendant de la pression.On s'intéresse enfin à la modélisation microscopique et macroscopique du trafic routier.Des schémas numériques originaux sont proposés afin de reproduire des phénomènes de persistance d'embouteillages<br>This thesis is dedicated to the description and the mathematical analysis of heterogeneities and congestion phenomena in fluid mechanics models.A rigorous link between soft congestion models, based on the compressible Navier--Stokes equations which take into account short--range repulsive forces between elementary components; and hard congestion models which describe the transitions between free/compressible zones and congested/incompressible zones.We are interested then in the macroscopic modelling of mixtures composed solid particles immersed in a fluid.We provide a first mathematical answer to the question of the transition between the suspension regime dictated by hydrodynamical interactions and the granular regime dictated by the contacts between the solid particles.The method highlights the crucial role played by the memory effects in the granular regime.This approach enables also a new point of view concerning fluids with pressure-dependent viscosities.We finally deal with the microscopic and the macroscopic modelling of vehicular traffic.Original numerical schemes are proposed to robustly reproduce persistent traffic jams

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Benjelloun, Saad. "Quelques problèmes d'écoulement multi-fluide : analyse mathématique, modélisation numérique et simulation." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2012. http://tel.archives-ouvertes.fr/tel-00764374.

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La présente thèse comporte trois parties indépendantes. La première partie présente une preuve d'existence de solutions faibles globales pour un modèle de sprays de type Vlasov-Navier-Stokes-incompressible avec densité variable. Ce modèle est obtenu par une limite formelle à partir d'un modèle Vlasov-Navier-Stokes-incompressible avec fragmentation, où seules deux valeurs de rayons de particules sont considérées : un rayon r1 pour les particules avant fragmentation, et un rayon r2 plus petit pour les particules obtenues par fragmentation. Le modèle asymptotique est obtenu dans la limite r2 tendant vers zéro. La démonstration s'appuie sur des techniques de régularisation et de troncature en vitesse, sur le théorème de Schauder et enfin sur une méthode de compacité de Lions-Di-Perna pour l'élimination des régularisations introduites dans le système initial. La deuxième partie concerne la modélisation de l'impact d'une vague de liquide sur une paroi. L'objectif de cette partie est d'obtenir un modèle pour la fuite du gaz environnant sur les "côtés" de la vague. Un modèle numérique est réalisé en remplaçant la vague liquide par une masse solide indéformable et un schéma VFFC-ALE est conçu pour la simulation numérique du modèle. La mise sans dimension des équations permet de montrer les nombres sans dimension qui régissent le phénomène de fuite. La vitesse moyenne de fuite est comparée à la vitesse dans le cas d'un fluide incompressible (pour lequel on a une expression exacte). Enfin, via la simulation numérique, une étude paramétrique est réalisée en fonction des nombres sans dimensions. Dans la troisième partie on présente une méthode numérique pour la simulation d'un modèle Vlasov-Boltzmann-Euler pour les sprays. Cette méthode couple le schéma VFFC à la méthode PIC (Particle In Cell). Les résultats présentés concernent l'écoulement d'un spray dans un pipeline courbe qu'on modélise par un système Vlasov-Boltzmann-Euler quasi-1D.

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Noisette, Florent. "Interactions avec la frontière pour des équations d’évolutions non-linéaires, non-locales." Electronic Thesis or Diss., Bordeaux, 2024. http://www.theses.fr/2024BORD0356.

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Ce manuscrit est découpé en deux parties et 9 chapitres. Dans la deuxième partie, j’ai mis les six résultats principaux prouvé pendant ma thèse :• Chapitre 5 : unicité des solutions d’Euler 2D avec sources et puits;• Chapitre 6 : unicité des solutions de l’équation de Camassa-Holm avec flot entrant et sortant ;• Chapitre 7 : un algorithme pour la simulation numérique de la croissance de micro-algues;• Chapitre 8 : Dérivée de forme de l’opérateur Dirichlet vers Neumann sur une variété bornée; ETcaractère bien posé d’une équation sur les protrusions céllulaires;• Chapitre 9 : régularité de l’opérateur de Dirichlet vers Neumann sur une variété Hs.Les deux premiers résultats ont été réalisés sous la tutelle de Franck Sueur au début de ma thèse, voir [ NS21 ] et [ Noi23 ] pour leur versions individuelles. Le troisième a été écrit à la suite du CEMRACS, en collaboration avec Mickaël Bestard et Léo Meyer, sous la direction de Bastien Polizzi, ainsi que Thierry Goudon et Sebastian Minjeaud, que vous trouverez aussi dans [ Ber+23 ]. Tout en étant plus éloigné du reste de mes travaux, l’algorithme sur lequel nous avons travaillé est interessant et je suis content de l’inclure ici. Les deux derniers résultats ont été écrits sous la tutelle de David Lannes, en cours de publication [ Noi24 ]. En première partie, je présente chacun des résultats, dans l’ordre. Le chapitre 1 va introduire l’équation et la littérature nécessaire pour comprendre le résultat du chapitre 5, le chapitre 2 fait de même pour le résultat du chapitre 6, et ainsi de suite jusqu’au chapitre 4, qui lui introduit à la fois les réultatats du chapitre 8 et 9<br>The main results of my PhD thesis are :• Uniqueness of bounded vorticity solution for the 2D euler equation with sources and sinks• Uniqueness of bounded momentum solution of the CH equation with in and out-flow• An algorythm for the simulation of growth of Micro algae• shape derivative of the Dirichlet to neumann operator on a generic bounded domain• regularity of the Dirichlet to Neumann operator on a generic H^s manifold

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Doyeux, Vincent. "Modelisation et simulation de systemes multi-fluides. Application aux ecoulements sanguins." Phd thesis, Université de Grenoble, 2014. http://tel.archives-ouvertes.fr/tel-00939930.

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Dans ce travail, nous développons un cadre de calcul dédié à la simulation d'écoulements à plusieurs fluides. Nous présentons des validations et vérifications de ces méthodes sur des problèmes de capture d'interfaces et de simulations de bulles visqueuses. Nous montrons ensuite que ce cadre de calcul est adapté à la simulation d'objet rigides en écoulement. Puis, nous étendons ces méthodes à la simulation d'objets déformables simulant le comportement des globules rouges : les vésicules. Nous validons aussi ces simulations. Enfin nous appliquons les précédents modèles à des problèmes ouverts de microfluidique tels que la séparation d'une suspension dans une bifurcation microfluidique et la rhéologie en milieu confiné.

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Martin, Sébastien. "Modélisation et analyse mathématique de problèmes issus de la mécanique des fluides : applications à la tribologie et aux sciences du vivant." Habilitation à diriger des recherches, Université Paris Sud - Paris XI, 2012. http://tel.archives-ouvertes.fr/tel-00765580.

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Ce mémoire presente une synthèse de travaux de recherche consacrés à l'analyse de problèmes mathématiques issus de la mécanique des fluides. En particulier, par le mélange de modélisation, d'analyse théorique et numérique d' équations aux dérivées partielles ainsi que de calcul scientifique, les champs applicatifs de ces travaux ont porté essentiellement sur deux grandes thématiques : la mécanique des films minces et les biosciences. Cette synthèse s'articule autour de trois chapitres : 1) la lubrification hydrodynamique, 2) les lois de conservation scalaires sur un domaine borné et 3) la modélisation mathématique appliquée aux sciences du vivant qui présente, à son tour, deux axes distincts : la modélisation du système respiratoire et, en particulier, des échanges gazeux dans l'arbre bronchique et la simulation de suspensions biomimétiques actives ou passives dans un fluide de Stokes.

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Books on the topic "PDEs in fluid mechanics"

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Layton, Anita T., and Sarah D. Olson. Biological fluid dynamics: Modeling, computations, and applications : AMS Special Session, Biological Fluid Dynamics : Modeling, Computations, and Applications : October 13, 2012, Tulane University, New Orleans, Louisiana. American Mathematical Society, 2014.

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Spurk, Joseph H. Fluid mechanics. 2nd ed. Springer, 2008.

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Durst, Franz. Fluid Mechanics. Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-71343-2.

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Spurk, Joseph H. Fluid Mechanics. Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-642-58277-6.

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Boxer, G. Fluid Mechanics. Macmillan Education UK, 1988. http://dx.doi.org/10.1007/978-1-349-09805-7.

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Spurk, Joseph H., and Nuri Aksel. Fluid Mechanics. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-30259-7.

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Widden, Martin. Fluid Mechanics. Macmillan Education UK, 1996. http://dx.doi.org/10.1007/978-1-349-11334-7.

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Douglas, J. F. Fluid mechanics. 3rd ed. Longman Scientific & Technical, 1995.

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Brewster, Hilary D. Fluid mechanics. Oxford Book Co., 2009.

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White, Frank M. Fluid mechanics. 7th ed. McGraw Hill, 2011.

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Book chapters on the topic "PDEs in fluid mechanics"

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Mkhatshwa, Musawenkhosi, Sandile Motsa, and Precious Sibanda. "Overlapping Multi-domain Bivariate Spectral Method for Systems of Nonlinear PDEs with Fluid Mechanics Applications." In Advances in Fluid Dynamics. Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-4308-1_54.

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Ranjan, Aditya, Vijay S. Duryodhan, and Nagesh D. Patil. "On the Replication of Human Skin Texture and Hydration on a PDMS-Based Artificial Human Skin Model." In Fluid Mechanics and Fluid Power, Volume 4. Springer Nature Singapore, 2024. http://dx.doi.org/10.1007/978-981-99-7177-0_58.

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Bresch, Didier, and Pierre-Emmanuel Jabin. "Global Weak Solutions of PDEs for Compressible Media: A Compactness Criterion to Cover New Physical Situations." In Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-52042-1_2.

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Ward, Michael J., and Mary-Catherine Kropinski. "Asymptotic Methods For PDE Problems In Fluid Mechanics and Related Systems With Strong Localized Perturbations In Two-Dimensional Domains." In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances. Springer Vienna, 2010. http://dx.doi.org/10.1007/978-3-7091-0408-8_2.

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Diening, Lars, Petteri Harjulehto, Peter Hästö, and Michael Růžička. "PDEs and Fluid Dynamics." In Lebesgue and Sobolev Spaces with Variable Exponents. Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-18363-8_14.

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Boffi, Daniele, Frédéric Hecht, and Olivier Pironneau. "Distributed Lagrange Multiplier for Fluid-Structure Interactions." In Numerical Methods for PDEs. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-94676-4_5.

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Larson, Mats G., and Fredrik Bengzon. "Fluid Mechanics." In Texts in Computational Science and Engineering. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-33287-6_12.

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Betounes, David. "Fluid Mechanics." In Partial Differential Equations for Computational Science. Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-2198-2_10.

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Lawson, Thomas B. "Fluid Mechanics." In Fundamentals of Aquacultural Engineering. Springer US, 1995. http://dx.doi.org/10.1007/978-1-4615-7047-9_6.

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Ng, Xian Wen. "Fluid Mechanics." In Engineering Problems for Undergraduate Students. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13856-1_5.

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Conference papers on the topic "PDEs in fluid mechanics"

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Akhtar, Imran, Jeff Borggaard, John A. Burns, and Lizette Zietsman. "Using Functional Gains for Optimal Sensor Placement in Fluid-Structure Interaction." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-13090.

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Functional gains are integral kernels of the standard feedback operator and are useful in control of partial differential equations (PDEs). These functional gains provide physical insight into how the control mechanism is operating. In some cases, these functional gains can provide information about the optimal placement of actuators and sensors. The study is motivated by fluid flow control and focuses on the computation of these functions. However, for practical purposes, one must be able to compute these functions for a wide variety of PDEs. For higher dimensional systems, computing these gains is at least as challenging as the original simulation problem. To reduce the complexity of the governing equations, reduced-order models are often developed by reducing the PDEs to ordinary-differential equations (ODEs). In this study, we use proper orthogonal decomposition (POD)-Galerkin based approach and develop a reduced-order model of a bluff body wake. We solve the incompressible Navier-Stokes equations, simulate the flow past a circular cylinder, and record the snapshots of the flow field. We compute the POD eigenfunctions and project the Navier-Stokes equations onto these few of these eigenfunctions to develop a reduced-order model. Later, we modify the model by introducing a control function simulating suction actuation on the cylinder surface. We linearize the model about the mean flow and apply feedback control to suppress vortex shedding. We then compute the functional gains for the applied control. We identify these gains at various stations in the wake region and suggest optimum locations for the sensors.

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Polly, James B., and J. M. McDonough. "Application of the Poor Man’s Navier–Stokes Equations to Real-Time Control of Fluid Flow." In ASME 2011 International Mechanical Engineering Congress and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/imece2011-63564.

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Control of fluid flow is an important, and quite underutilized process possessing significant potential benefits ranging from avoidance of separation and stall on aircraft wings and reduction of friction factors in oil and gas pipelines to mitigation of noise from wind turbines. But the Navier–Stokes (N.–S.) equations governing fluid flow consist of a system of time-dependent, multi-dimensional, non-linear partial differential equations (PDEs) which cannot be solved in real time using current, or near-term foreseeable, computing hardware. The poor man’s Navier–Stokes (PMNS) equations comprise a discrete dynamical system that is algebraic—hence, easily (and rapidly) solved—and yet which retains many (possibly all) of the temporal behaviors of the full (PDE) N.–S. system at specific spatial locations. In this paper we outline derivation of these equations and present a short discussion of their basic properties. We then consider application of these equations to the problem of control by adding a control force. We examine the range of PMNS equation behaviors that can be achieved by changing values of this control force, and, in particular, consider controllability of this (non-linear) system via numerical experiments. Moreover, we observe that the derivation leading to the PMNS equations is very general, and, at least in principle, it can be applied to a wide variety of problems governed by PDEs and (possibly) time-delay ordinary differential equations such as, for example, models of machining processes.

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Debnath, Pinku, and K. M. Pandey. "Performance Investigation on Single Phase Pulse Detonation Engine Using Computational Fluid Dynamics." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-66274.

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Pulse detonation engines (PDEs) are new concept propulsion technologies and unsteady propulsion system that operates cyclically and typically consists of four stages, filling of fuel/air mixture, combustion, blow down and purging. Out of these four processes, combustion is the most crucial one since it produces reliable and repeatable detonation wave. Detonation is a supersonic combustion process which is essentially a shock front driven by the energy release from the reaction zone in the flow right behind it. It is based on supersonic mode of combustion and causes rapid burning of a fuel-air mixture, typically tens of thousands of times faster than in a flame, that utilize repetitive detonations to produce thrust or power. PDE offers the potential to provide increased performance while simultaneously reducing engine weight, cost, and complexity relative to conventional propulsion systems currently in service. It has the potential to drastically reduce the cost of orbit transfer vehicle system as well as space vehicle attitude control system and can be used for wide range of military, civil and commercial applications. Due to its obvious advantages, worldwide attention has been paid to the scientific and technical issues concerning PDE. The present study deals with the convergence and divergence nozzle effects on specific thrust and pressure of Pulse Detonation Engine (PDE) using computational fluid dynamics (CFD). Pulse Detonation Engine having 88.3cm length and 9.5cm diameter combustion chamber, convergence nozzle, detonation tube and divergence nozzle were design in Gambit 2.3.16. FLUENT 6.3 predict the flow physics of pressure and specific thrust (Fs), increase in divergence nozzle compared to convergence nozzle and specific thrust of detonation tube was changed with the change of flight Mach number. A three dimension computational unstructured grid was developed which gives the best meshing accuracy as well as computational results. RNG k-ε turbulence model was used for the mass flow rate, pressure and velocity contours analysis with standard wall function.

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Mathur, Sanjay R., Aarti Chigullapalli, and Jayathi Y. Murthy. "A Unified Unintrusive Discrete Approach to Sensitivity Analysis and Uncertainty Propagation in Fluid Flow Simulations." In ASME 2010 International Mechanical Engineering Congress and Exposition. ASMEDC, 2010. http://dx.doi.org/10.1115/imece2010-37789.

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In recent years, there has been growing interest in making computational fluid dynamics (CFD) predictions with quantifiable uncertainty. Tangent-mode sensitivity analysis and uncertainty propagation are integral components of the uncertainty quantification process. Generalized polynomial chaos (gPC) is a viable candidate for uncertainty propagation, and involves representing the dependant variables in the governing partial differential equations (pdes) as expansions in an orthogonal polynomial basis in the random variables. Deterministic coupled non-linear pdes are derived for the coefficients of the expansion, which are then solved using standard techniques. A significant drawback of this approach is its intrusiveness. In this paper, we develop a unified approach to automatic code differentiation and Galerkin-based gPC in a new finite volume solver, MEMOSA-FVM, written in C++. We exploit templating and operator overloading to perform standard mathematical operations, which are overloaded either to perform code differentiation or to address operations on polynomial expansions. The resulting solver is capable of either performing sensitivity or uncertainty propagation, with the choice being made at compile time. It is easy to read, looks like a deterministic CFD code, and can address new classes of physics automatically, without extensive re-implementation of either sensitivity or gPC equations. We perform tangent (forward) mode sensitivity analysis and Galerkin gPC-based uncertainty propagation in a variety of problems, and demonstrate the effectiveness of this approach.

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Cui, X. "Solving Coupled Partial Differential Equations in Porous/Fractured Geomaterials." In 58th U.S. Rock Mechanics/Geomechanics Symposium. ARMA, 2024. http://dx.doi.org/10.56952/arma-2024-0836.

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ABSTRACT: Exploring efficient and robust algorithms to solve the simultaneous Partial Differential Equations (PDEs) is essential to model the prevalent multiphysics processes in deep rock engineering activities, such as the thermo-hydro-mechanical coupling in nuclear waste disposal and geothermal exploitation. In this study, the staggered and monolithic solution schemes are developed in the context of poroelastic and fractured geomaterials, and the applicability of the two solution schemes is analyzed in detail. It is found that the degree of coupling between primary variables plays a pivotal role in determining the performance of the staggered and monolithic solution schemes. Two-way coupling can only be tackled by the monolithic solution scheme, while the staggered solution scheme is very robust and efficient to deal with one-way coupling. This study provides an overarching principle to solve simultaneous PDEs: Understand the degree of coupling between primary variables, and use the staggered and monolithic solution schemes to address one-way and two-way couplings, respectively. 1. INTRODUCTION Rock mechanics, as an applied geoscience, has its roots deeply entangled with various rock engineering activities. The pioneers and early generations of rock engineering practitioners established the framework and theory of rock mechanics mainly from on/near ground surface projects, such as rock foundations, rock slopes, and shallowly buried tunnels. In the last two decades, rock engineering activities started to go deeper and deeper to serve increasingly complex engineering purposes. Examples include nuclear waste disposal, hydraulic fracturing, geothermal exploitation, CO2 sequestration, and injection-induced earthquakes. The above new rock engineering activities pose great challenges to the existing rock mechanics theory, but at the same time, they provide a fertile ground for rock mechanics to extend and prosper. Future rock mechanics will interact more closely with computer science, seismology, petroleum engineering and many other disciplines to further highlight its interdisciplinary attributes. A common feature of rock engineering at a great depth is the potential coexistence of in-situ stress, high temperature, fluid pressure in pores/fractures, and chemical reactions. These factors are coupled together, and each of them plays a considerable role in the entire system. Therefore, deep rock engineering is a typical multiphysics system. It is not trivial to replicate such a complex working environment in the laboratory, neither can the field tests be readily conducted at such a depth. Numerical simulation, however, provides an insightful, economical and effective approach to understand the multiphysics system.

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Fopah Lele, Armand, Fréderic Kuznik, Holger Urs Rammelberg, Thomas Schmidt, and Wolfgang K. L. Ruck. "Modeling Approach of Thermal Decomposition of Salt-Hydrates for Heat Storage Systems." In ASME 2013 Heat Transfer Summer Conference collocated with the ASME 2013 7th International Conference on Energy Sustainability and the ASME 2013 11th International Conference on Fuel Cell Science, Engineering and Technology. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/ht2013-17022.

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Heat storage systems using reversible chemical solid-fluid reactions to store and release thermal energy operates in charging and discharging phases. During last three decades, discussions on thermal decomposition of several salt-hydrates were done (experimentally and numerically) [1,2]. A mathematical model of heat and mass transfer in fixed bed reactor for heat storage is proposed based on a set of partial differential equations (PDEs). Beside the physical phenomena, the chemical reaction is considered via the balances or conservations of mass, extent conversion and energy in the reactor. These PDEs are numerically solved by means of the finite element method using Comsol Multiphysics 4.3a. The objective of this paper is to describe an adaptive modeling approach and establish a correct set of PDEs describing the physical system and appropriate parameters for simulating the thermal decomposition process. In this paper, kinetic behavior as stated by the ICTAC committee [3] to understand transport phenomena and reactions mechanism in gas and solid phases is taking into account using the generalized Prout-Tompkins equation with modifications based on thermal analysis experiments. The model is then applied to two thermochemical materials CaCl2 and MgCl2 with experimental activation energies and a comparison is made with TGA-DSC measurement results.

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Müftü, Sinan. "Numerical Solution of the Equations Governing the Steady State of a Thin Cylindrical Web Supported by an Air Cushion." In ASME 1999 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/imece1999-0225.

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Abstract The numerical method used to solve the coupled nonlinear, partial differential equations (PDEs) representing the interaction between a thin, flexible, cylindrical web and an air cushion at steady state is analyzed. The web deflections are modeled by a cylindrical shell theory that allows moderately large deflections. The airflow is modeled in two-dimensions with a modified form of the Navier-Stokes and mass balance equations that have non-linear source terms. The coupled fluid/structure system is solved numerically in a stacked iteration scheme: The fluid equations are solved using pseudo-compressibility method with artificial viscosity; and the web equations are solved with a modified Newton-Raphson method. The convergence characteristics of the coupled system and the effects of the numerical parameters on the steady state solution are studied by numerical experiments.

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Alexanderian, Alen, William Reese, Ralph C. Smith, and Meilin Yu. "Efficient Uncertainty Quantification for Biotransport in Tumors With Uncertain Material Properties." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-86216.

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We consider modeling of single phase fluid flow in heterogeneous porous media governed by elliptic partial differential equations (PDEs) with random field coefficients. Our target application is biotransport in tumors with uncertain heterogeneous material properties. We numerically explore dimension reduction of the input parameter and model output. In the present work, the permeability field is modeled as a log-Gaussian random field, and its covariance function is specified. Uncertainties in permeability are then propagated into the pressure field through the elliptic PDE governing porous media flow. The covariance matrix of pressure is constructed via Monte Carlo sampling. The truncated Karhunen–Loève (KL) expansion technique is used to decompose the log-permeability field, as well as the random pressure field resulting from random permeability. We find that although very high-dimensional representation is needed to recover the permeability field when the correlation length is small, the pressure field is not sensitive to high-oder KL terms of input parameter, and itself can be modeled using a low-dimensional model. Thus a low-rank representation of the pressure field in a low-dimensional parameter space is constructed using the truncated KL expansion technique.

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Graber, Benjamin D., Athanasios P. Iliopoulos, John G. Michopoulos, John C. Steuben, Andrew J. Birnbaum, and Nicole A. Apetre. "Towards a Computational Framework for Hypervelocity-Induced Atmospheric Plasma Modeling." In ASME 2024 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2024. http://dx.doi.org/10.1115/detc2024-143763.

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Abstract The work presented in this paper estimates the spectral radiance emitted from plasma induced by the interaction of hypervelocity moving structural/material systems with the atmosphere. The motivation for this effort originates from the need to compute the radiative heat fluxes imparted to hypersonic vehicles to facilitate their design, control, and maintenance. In response to this need, a computational framework was established to predict the fluid dynamics fields around a hypervelocity vehicle that in turn is coupled with the plasma physics that enables the calculation of the plasma fields and species dynamics. This framework implements a one-way coupling between fluid dynamics and plasma physics models. The framework solves the fluid dynamics partial differential equations representing the conservation of mass, momentum, and energy. The computed pressure, velocity, and temperature fields are subsequently utilized to drive the plasma physics PDEs describing the mass transport and energetics of all the nitrogen-oxygen 11 species present according to the so-called Dunn plasma model. An application of this framework for a spherical body for a wide range of velocities is presented as a verification of the framework’s functionality. Typical distributions of the fluid and plasma dynamics are presented. Finally, the plasma radiance spectra are produced by employing statistical mechanics principles.

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Li, Guangfa, Yanglong Lu, and Dehao Liu. "Physics-Constrained Convolutional Recurrent Neural Networks for Solving Spatial-Temporal PDEs With Arbitrary Boundary Conditions." In ASME 2024 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2024. http://dx.doi.org/10.1115/detc2024-134569.

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Abstract The inception of physics-constrained or physics-informed machine learning represents a paradigm shift, addressing the challenges associated with data scarcity and enhancing model interpretability. This innovative approach incorporates the fundamental laws of physics as constraints, guiding the training process of machine learning models. In this work, the physics-constrained convolutional recurrent neural network is further extended for solving spatial-temporal partial differential equations with arbitrary boundary conditions. Two notable advancements are introduced: the implementation of boundary conditions as soft constraints through finite difference-based differentiation, and the establishment of an adaptive weighting mechanism for the optimal allocation of weights to various losses. These enhancements significantly augment the network’s ability to manage intricate boundary conditions and expedite the training process. The efficacy of the proposed model is validated through its application to two-dimensional problems in heat transfer, phase transition, and fluid dynamics, which are pivotal in materials modeling. Compared to traditional physics-constrained neural networks, the physics-constrained convolutional recurrent neural network demonstrates a tenfold increase in prediction accuracy within a similar computational budget. Moreover, the model’s exceptional performance in extrapolating solutions for the Burgers’ equation underscores its utility. Consequently, this research establishes the physics-constrained recurrent neural network as a viable surrogate model for sophisticated spatiotemporal PDE systems, particularly beneficial in scenarios plagued by sparse and noisy datasets.

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Reports on the topic "PDEs in fluid mechanics"

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Monin, A. S., and A. M. Yaglom. Statistical Fluid Mechanics: The Mechanics of Turbulence. Defense Technical Information Center, 1999. http://dx.doi.org/10.21236/ada398728.

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Puterbaugh, Steven L., David Car, and S. Todd Bailie. Turbomachinery Fluid Mechanics and Control. Defense Technical Information Center, 2010. http://dx.doi.org/10.21236/ada514567.

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Martinez-Sanchez, Manuel. Physical Fluid Mechanics in MPD Thrusters. Defense Technical Information Center, 1987. http://dx.doi.org/10.21236/ada190309.

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Anderson, D. M., G. B. McFadden, and A. A. Wheeler. Diffuse-interface methods in fluid mechanics. National Institute of Standards and Technology, 1997. http://dx.doi.org/10.6028/nist.ir.6018.

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Car, David, and Steven L. Puterbaugh. Fluid Mechanics of Compression System Flow Control. Defense Technical Information Center, 2005. http://dx.doi.org/10.21236/ada444617.

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Bdzil, John Bohdan. Fluid Mechanics of an Obliquely Mounted MIV Gauge. Office of Scientific and Technical Information (OSTI), 2018. http://dx.doi.org/10.2172/1429987.

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Lipfert, F., M. Daum, G. Hendrey, and K. Lewin. Fluid mechanics and spatial performance of face arrays. Office of Scientific and Technical Information (OSTI), 1989. http://dx.doi.org/10.2172/5292902.

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Seume, J., G. Friedman, and T. W. Simon. Fluid mechanics experiments in oscillatory flow. Volume 1. Office of Scientific and Technical Information (OSTI), 1992. http://dx.doi.org/10.2172/10181069.

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Homsy, George M. Fundamental Studies of Fluid Mechanics: Stability in Porous Media. Office of Scientific and Technical Information (OSTI), 2014. http://dx.doi.org/10.2172/1120125.

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Keller, H. B., and P. G. Saffman. Analysis, scientific computing and fundamental studies in fluid mechanics. Office of Scientific and Technical Information (OSTI), 1991. http://dx.doi.org/10.2172/5025553.

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