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Journal articles on the topic 'Fluid dynamical problems'
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1
Nobumasa, Sugimoto. "IL12 THERMOACOUSTIC INSTABILITY AND ITS RELATED FLUID DYNAMICAL PROBLEMS." Proceedings of the International Conference on Jets, Wakes and Separated Flows (ICJWSF) 2013.4 (2013): _IL12–1_—_IL12–12_. http://dx.doi.org/10.1299/jsmeicjwsf.2013.4._il12-1_.
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Lim, H. A. "Lattice-gas automaton simulations of simple fluid dynamical problems." Mathematical and Computer Modelling 14 (1990): 720–27. http://dx.doi.org/10.1016/0895-7177(90)90276-s.
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Zeytounian, R. Kh. "Well-posedness of problems in fluid dynamics (a fluid-dynamical point of view)." Russian Mathematical Surveys 54, no. 3 (June 30, 1999): 479–564. http://dx.doi.org/10.1070/rm1999v054n03abeh000152.
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Koumboulis, F. N., M. G. Skarpetis, and B. G. Mertzios. "Numerical integration of fluid dynamics problems by discrete dynamical systems." Chaos, Solitons & Fractals 11, no. 1-3 (January 2000): 193–206. http://dx.doi.org/10.1016/s0960-0779(98)00284-7.
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Zarnescu, Arghir. "Mathematical problems of nematic liquid crystals: between dynamical and stationary problems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, no. 2201 (May 24, 2021): 20200432. http://dx.doi.org/10.1098/rsta.2020.0432.
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Mathematical studies of nematic liquid crystals address in general two rather different perspectives: that of fluid mechanics and that of calculus of variations. The former focuses on dynamical problems while the latter focuses on stationary ones. The two are usually studied with different mathematical tools and address different questions. The aim of this brief review is to give the practitioners in each area an introduction to some of the results and problems in the other area. Also, aiming to bridge the gap between the two communities, we will present a couple of research topics that generate natural connections between the two areas. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.
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Rahman, Aminur, and Denis Blackmore. "Walking droplets through the lens of dynamical systems." Modern Physics Letters B 34, no. 34 (November 9, 2020): 2030009. http://dx.doi.org/10.1142/s0217984920300094.
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Over the past decade the study of fluidic droplets bouncing and skipping (or “walking”) on a vibrating fluid bath has gone from an interesting experiment to a vibrant research field. The field exhibits challenging fluids problems, potential connections with quantum mechanics, and complex nonlinear dynamics. We detail advancements in the field of walking droplets through the lens of Dynamical Systems Theory, and outline questions that can be answered using dynamical systems analysis. The paper begins by discussing the history of the fluidic experiments and their resemblance to quantum experiments. With this physics backdrop, we paint a portrait of the complex nonlinear dynamics present in physical models of various walking droplet systems. Naturally, these investigations lead to even more questions, and some unsolved problems that are bound to benefit from rigorous Dynamical Systems Analysis are outlined.
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Geiser, Jürgen. "Modelling and analysis of multiscale systems related to fluid dynamical problems." Mathematical and Computer Modelling of Dynamical Systems 24, no. 4 (July 4, 2018): 315–18. http://dx.doi.org/10.1080/13873954.2018.1488743.
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Wang, Hao Cheng, and Jian Liu. "On Dynamical Simulations in Abrasive Flow Finishing." Advanced Materials Research 320 (August 2011): 75–80. http://dx.doi.org/10.4028/www.scientific.net/amr.320.75.
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In this paper, we point out several problems on fluid mechanics existing in the abrasive flow finishing, and study the dynamic simulations methods in the area. A case study is conducted on the process of free abrasive flow finishing, where we complete the dynamic simulations on the kinematic characteristics by a model of two-phase fluid. It is shown that the theory of two-phase fluid can practically direct the design of polishing machine, and the selection as well as the optimization of parameters for polishing technique.
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Moon, F. C. "Nonlinear Dynamical Systems." Applied Mechanics Reviews 38, no. 10 (October 1, 1985): 1284–86. http://dx.doi.org/10.1115/1.3143693.
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New discoveries have been made recently about the nature of complex motions in nonlinear dynamics. These new concepts are changing many of the ideas about dynamical systems in physics and in particular fluid and solid mechanics. One new phenomenon is the apparently random or chaotic output of deterministic systems with no random inputs. Another is the sensitivity of the long time dynamic history of many systems to initial starting conditions even when the motion is not chaotic. New mathematical ideas to describe this phenomenon are entering the field of nonlinear vibrations and include ideas from topology and analysis such as Poincare´ maps, fractal dimensions, Cantor sets and strange attractors. These new ideas are already making their way into the engineering vibrations laboratory. Further research in this field is needed to extend these new ideas to multi-degree of freedom and continuum vibration problems. Also the loss of predictability in certain nonlinear problems should be studied for its impact on the field of numerical simulation in mechanics of nonlinear materials and structures.
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Salmon, John K., and Michael S. Warren. "Fast Parallel Tree Codes for Gravitational and Fluid Dynamical N-Body Problems." International Journal of Supercomputer Applications and High Performance Computing 8, no. 2 (June 1994): 129–42. http://dx.doi.org/10.1177/109434209400800205.
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Dijkstra, Henk A., Fred W. Wubs, Andrew K. Cliffe, Eusebius Doedel, Ioana F. Dragomirescu, Bruno Eckhardt, Alexander Yu Gelfgat, et al. "Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation." Communications in Computational Physics 15, no. 1 (January 2014): 1–45. http://dx.doi.org/10.4208/cicp.240912.180613a.
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AbstractWe provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as ‘tipping points’, is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods arementioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.
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Bellomo, N., P. LeTallec, and B. Perthame. "Nonlinear Boltzmann Equation Solutions and Applications to Fluid Dynamics." Applied Mechanics Reviews 48, no. 12 (December 1, 1995): 777–94. http://dx.doi.org/10.1115/1.3005093.
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This paper provides a review of the mathematical results on the solution of the nonlinear Boltzmann equation. The survey deals both with analytical and computational aspects: Mathematical formulation of problems, initial and/or boundary value problems, a survey of the qualitative analysis of solutions, and computational treatment of fluid dynamical problems. A discussion of some problems deserving further study concludes this work.
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SCHMID, PETER J. "Dynamic mode decomposition of numerical and experimental data." Journal of Fluid Mechanics 656 (July 1, 2010): 5–28. http://dx.doi.org/10.1017/s0022112010001217.
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The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders.
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Hasslacher, Brosl, and David A. Meyer. "Modeling Dynamical Geometry with Lattice-Gas Automata." International Journal of Modern Physics C 09, no. 08 (December 1998): 1597–605. http://dx.doi.org/10.1142/s0129183198001448.
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Conventional lattice-gas automata consist of particles moving discretely on a fixed lattice. While such models have been quite successful for a variety of fluid flow problems, there are other systems, e.g., flow in a flexible membrane or chemical self-assembly, in which the geometry is dynamical and coupled to the particle flow. Systems of this type seem to call for lattice gas models with dynamical geometry. We construct such a model on one-dimensional (periodic) lattices and describe some simulations illustrating its nonequilibrium dynamics.
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VLADIMIROV, V. A., and K. I. ILIN. "On the stability of the dynamical system ‘rigid body + inviscid fluid’." Journal of Fluid Mechanics 386 (May 10, 1999): 43–75. http://dx.doi.org/10.1017/s0022112099004267.
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In this paper we study a dynamical system consisting of a rigid body and an inviscid incompressible fluid. Two general configurations of the system are considered: (a) a rigid body with a cavity completely filled with a fluid and (b) a rigid body surrounded by a fluid. In the first case the fluid is confined to an interior (for the body) domain and in the second case it occupies an exterior domain, which may, in turn, be bounded by some fixed rigid boundary or may extend to infinity. The aim of the paper is twofold: (i) to develop Arnold's technique for the system ‘body + fluid’ and (ii) to obtain sufficient conditions for the stability of steady states of the system. We first establish an energy-type variational principle for an arbitrary steady state of the system. Then we generalize this principle for states that are steady either in translationally moving in some fixed direction or rotating around some fixed axis coordinate system. The second variations of the corresponding functionals are calculated. The general results are applied to a number of particular stability problems. The first is the stability of a steady translational motion of a two-dimensional body in an irrotational flow. Here we have found that (for a quite wide class of bodies) the presence of non-zero circulation about the body does not affect its stability – a result that seems to be new. The second problem concerns the stability of a steady rotation of a force-free rigid body with a cavity containing an ideal fluid. Here we rediscover the stability criterion of Rumyantsev (see Moiseev & Rumyantsev 1965). The complementary problem – when a body is surrounded by a fluid and both body and fluid rotate with constant angular velocity around a fixed axis passing through the centre of mass of the body – is also considered and the corresponding sufficient conditions for stability are obtained.
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Shao, Mingyu, Chicheng Ma, Shuaizhao Hu, Chuansong Sun, and Dong Jing. "Effects of Time-Varying Fluid on Dynamical Characteristics of Cantilever Beams: Numerical Simulations and Experimental Measurements." Mathematical Problems in Engineering 2020 (December 21, 2020): 1–18. http://dx.doi.org/10.1155/2020/6679443.
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In order to obtain the effects of time-varying fluid on dynamical characteristics of cantilever beams, this paper gives a comprehensive study of cantilever beams vibrating in a fluid with variable depth. The mathematical model of the cantilever beams in time-varying fluid is derived by combining Euler–Bernoulli beam theory and velocity potential theory, and the influence of the time-varying fluid is discussed. Then, a two-way fluid-structure interaction (FSI) numerical simulation procedure is proposed to calculate the transient responses of the beam. The validity and accuracy are verified according to the comparison among theoretical analysis, numerical simulations, and experimental measurements. Results show that, besides the added mass effect, a damping-like term is also induced due to the motion of the fluid, which is proportional to the moving velocity of the fluid. Both the added mass and the added damping increase with the increment of the width of the beam. The surrounding fluid near the free end affects the beam more significantly. As a negative damping is caused while the fluid decreases, resulting in a much slower decay of the time responses. Therefore, the added damping should not be neglected in the analysis of the FSI problems with time-varying fluid.
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Griffith, Boyce E., and Sookkyung Lim. "Simulating an Elastic Ring with Bend and Twist by an Adaptive Generalized Immersed Boundary Method." Communications in Computational Physics 12, no. 2 (August 2012): 433–61. http://dx.doi.org/10.4208/cicp.190211.060811s.
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AbstractMany problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary (IB) method. In the IB approach to such problems, the elastic forces generated by the immersed structure are applied to the surrounding fluid, and the motion of the immersed structure is determined by the local motion of the fluid. Recently, the IB method has been extended to treat more general elasticity models that include both positional and rotational degrees of freedom. For such models, force and torque must both be applied to the fluid. The positional degrees of freedom of the immersed structure move according to the local linear velocity of the fluid, whereas the rotational degrees of freedom move according to the local angular velocity. This paper introduces a spatially adaptive, formally second-order accurate version of this generalized immersed boundary method. We use this adaptive scheme to simulate the dynamics of an elastic ring immersed in fluid. To describe the elasticity of the ring, we use an unconstrained version of Kirchhoff rod theory. We demonstrate empirically that our numerical scheme yields essentially second-order convergence rates when applied to such problems. We also study dynamical instabilities of such fluid-structure systems, and we compare numerical results produced by our method to classical analytic results from elastic rod theory.
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Cheemaa, N., S. Chen, and A. R. Seadawy. "Chiral soliton solutions of perturbed chiral nonlinear Schrödinger equation with its applications in mathematical physics." International Journal of Modern Physics B 34, no. 31 (November 26, 2020): 2050301. http://dx.doi.org/10.1142/s0217979220503014.
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In this article, we have discussed the analytical treatment of perturbed chiral nonlinear Schrödinger equation with the help of our newly developed method extended modified auxiliary equation mapping method (EMAEMM). By using this newly proposed technique we have found some quite general and new variety of exact traveling wave solutions, which are collecting some kind of semi half bright, dark, bright, semi half dark, doubly periodic, combined, periodic, half hark, and half bright via three parametric values, which is the primary key point of difference of our technique. These results are highly applicable to develop new theories of quantum mechanics, biomedical problems, soliton dynamics, plasma physics, nuclear physics, optical physics, fluid dynamics, biomedical problems, electromagnetism, industrial studies, mathematical physics, and in many other natural and physical sciences. For detailed physical dynamical representation of our results we have shown them with graphs in different dimensions using Mathematica 10.4 to get complete understanding in a more efficient manner to observe the behavior of different new dynamical shapes of solutions.
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Goldstein, Raymond E. "Batchelor Prize Lecture Fluid dynamics at the scale of the cell." Journal of Fluid Mechanics 807 (October 17, 2016): 1–39. http://dx.doi.org/10.1017/jfm.2016.586.
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The world of cellular biology provides us with many fascinating fluid dynamical phenomena that lie at the heart of physiology, development, evolution and ecology. Advances in imaging, micromanipulation and microfluidics over the past decade have made possible high-precision measurements of such flows, providing tests of microhydrodynamic theories and revealing a wealth of new phenomena calling out for explanation. Here I summarize progress in four areas within the field of ‘active matter’: cytoplasmic streaming in plant cells, synchronization of eukaryotic flagella, interactions between swimming cells and surfaces and collective behaviour in suspensions of microswimmers. Throughout, I emphasize open problems in which fluid dynamical methods are key ingredients in an interdisciplinary approach to the mysteries of life.
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HENNINGSON, DAN S. "Description of complex flow behaviour using global dynamic modes." Journal of Fluid Mechanics 656 (July 20, 2010): 1–4. http://dx.doi.org/10.1017/s0022112010002776.
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A novel method for performing spectral analysis of a fluid flow solely based on snapshot sequences from numerical simulations or experimental data is presented by Schmid (J. Fluid Mech., 2010, this issue, vol. 656, pp. 5–28). Dominant frequencies and wavenumbers are extracted together with dynamic modes which represent the associated flow structures. The mathematics underlying this decomposition is related to the Koopman operator which provides a linear representation of a nonlinear dynamical system. The procedure to calculate the spectra and dynamic modes is based on Krylov subspace methods; the dynamic modes reduce to global linear eigenmodes for linearized problems or to Fourier modes for (nonlinear) periodic problems. Schmid (2010) also generalizes the analysis to the propagation of flow variables in space which produces spatial growth rates with associated dynamic modes, and an application of the decomposition to subdomains of the flow region allows the extraction of localized stability information. For finite-amplitude flows this spectral analysis identifies relevant frequencies more effectively than global eigenvalue analysis and decouples frequency information more clearly than proper orthogonal decomposition.
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Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. "Discovering governing equations from data by sparse identification of nonlinear dynamical systems." Proceedings of the National Academy of Sciences 113, no. 15 (March 28, 2016): 3932–37. http://dx.doi.org/10.1073/pnas.1517384113.
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Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.
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Amabili, Marco, and Michael P. Paı¨doussis. "Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction." Applied Mechanics Reviews 56, no. 4 (July 1, 2003): 349–81. http://dx.doi.org/10.1115/1.1565084.
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This literature review focuses mainly on geometrically nonlinear (finite amplitude) free and forced vibrations of circular cylindrical shells and panels, with and without fluid-structure interaction. Work on shells and curved panels of different geometries is but briefly discussed. In addition, studies dealing with particular dynamical problems involving finite deformations, eg, dynamic buckling, stability, and flutter of shells coupled to flowing fluids, are also discussed. This review is structured as follows: after a short introduction on some of the fundamentals of geometrically nonlinear theory of shells, vibrations of shells and panels in vacuo are discussed. Free and forced vibrations under radial harmonic excitation (Section 2.2), parametric excitation (axial tension or compression and pressure-induced excitations) (Section 2.3), and response to radial transient loads (Section 2.4) are reviewed separately. Studies on shells and panels in contact with dense fluids (liquids) follow; some of these studies present very interesting results using methods also suitable for shells and panels in vacuo. Then, in Section 4, shells and panels in contact with light fluids (gases) are treated, including the problem of stability (divergence and flutter) of circular cylindrical panels and shells coupled to flowing fluid. For shells coupled to flowing fluid, only the case of axial flow is reviewed in this paper. Finally, papers dealing with experiments are reviewed in Section 5. There are 356 references cited in this article.
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Wang, Lihua, and Zheng Zhong. "Radial Basis Collocation Method for the Dynamics of Rotating Flexible Tube Conveying Fluid." International Journal of Applied Mechanics 07, no. 03 (June 2015): 1550045. http://dx.doi.org/10.1142/s1758825115500453.
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A Meshfree Radial Basis Collocation Method (RBCM) associated with explicit and implicit time integration scheme is formulated to study the coupling dynamics of a rotating flexible tube conveying fluid, which involves a partial differential equation (PDE) with variable coefficients. Dispersion studies are performed and they indicate that the proposed RBCM has a very small dispersion error compared with conventional FEM and Galerkin-based meshfree methods. Numerical examples are conducted for the influence of initial flow rate of the fluid, discretization and shape parameter on the dispersion error. The critical time step is obtained from a Von Neumann stability analysis. For the eigenproblem, Hermite-type RBCM is proposed in order to construct square matrices and eigenvalue analysis gives the frequencies of the system. Subsequently, the influence of angular velocity, flow rate of the fluid and the time variation on the fundamental frequencies is studied. Though proposed for studying the dynamics of a rotating flexible tube conveying fluid, this solution scheme is applicable to other dynamical problems which have similar PDEs with variable coefficients.
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D'yachenko, A. I., and G. A. Lyubimov. "System of equations for describing dynamical problems associated with the mechanics of lung parenchyma." Fluid Dynamics 23, no. 3 (1988): 340–47. http://dx.doi.org/10.1007/bf01054738.
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Xie, Xuping, Guannan Zhang, and Clayton G. Webster. "Non-Intrusive Inference Reduced Order Model for Fluids Using Deep Multistep Neural Network." Mathematics 7, no. 8 (August 19, 2019): 757. http://dx.doi.org/10.3390/math7080757.
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In this effort we propose a data-driven learning framework for reduced order modeling of fluid dynamics. Designing accurate and efficient reduced order models for nonlinear fluid dynamic problems is challenging for many practical engineering applications. Classical projection-based model reduction methods generate reduced systems by projecting full-order differential operators into low-dimensional subspaces. However, these techniques usually lead to severe instabilities in the presence of highly nonlinear dynamics, which dramatically deteriorates the accuracy of the reduced-order models. In contrast, our new framework exploits linear multistep networks, based on implicit Adams–Moulton schemes, to construct the reduced system. The advantage is that the method optimally approximates the full order model in the low-dimensional space with a given supervised learning task. Moreover, our approach is non-intrusive, such that it can be applied to other complex nonlinear dynamical systems with sophisticated legacy codes. We demonstrate the performance of our method through the numerical simulation of a two-dimensional flow past a circular cylinder with Reynolds number Re = 100. The results reveal that the new data-driven model is significantly more accurate than standard projection-based approaches.
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Drapaca, C. S., S. Sivaloganathan, G. Tenti, and J. M. Drake. "Dynamical Morphology of the Brain's Ventricular Cavities in Hydrocephalus." Journal of Theoretical Medicine 6, no. 3 (2005): 151–60. http://dx.doi.org/10.1080/10273660500143631.
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Although interest in the biomechanics of the brain goes back over centuries, mathematical models of hydrocephalus and other brain abnormalities are still in their infancy and a much more recent phenomenon. This is rather surprising, since hydrocephalus is still an endemic condition in the pediatric population with an incidence of approximately 1–3 per 1000 births. Treatment has dramatically improved over the last three decades, thanks to the introduction of cerebrospinal fluid (CSF) shunts. Their use, however, is not without problems and the shunt failure at two years remains unacceptably high at 50%. The most common factor causing shunt failure is obstruction, especially of the proximal catheters. There is currently no agreement among neurosurgeons as to the optimal catheter tip position; however, common sense suggests that the lowest risk location is the place that remains larger after ventricular decompression drainage. Thus, success in this direction will depend on the development of a quantitative theory capable of predicting the ultimate shape of the ventricular wall. In this paper, we report on some recent progress towards the solution to this problem.
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Keskinen, R. P. "Transient Hydroelastic Vibration of Piping With Local Nonlinearities." Journal of Pressure Vessel Technology 107, no. 4 (November 1, 1985): 350–55. http://dx.doi.org/10.1115/1.3264463.
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A mode superposition algorithm is presented to solve fluid and structural dynamics problems in piping systems with a local cross-sectional material nonlinearity, such as cavitation of fluid or circumferential cracking of the pipe material. Two families of eigenmodes are used to decompose the total response into so-called compatibility-controlling and resistance-controlling responses which satisfy the governing partial differential equations. The responses are simultaneously solved in time by means of convolution integral techniques. Either response is always predicting for the other an additional excitation—load or displacement—at the nonlinear cross section in such a manner that the resulting boundary conditions fully simulate the inelastic material behavior. The algorithm is applied to a test problem of waterhammer-induced coupled acoustic and mechanical piping vibrations with cavitating fluid. The coupling is shown to reduce the dynamical loading of the pipe and to eliminate unrealistic beating of closely spaced acoustic and mechanical eigenmodes appearing in uncoupled analysis.
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Paidoussis, Michael P. "Flow-induced Instabilities of Cylindrical Structures." Applied Mechanics Reviews 40, no. 2 (February 1, 1987): 163–75. http://dx.doi.org/10.1115/1.3149530.
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A kaleidoscopic view of the many diverse and interesting instabilities are presented, to which cylindrical structures are susceptible when in contact with flowing fluids. The physical mechanisms involved are discussed in each case, to the extent that they are understood, and the degree of success of available mathematical models is assessed. Four classes of problems are dealt with, according to the disposition of the flow vis-a`-vis the cylindrical structures: (a) instabilities induced by internal flows in tubular structures; (b) instabilities of solitary or clustered cylinders due to external axial flow; (c) annular-flow-induced instabilities of coaxial beams and shells; (d) instabilities of arrays of cylinders subject to cross-flow. In the first class of problems, the stability of straight tubular beams and cylindrical shells conveying fluid is discussed first, followed by the stability of curved pipes containing flow. In the second class of problems, the instabilities of solitary and clustered cylinders subjected to an external axial flow are treated, and their dynamical behavior is compared to that of systems with internal flow. The third class of problems involves annular flow in coaxial systems of beams and/or shells. Cross-flow-induced instabilities of clustered cylinders, in the form of arrays of different geometrical patterns, are the last class of problems considered; they are fundamentally distinct from the foregoing in terms of the fluid mechanics of the problem, for in this case the flow field is not irrotational—not even approximately.
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Gomoyunov, M. I., and D. A. Serkov. "Non-anticipative strategies in guarantee optimization problems under functional constraints on disturbances." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 30, no. 4 (December 2020): 553–71. http://dx.doi.org/10.35634/vm200402.
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For a dynamical system controlled under conditions of disturbances, a problem of optimizing the guaranteed result is considered. A feature of the problem is the presence of functional constraints on disturbances, under which, in general, the set of admissible disturbances is not closed with respect to the operation of “gluing up” of two of its elements. This circumstance does not allow to apply directly the methods developed within the differential games theory for studying the problem and, thus, leads to the necessity of modifying them appropriately. The paper provides a new notion of a non-anticipative control strategy. It is proved that the corresponding functional of the optimal guaranteed result satisfies the dynamic programming principle. As a consequence, so-called properties of $u$- and $v$-stability of this functional are established, which may allow, in the future, to obtain a constructive solution of the problem in the form of feedback (positional) controls.
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Blizorukova, M. S., and V. I. Maksimov. "Reconstruction of the right-hand part of a distributed differential equation using a positional controlled model." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 30, no. 4 (December 2020): 533–52. http://dx.doi.org/10.35634/vm200401.
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In this paper, we consider the stable reconstruction problem of the unknown input of a distributed system of second order by results of inaccurate measurements of its solution. The content of the problem considered is as follows. We consider a distributed equation of second order. The solution of the equation depends on the input varying in the time. The input, as well as the solution, is not given in advance. At discrete times the solution of the equation is measured. These measurements are not accurate in general. It is required to design an algorithm for approximate reconstruction of the input that has dynamical and stability properties. The dynamical property means that the current values of approximations of the input are produced on-line, and the stability property means that the approximations are arbitrarily accurate for a sufficient accuracy of measurements. The problem refers to the class of inverse problems. The algorithm presented in the paper is based on the constructions of a stable dynamical inversion and on the combination of the methods of ill-posed problems and positional control theory.
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Godoy-Diana, R., and B. Thiria. "On the diverse roles of fluid dynamic drag in animal swimming and flying." Journal of The Royal Society Interface 15, no. 139 (February 2018): 20170715. http://dx.doi.org/10.1098/rsif.2017.0715.
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Questions of energy dissipation or friction appear immediately when addressing the problem of a body moving in a fluid. For the most simple problems, involving a constant steady propulsive force on the body, a straightforward relation can be established balancing this driving force with a skin friction or form drag, depending on the Reynolds number and body geometry. This elementary relation closes the full dynamical problem and sets, for instance, average cruising velocity or energy cost. In the case of finite-sized and time-deformable bodies though, such as flapping flyers or undulatory swimmers, the comprehension of driving/dissipation interactions is not straightforward. The intrinsic unsteadiness of the flapping and deforming animal bodies complicates the usual application of classical fluid dynamic forces balance. One of the complications is because the shape of the body is indeed changing in time, accelerating and decelerating perpetually, but also because the role of drag (more specifically the role of the local drag) has two different facets, contributing at the same time to global dissipation and to driving forces. This causes situations where a strong drag is not necessarily equivalent to inefficient systems. A lot of living systems are precisely using strong sources of drag to optimize their performance. In addition to revisiting classical results under the light of recent research on these questions, we discuss in this review the crucial role of drag from another point of view that concerns the fluid–structure interaction problem of animal locomotion. We consider, in particular, the dynamic subtleties brought by the quadratic drag that resists transverse motions of a flexible body or appendage performing complex kinematics, such as the phase dynamics of a flexible flapping wing, the propagative nature of the bending wave in undulatory swimmers, or the surprising relevance of drag-based resistive thrust in inertial swimmers.
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HOLMES, PHILIP. "NINETY PLUS THIRTY YEARS OF NONLINEAR DYNAMICS: LESS IS MORE AND MORE IS DIFFERENT." International Journal of Bifurcation and Chaos 15, no. 09 (September 2005): 2703–16. http://dx.doi.org/10.1142/s0218127405013678.
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I review the early (1885–1975) and more recent history of dynamical systems theory, identifying key principles and themes, including those of dimension reduction, normal form transformation and unfolding of degenerate cases. I end by briefly noting recent extensions and applications in nonlinear fluid and solid mechanics, with a nod toward mathematical biology. I argue throughout that this essentially mathematical theory was largely motivated by nonlinear scientific problems, and that after a long gestation it is propagating throughout the sciences and technology.
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Koshel, Konstantin, Eugene Ryzhov, and Xavier Carton. "Vortex Interactions Subjected to Deformation Flows: A Review." Fluids 4, no. 1 (January 18, 2019): 14. http://dx.doi.org/10.3390/fluids4010014.
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Deformation flows are the flows incorporating shear, strain and rotational components. These flows are ubiquitous in the geophysical flows, such as the ocean and atmosphere. They appear near almost any salience, such as isolated coherent structures (vortices and jets) and various fixed obstacles (submerged obstacles and continental boundaries). Fluid structures subject to such deformation flows may exhibit drastic changes in motion. In this review paper, we focus on the motion of a small number of coherent vortices embedded in deformation flows. Problems involving isolated one and two vortices are addressed. When considering a single-vortex problem, the main focus is on the evolution of the vortex boundary and its influence on the passive scalar motion. Two vortex problems are addressed with the use of point vortex models, and the resulting stirring patterns of neighbouring scalars are studied by a combination of numerical and analytical methods from the dynamical system theory. Many dynamical effects are reviewed with emphasis on the emergence of chaotic motion of the vortex phase trajectories and the scalars in their immediate vicinity.
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Davey, A., and H. Salwen. "On the stability of flow in an elliptic pipe which is nearly circular." Journal of Fluid Mechanics 281 (December 25, 1994): 357–69. http://dx.doi.org/10.1017/s0022112094003149.
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In an earlier paper (Davey 1978) the first author investigated the linear stability of flow in a straight pipe whose cross-section was an ellipse, of small ellipticity e, by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. That paper contained some serious errors which we correct herein. We show analytically that for the most important mode n = 1, for which the circular problem has a double eigenvalue c0 as the ‘swirl’ can be in either direction, the ellipticity splits the double eigenvalue into two separate eigenvalues c0 ± e2c12, to leading order, when the cross-sectional area of the pipe is kept fixed. The imaginary part of c12 is non-zero and so the ellipticity always makes the flow less stable. This specific problem is generic to a much wider class of fluid dynamical problems which are made less stable when the symmetry group of the dynamical system is reduced from S1 to Z2.In the Appendix, P. G. Drazin describes simply the qualitative structure of this problem, and other problems with the same symmetries, without technical detail.
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Gu, Yaqing, and Dean S. Oliver. "An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation." SPE Journal 12, no. 04 (November 1, 2007): 438–46. http://dx.doi.org/10.2118/108438-pa.
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Summary The dynamical equations for multiphase flow in porous media are highly nonlinear and the number of variables required to characterize the medium is usually large, often two or more variables per simulator gridblock. Neither the extended Kalman filter nor the ensemble Kalman filter is suitable for assimilating data or for characterizing uncertainty for this type of problem. Although the ensemble Kalman filter handles the nonlinear dynamics correctly during the forecast step, it sometimes fails badly in the analysis (or updating) of saturations. This paper focuses on the use of an iterative ensemble Kalman filter for data assimilation in nonlinear problems, especially of the type related to multiphase ow in porous media. Two issues are key:iteration to enforce constraints andensuring that the resulting ensemble is representative of the conditional pdf (i.e., that the uncertainty quantification is correct). The new algorithm is compared to the ensemble Kalman filter on several highly nonlinear example problems, and shown to be superior in the prediction of uncertainty. Introduction For linear problems, the Kalman filter is optimal for assimilating measurements to continuously update the estimate of state variables. Kalman filters have occasionally been applied to the problem of estimating values of petroleum reservoir variables (Eisenmann et al. 1994; Corser et al. 2000), but they are most appropriate when the problems are characterized by a small number of variables and when the variables to be estimated are linearly related to the observations. Most data assimilation problems in petroleum reservoir engineering are highly nonlinear and are characterized by many variables, often two or more variables per simulator gridblock. The problem of weather forecasting is in many respects similar to the problem of predicting future petroleum reservoir performance. The economic impact of inaccurate predictions is substantial in both cases, as is the difficulty of assimilating very large data sets and updating very large numerical models. One method that has been recently developed for assimilating data in weather forecasting is ensemble Kalman filtering (Evensen 1994; Houtekamer and Mitchell 1998; Anderson and Anderson 1999; Hamill et al. 2000; Houtekamer and Mitchell 2001; Evensen 2003). It has been demonstrated to be useful for weather prediction over the North Atlantic. The method is now beginning to be applied for data assimilation in groundwater hydrology (Reichle et al. 2002; Chen and Zhang 2006) and in petroleum engineering (Nævdal et al. 2002, 2005; Gu and Oliver 2005; Liu and Oliver 2005a; Wen and Chen 2006, 2007; Zafari and Reynolds 2007; Gao et al. 2006; Lorentzen et al. 2005; Skjervheim et al. 2007; Dong et al. 2006), but the applications to state variables whose density functions are bimodal has proved problematic (Gu and Oliver 2006). For applications to nonlinear assimilation problems, it is useful to think of the ensemble Kalman filter as a least squares method that obtains an averaged gradient for minimization not from a variational approach but from an empirical correlation between model variables (Anderson 2003; Zafari et al. 2006). In addition to providing a mean estimate of the variables, a Monte Carlo estimate of uncertainty can be obtained directly from the variability in the ensemble.
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Chinyoka, Tirivanhu, and Daniel Oluwole Makinde. "Unsteady and porous media flow of reactive non-Newtonian fluids subjected to buoyancy and suction/injection." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 7 (September 7, 2015): 1682–704. http://dx.doi.org/10.1108/hff-10-2014-0329.
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Purpose – The purpose of this paper is to examine the unsteady pressure-driven flow of a reactive third-grade non-Newtonian fluid in a channel filled with a porous medium. The flow is subjected to buoyancy, suction/injection asymmetrical and convective boundary conditions. Design/methodology/approach – The authors assume that exothermic chemical reactions take place within the flow system and that the asymmetric convective heat exchange with the ambient at the surfaces follow Newton’s law of cooling. The authors also assume unidirectional suction injection flow of uniform strength across the channel. The flow system is modeled via coupled non-linear partial differential equations derived from conservation laws of physics. The flow velocity and temperature are obtained by solving the governing equations numerically using semi-implicit finite difference methods. Findings – The authors present the results graphically and draw qualitative and quantitative observations and conclusions with respect to various parameters embedded in the problem. In particular the authors make observations regarding the effects of bouyancy, convective boundary conditions, suction/injection, non-Newtonian character and reaction strength on the flow velocity, temperature, wall shear stress and wall heat transfer. Originality/value – The combined fluid dynamical, porous media and heat transfer effects investigated in this paper have to the authors’ knowledge not been studied. Such fluid dynamical problems find important application in petroleum recovery.
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Wiggins, S., and A. M. Mancho. "Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori." Nonlinear Processes in Geophysics 21, no. 1 (February 4, 2014): 165–85. http://dx.doi.org/10.5194/npg-21-165-2014.
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Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.
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Li, Haijing, and Federico Toschi. "Plasma-induced catalysis: towards a numerical approach." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2175 (June 22, 2020): 20190396. http://dx.doi.org/10.1098/rsta.2019.0396.
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A lattice Boltzmann (LB) model is developed, validated and used to study simplified plasma/flow problems in complex geometries. This approach solves a combined set of equations, namely the Navier–Stokes equations for the momentum field, the advection–diffusion and the Nernst–Planck equations for electrokinetic and the Poisson equation for the electric field. This model allows us to study the dynamical interaction of the fluid/plasma density, velocity, concentration and electric field. In this work, we discuss several test cases for our numerical model and use it to study a simplified plasma fluid flowing and reacting inside a packed bed reactor. Inside the packed bed, electric breakdown reactions take place due to the electric field, making neutral species ionize. The presence of the packed beads can help enhance the reaction efficiency by locally increasing the electric field, and the size of packed beads and the pressure drop of the packed bed do influence the outflux. Hence trade-offs exist between reaction efficiency and packing porosity, the size of packing beads and the pressure drop of the packed bed. Our model may be used as a guidance to achieve higher reaction efficiencies by optimizing the relevant parameters. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.
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Grafke, Tobias, Rainer Grauer, and Stephan Schindel. "Efficient Computation of Instantons for Multi-Dimensional Turbulent Flows with Large Scale Forcing." Communications in Computational Physics 18, no. 3 (September 2015): 577–92. http://dx.doi.org/10.4208/cicp.031214.200415a.
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AbstractExtreme events play a crucial role in fluid turbulence. Inspired by methods from field theory, these extreme events, their evolution and probability can be computed with help of the instanton formalism as minimizers of a suitable action functional. Due to the high number of degrees of freedom in multi-dimensional fluid flows, traditional global minimization techniques quickly become prohibitive in their memory requirements. We outline a novel method for finding the minimizing trajectory in a wide class of problems that typically occurs in turbulence setups, where the underlying dynamical system is a non-gradient, non-linear partial differential equation, and the forcing is restricted to a limited length scale. We demonstrate the efficiency of the algorithm in terms of performance and memory by computing high resolution instanton field configurations corresponding to viscous shocks for 1D and 2D compressible flows.
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Momen, Mostafa, Zhong Zheng, Elie Bou-Zeid, and Howard A. Stone. "Inertial gravity currents produced by fluid drainage from an edge." Journal of Fluid Mechanics 827 (August 29, 2017): 640–63. http://dx.doi.org/10.1017/jfm.2017.480.
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We present theoretical, numerical and experimental studies of the release of a finite volume of fluid instantaneously from an edge of a rectangular domain for high Reynolds number flows. For the cases we considered, the results indicate that approximately half of the initial volume exits during an early adjustment period. Then, the inertial gravity current reaches a self-similar phase during which approximately 40 % of its volume drains and its height decreases as $\unicode[STIX]{x1D70F}^{-2}$, where $\unicode[STIX]{x1D70F}$ is a dimensionless time that is derived with the typical gravity wave speed and the horizontal length of the domain. Based on scaling arguments, we reduce the shallow-water partial differential equations into two nonlinear ordinary differential equations (representing the continuity and momentum equations), which are solved analytically by imposing a zero velocity boundary condition at the closed end wall and a critical Froude number condition at the open edge. The solutions are in good agreement with the performed experiments and direct numerical simulations for various geometries, densities and viscosities. This study provides new insights into the dynamical behaviour of a fluid draining from an edge in the inertial regime. The solutions may be useful for environmental, geophysical and engineering applications such as open channel flows, ventilations and dam-break problems.
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ROBINSON, JAMES C. "Parametrization of global attractors, experimental observations, and turbulence." Journal of Fluid Mechanics 578 (April 26, 2007): 495–507. http://dx.doi.org/10.1017/s0022112007005137.
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This paper is concerned with rigorous results in the theory of turbulence and fluid flow. While derived from the abstract theory of attractors in infinite-dimensional dynamical systems, they shed some light on the conventional heuristic theories of turbulence, and can be used to justify a well-known experimental method.Two results are discussed here in detail, both based on parametrization of the attractor. The first shows that any two fluid flows can be distinguished by a sufficient number of point observations of the velocity. This allows one to connect rigorously the dimension of the attractor with the Landau–Lifschitz ‘number of degrees of freedom’, and hence to obtain estimates on the ‘minimum length scale of the flow’ using bounds on this dimension. While for two-dimensional flows the rigorous estimate agrees with the heuristic approach, there is still a gap between rigorous results in the three-dimensional case and the Kolmogorov theory.Secondly, the problem of using experiments to reconstruct the dynamics of a flow is considered. The standard way of doing this is to take a number of repeated observations, and appeal to the Takens time-delay embedding theorem to guarantee that one can indeed follow the dynamics ‘faithfully’. However, this result relies on restrictive conditions that do not hold for spatially extended systems: an extension is given here that validates this important experimental technique for use in the study of turbulence.Although the abstract results underlying this paper have been presented elsewhere, making them specific to the Navier–Stokes equations provides answers to problems particular to fluid dynamics, and motivates further questions that would not arise from within the abstract theory itself.
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Derfoufi, Soufiane, Fayçal Moufekkir, and Ahmed Mezrhab. "Numerical assessment of the mixed convection and volumetric radiation in a vertical channel with MRT-LBM." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 3 (March 5, 2018): 745–62. http://dx.doi.org/10.1108/hff-04-2017-0161.
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Purpose The paper aims to focus on modeling of combined mixed convection and volumetric radiation within a vertical channel using a hybrid thermal lattice Boltzmann method (LBM). The multiple relaxation time LBM (MRT-LBM) is used to compute the dynamical field. The thermal field is determined by a finite difference method (FDM), and the simple relaxation time-LBM (SRT-LBM) serves to calculate the radiative part. The geometry considered concerns a vertical channel defined by two diffuse and isothermal walls. The active fluid represents a gray gas participating in absorption, emission and isotropically scattering. The parametrical study conducted aims to highlight the effect of Richardson number (Ri), Planck number (Pl) and the optical thickness (τ) on dynamical and thermal fields. It is found that radiation affects greatly heat transfer. Design/methodology/approach MRT-LBM is used to compute the dynamical field. The thermal field is determined by FDM, and SRT-LBM serves to calculate the radiative part. Findings This study has shown the strong capability of this approach to simulate similar problems. The Planck number largely affects the streamlines and isotherms distribution. Also, it causes disappearance of reversal flow, undesirable in most industrial applications, for low Planck numbers. The optical thickness causes the disappearance of reversal flow, in the case in which it appears, for lower opacity. However, for higher opacity it leads to a recurrence of reversed flow. Originality/value The use of a new original method composed of MRT-LBM to solve the fluid velocity, FDM to handle the temperature equation and extended SRT-LBM to compute the radiative part of the energy equation.
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Boldrini, José Luiz, Exequiel Mallea-Zepeda, and Marko Antonio Rojas-Medar. "Optimal boundary control for the stationary Boussinesq equations with variable density." Communications in Contemporary Mathematics 22, no. 05 (May 29, 2019): 1950031. http://dx.doi.org/10.1142/s0219199719500317.
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Certain classes of optimal boundary control problems for the Boussinesq equations with variable density are studied. Controls for the velocity vector and temperature are applied on parts of the boundary of the domain, while Dirichlet and Navier friction boundary conditions for the velocity and Dirichlet and Robin boundary conditions for the temperature are assumed on the remaining parts of the boundary. As a first step, we prove a result on the existence of weak solution of the dynamical equations; this is done by first expressing the fluid density in terms of the stream-function. Then, the boundary optimal control problems are analyzed, and the existence of optimal solutions are proved; their corresponding characterization in terms of the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.
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Maksimov, V. I. "On the application of regularized extremal shift to the investigation of some problems of dynamical identification and robust control for systems with delay." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, no. 2 (April 2008): 83–86. http://dx.doi.org/10.20537/vm080230.
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Sánchez-Martín, P., J. J. Masdemont, and M. Romero-Gómez. "From manifolds to Lagrangian coherent structures in galactic bar models." Astronomy & Astrophysics 618 (October 2018): A72. http://dx.doi.org/10.1051/0004-6361/201833451.
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We study the dynamics near the unstable Lagrangian points in galactic bar models using dynamical system tools in order to determine the global morphology of a barred galaxy. We aim at the case of non-autonomous models, in particular with secular evolution, by allowing the bar pattern speed to decrease with time. We have extended the concept of manifolds widely used in the autonomous problem to the Lagrangian coherent structures (LCS), widely used in fluid dynamics, which behave similar to the invariant manifolds driving the motion. After adapting the LCS computation code to the galactic dynamics problem, we apply it to both the autonomous and non-autonomous problems, relating the results with the manifolds and identifying the objects that best describe the motion in the non-autonomous case. We see that the strainlines coincide with the first intersection of the stable manifold when applied to the autonomous case, while, when the secular model is used, the strainlines still show the regions of maximal repulsion associated to both the corresponding stable manifolds and regions with a steep change of energy. The global morphology of the galaxy predicted by the autonomous problem remains unchanged.
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Eke, F. O., and Song-Min Wang. "Equations of Motion of Two-Phase Variable Mass Systems With Solid Base." Journal of Applied Mechanics 61, no. 4 (December 1, 1994): 855–60. http://dx.doi.org/10.1115/1.2901568.
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This paper develops dynamical equations for variable mass systems that can be viewed, at any given instant, as comprising a solid phase and a fluid phase. The equations of translational and rotational motion are presented, and several versions of each are given. It is shown that some versions have major advantages over others because they involve parameters that are relatively easy to estimate in practical problems, and make close-form solutions possible without the usual penalty of drastic simplifying assumptions. A simple rocket example is presented, and shows that instability cannot be ruled out for such systems. It is shown that system and combustion chamber geometry play a crucial role in the attitude stability of such systems.
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HANSRAJ, SUDAN, and DANIEL KRUPANANDAN. "ALGORITHMIC CONSTRUCTION OF EXACT SOLUTIONS FOR NEUTRAL STATIC PERFECT FLUID SPHERES." International Journal of Modern Physics D 22, no. 09 (June 26, 2013): 1350052. http://dx.doi.org/10.1142/s0218271813500521.
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Although it ranks amongst the oldest of problems in classical general relativity, the challenge of finding new exact solutions for spherically symmetric perfect fluid spacetimes is still ongoing because of a paucity of solutions which exhibit the necessary qualitative features compatible with observational evidence. The problem amounts to solving a system of three partial differential equations in four variables, which means that any one of four geometric or dynamical quantities must be specified at the outset and the others should follow by integration. The condition of pressure isotropy yields a differential equation that may be interpreted as second-order in one of the space variables or also as first-order Ricatti type in the other space variable. This second option has been fruitful in allowing us to construct an algorithm to generate a complete solution to the Einstein field equations once a geometric variable is specified ab initio. We then demonstrate the construction of previously unreported solutions and examine these for physical plausibility as candidates to represent real matter. In particular we demand positive definiteness of pressure, density as well as a subluminal sound speed. Additionally, we require the existence of a hypersurface of vanishing pressure to identify a radius for the closed distribution of fluid. Finally, we examine the energy conditions. We exhibit models which display all of these elementary physical requirements.
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BIELLO, JOSEPH A., KENNETH I. SALDANHA, and NORMAN R. LEBOVITZ. "Instabilities of exact, time-periodic solutions of the incompressible Euler equations." Journal of Fluid Mechanics 404 (February 10, 2000): 269–87. http://dx.doi.org/10.1017/s0022112099007089.
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We consider the linear stability of exact, temporally periodic solutions of the Euler equations of incompressible, inviscid flow in an ellipsoidal domain. The problem of linear stability is reduced, without approximation, to a hierarchy of finite-dimensional Floquet problems governing fluid-dynamical perturbations of differing spatial scales and symmetries. We study two of these Floquet problems in detail, emphasizing parameter regimes of special physical significance. One of these regimes includes periodic flows differing only slightly from steady flows. Another includes long-period flows representing the nonlinear outcome of an instability of steady flows. In both cases much of the parameter space corresponds to instability, excepting a region adjacent to the spherical configuration. In the second case, even if the ellipsoid departs only moderately from a sphere, there are filamentary regions of instability in the parameter space. We relate this and other features of our results to properties of reversible and Hamiltonian systems, and compare our results with related studies of periodic flows.
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Epifanio, C. C., and R. Rotunno. "The Dynamics of Orographic Wake Formation in Flows with Upstream Blocking." Journal of the Atmospheric Sciences 62, no. 9 (September 1, 2005): 3127–50. http://dx.doi.org/10.1175/jas3523.1.
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Abstract The development of orographic wakes and vortices is revisited from the dynamical perspective of a three-dimensional (3D) vorticity-vector potential formulation. Particular emphasis is given to the role of upstream blocking in the formation of the wake. Scaling arguments are first presented to explore the limiting form of the 3D vorticity inversion for the case of flow at small dynamical aspect ratio δ. It is shown that in the limit of small δ the inversion is determined completely by the two horizontal vorticity components—that is, the part of the velocity induced by the vertical component of vorticity vanishes in the small-δ limit. This result leads to an approximate formulation of small-δ fluid mechanics in which the three governing prognostic variables are the two horizontal vorticity components and the potential temperature. The remainder of the study then revisits the problem of orographic wake formation from the perspective of this small-δ vorticity dynamics framework. Previous studies have suggested that one of the potential routes to stratified wake formation is through the blocking of flow on the upstream side of the barrier. This apparent link between blocking and wake formation is shown to be relatively straightforward in the small-δ vorticity context. In particular, it is shown that blocking of the flow inevitably leads to a horizontal vorticity distribution that favors deceleration of the leeside flow at the ground. This process of leeside flow deceleration, as well as the subsequent time evolution of the wake, is illustrated through a series of numerical initial-value problems involving flows past 2D and 3D barriers. It is proposed that the initiation of the wake flow in these stratified problems resembles the flow produced by a retracting piston in shallow-water theory.
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Chapman, C. J. "The asymptotic theory of dispersion relations containing Bessel functions of imaginary order." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2148 (September 26, 2012): 4008–23. http://dx.doi.org/10.1098/rspa.2012.0459.
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This paper presents a method of analysing wave-field dispersion relations in which Bessel functions of imaginary order occur. Such dispersion relations arise in applied studies in oceanography and astronomy, for example. The method involves the asymptotic theory developed by Dunster in 1990, and leads to simple analytical approximations containing only trigonometric and exponential functions. Comparisons with accurate numerical calculations show that the resulting approximations to the dispersion relation are highly accurate. In particular, the approximations are powerful enough to reveal the fine structure in the dispersion relation and so identify different wave regimes corresponding to different balances of physical processes. Details of the method are presented for the fluid-dynamical problem that stimulated this analysis, namely the dynamics of an internal ocean wave in the presence of an aerated surface layer; the method identifies and gives different approximations for the subcritical, supercritical and critical regimes. The method is potentially useful in a wide range of problems in wave theory and stability theory. A mathematical theme of the paper is that of the removable singularity.