Academic literature on the topic 'Equation de filaments de vorticités'
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Journal articles on the topic "Equation de filaments de vorticités"
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ONG, LAWRENCE, and JAMES M. WALLACE. "Joint probability density analysis of the structure and dynamics of the vorticity field of a turbulent boundary layer." Journal of Fluid Mechanics 367 (July 25, 1998): 291–328. http://dx.doi.org/10.1017/s002211209800158x.
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An experimental study of a turbulent boundary layer at Rθ≈1070 and Rτ≈543 was conducted. Detailed measurements of the velocity vector and the velocity gradient tensor within the near-wall region were performed at various distances from the wall, ranging from approximately y+=14 to y+=89. The measured mean statistical properties of the fluctuating velocity and vorticity components agree well with previous experimental and numerically simulated data. These boundary layer measurements were used in a joint probability density analysis of the various component vorticity and vorticity–velocity gradient products that appear in the instantaneous vorticity and enstrophy transport equations. The vorticity filaments that contribute most to the vorticity covariance Ω[bar]xΩ [bar]y in this region were found to be oriented downstream with angles of inclination to the wall, when projected on the streamwise (x, y)-plane, that decrease with distance moving from the buffer to the logarithmic layer. When projected on the planview (x, z)- and cross-stream (y, z)-planes, the vorticity filaments that most contribute to the vorticity covariances Ω [bar]xΩ [bar]z and Ω [bar]yΩ [bar]z have angles of inclination to the z-ordinate axis that increase with distance from it. All the elements of the ΩiΩj ∂Ui/∂xj term in the enstrophy transport equation, i.e. the term that describes the rate of increase or decrease of the enstrophy by vorticity filament stretching or compression by the strain-rate field, have been examined. On balance, the average stretching of the vorticity filaments is greater than compression at all y+ locations examined here. However, some individual velocity gradient components compress the vorticity filaments, on average, more than they stretch them.
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Ostrikov, N. N., and E. M. Zhmulin. "Vortex dynamics of viscous fluid flows. Part 1. Two-dimensional flows." Journal of Fluid Mechanics 276 (October 10, 1994): 81–111. http://dx.doi.org/10.1017/s0022112094002478.
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The method of product integration is applied to the vortex dynamics of two-dimensional incompressible viscous media. In the cases of both unbounded and bounded flows under the no-slip boundary condition, the analytic solutions of the Cauchy problem are obtained for the Helmholtz equation in the form of linear and nonlinear product integrals. The application of product integrals allows the generalization in a natural way of the vortex dynamics concept to the case of viscous flows. However, this new approach requires the reconsideration of some traditional notions of vortex dynamics. Two lengthscales are introduced in the form of a micro- and a macro-scale. Elementary ‘vortex objects’ are defined as two types of singular vortex filaments with equal but opposite intensities. The vorticity is considered as the macro-value proportional to the concentration of elementary vortex filaments inhabiting the micro-level. The vortex motion of a viscous medium is represented as the stochastic motion of an infinite set of elementary vortex filaments on the micro-level governed by the stochastic differential equations, where the stochastic velocity component of every filament simulates the viscous diffusion of vorticity, and the regular component is the macro-value induced according to the Biot–Savart law and simulates the convective transfer of vorticity.In flows with boundaries, the production of elementary vortex filaments at the boundary is introduced to satisfy the no-slip condition. This phenomenon is described by the application of the generalized Markov processes theory. The integral equation for the production intensity of elementary vortex filaments is derived and solved using the no-slip condition reformulated in terms of vorticity. Additional conditions on this intensity are determined to avoid the many-valuedness of the pressure in a multi-connected flow domain. This intensity depends on the vorticity in the flow and the boundary velocity at every time instant, together with boundary acceleration.As a result, the successive and accurate application of the product-integral method allows the study of vortex dynamics in a viscous fluid according to the concepts of Helmholtz and Kelvin.
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KIMURA, Y., and J. R. HERRING. "Gradient enhancement and filament ejection for a non-uniform elliptic vortex in two-dimensional turbulence." Journal of Fluid Mechanics 439 (July 23, 2001): 43–56. http://dx.doi.org/10.1017/s0022112001004529.
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The axisymmetrization of a two-dimensional non-uniform elliptic vortex is studied in terms of the growth of palinstrophy, the squared vorticity gradient. First, it is pointed out that the equation for palinstrophy growth, if written in terms of the strain rate tensor, has a similar form to that of enstrophy growth in three-dimensions – the vortex-stretching equation. Then palinstrophy production is analysed, particularly for non-uniform elliptic vortices. It is shown analytically and verified numerically that a non-uniform elliptic vortex in general has a quadrupole structure for palinstrophy production, and that in the positive production regions, vortex filaments are ejected following the gradient enhancement process for vorticity. Numerical simulations are conducted for two different initial conditions, compact support and Gaussian vorticity distributions. These are characterized by distinctly different features of filament ejection and energy spectra. For both cases, the total palinstrophy production is a good indicator of the development of small-scale vorticity. In particular for the compact support case, a possible intermittency mechanism in the filament ejection process is proposed.
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Klein, Rupert, and Omar M. Knio. "Asymptotic vorticity structure and numerical simulation of slender vortex filaments." Journal of Fluid Mechanics 284 (February 10, 1995): 275–321. http://dx.doi.org/10.1017/s002211209500036x.
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A new asymptotic analysis of slender vortices in three dimensions, based solely on the vorticity transport equation and the non-local vorticity–velocity relation gives new insight into the structure of slender vortex filaments. The approach is quite different from earlier analyses using matched asymptotic solutions for the velocity field and it yields additional information. This insight is used to derive three different modifications of the thin-tube version of a numerical vortex element method. Our modifications remove an O(1) error from the node velocities of the standard thin-tube model and allow us to properly account for any prescribed physical vortex core structure independent of the numerical vorticity smoothing function. We demonstrate the performance of the improved models by comparison with asymptotic solutions for slender vortex rings and for perturbed slender vortex filaments in the Klein–Majda regime, in which the filament geometry is characterized by small-amplitude–short-wavelength displacements from a straight line. These comparisons represent a stringent mutual test for both the proposed modified thin-tube schemes and for the Klein–Majda theory. Importantly, we find a convincing agreement of numerical and asymptotic predictions for values of the Klein–Majda expansion parameter ε as large as ½. Thus, our results support their findings in earlier publications for realistic physical vortex core sizes.
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KUVSHINOV, B. N., V. P. LAKHIN, F. PEGORARO, and T. J. SCHEP. "Hamiltonian vortices and reconnection in a magnetized plasma." Journal of Plasma Physics 59, no. 4 (June 1998): 727–36. http://dx.doi.org/10.1017/s0022377898006655.
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Hamiltonian vortices and reconnection in magnetized plasmas are investigated analytically and numerically using a two-fluid model. The equations are written in the Lagrangian form of three fields that are advected with different velocities. This system can be considered as a generalization and extension of the two-dimensional Euler equation for an ordinary fluid. It is pointed out that these equations allow solutions in the form of singular current-vortex filaments, drift-Alfvén vortices and magnetic islands, and admit collisionless magnetic reconnection where magnetic flux is converted into electron momentum and ion vorticity.
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Li, Siran. "Regularity of Desingularized Models for Vortex Filaments in Incompressible Viscous Flows: A Geometrical Approach." Quarterly Journal of Mechanics and Applied Mathematics 73, no. 3 (May 20, 2020): 217–30. http://dx.doi.org/10.1093/qjmam/hbaa008.
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Summary We establish the regularity of weak solutions for the vorticity equation associated to a family of desingularized models for vortex filament dynamics in 3D incompressible viscous flows. These generalize the classical model ‘of an allowance for the thickness of the vortices’ due to Louis Rosenhead in 1930. Our approach is based on an interplay between the geometry of vorticity and analytic inequalities in Sobolev spaces.
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Dauhajre, Daniel P., and James C. McWilliams. "Diurnal Evolution of Submesoscale Front and Filament Circulations." Journal of Physical Oceanography 48, no. 10 (October 2018): 2343–61. http://dx.doi.org/10.1175/jpo-d-18-0143.1.
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AbstractThe local circulation of submesoscale fronts and filaments can be partly understood through a horizontal momentum balance of Coriolis, a horizontal pressure gradient, and vertical diffusivity in a turbulent boundary layer, known as the turbulent thermal wind balance (TTW or T2W). T2W often reproduces the instantaneous relative vorticity and divergence of submesoscale circulations in open-ocean and shelf settings. However, a diurnal cycle in submesoscale vorticity and divergence is characterized by a non-T2W phasing: a maximum in divergence magnitude lags the maximum in vertical diffusivity (with vorticity lagging divergence). Here, an idealized model is used to solve the transient turbulent thermal wind (T3W) equations and to investigate the diurnal evolution of front and filament circulation in a 2D plane. Relative to a steady-state circulation, transient evolution can cause both instantaneous strengthening and a weaker diurnal average of the secondary circulation. The primary mechanisms controlling the diurnal variability exist in a 1D Ekman layer that imprints onto the 2D circulation. In midlatitudes, acceleration during separate phases of the diurnal cycle (from night to day and from day to night) is dominated by distinct inertial oscillation and vertically diffusive mechanisms, respectively. However, the manifestation of these dual accelerations is sensitive to latitude, boundary layer depth, and the strength of the forcing. A simple 1D model predicts the diurnal phasing of submesoscale divergence and vorticity in realistic primitive equation simulations of the southwestern Pacific and coastal California.
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ANDERSEN, TIMOTHY D., and CHJAN C. LIM. "A length-scale formula for confined quasi-two-dimensional plasmas." Journal of Plasma Physics 75, no. 4 (August 2009): 437–54. http://dx.doi.org/10.1017/s0022377809008137.
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AbstractTypically a magnetohydrodynamical model for neutral plasmas must take into account both the ionic and the electron fluids and their interaction. However, at short time scales, the ionic fluid appears to be stationary compared to the electron fluid. On these scales, we need consider only the electron motion and associated field dynamics, and a single fluid model called the electron magnetohydrodynamical model which treats the ionic fluid as a uniform neutralizing background applies. Using Maxwell's equations, the vorticity of the electron fluid and the magnetic field can be combined to give a generalized vorticity field, and one can show that Euler's equations govern its behavior. When the vorticity is concentrated into slender, periodic, and nearly parallel (but slightly three-dimensional) filaments, one can also show that Euler's equations simplify into a Hamiltonian system and treat the system in statistical equilibrium, where the filaments act as interacting particles. In this paper, we show that, under a mean-field approximation, as the number of filaments becomes infinite (with appropriate scaling to keep the vorticity constant) and given that angular momentum is conserved, the statistical length scale, R, of this system in the Gibbs canonical ensemble follows an explicit formula, which we derive. This formula shows how the most critical statistic of an electron plasma of this type, its size, varies with angular momentum, kinetic energy, and filament elasticity (a measure of the interior structure of each filament) and in particular it shows how three-dimensional effects cause significant increases in the system size from a perfectly parallel, two-dimensional, one-component Coulomb gas.
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Ragone, Francesco, and Gualtiero Badin. "A study of surface semi-geostrophic turbulence: freely decaying dynamics." Journal of Fluid Mechanics 792 (March 4, 2016): 740–74. http://dx.doi.org/10.1017/jfm.2016.116.
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In this study we give a characterization of semi-geostrophic turbulence by performing freely decaying simulations for the case of constant uniform potential vorticity, a set of equations known as the surface semi-geostrophic approximation. The equations are formulated as conservation laws for potential temperature and potential vorticity, with a nonlinear Monge–Ampère type inversion equation for the streamfunction, expressed in a transformed coordinate system that follows the geostrophic flow. We perform model studies of turbulent surface semi-geostrophic flows in a domain doubly periodic in the horizontal and limited in the vertical by two rigid lids, allowing for variations of potential temperature at one of the boundaries, and we compare the results with those obtained in the corresponding surface quasi-geostrophic case. The results show that, while the surface quasi-geostrophic dynamics is dominated by a symmetric population of cyclones and anticyclones, the surface semi-geostrophic dynamics features a more prominent role of fronts and filaments. The resulting distribution of potential temperature is strongly skewed and peaked at non-zero values at and close to the active boundary, while symmetry is restored in the interior of the domain, where small-scale frontal structures do not penetrate. In surface semi-geostrophic turbulence, energy spectra are less steep than in the surface quasi-geostrophic case, with more energy concentrated at small scales for increasing Rossby number. The energy related to frontal structures, the lateral strain rate and the vertical velocities are largest close to the active boundary. These results show that the semi-geostrophic model could be of interest for studying the lateral mixing of properties in geophysical flows.
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Stout, Eric, and Fazle Hussain. "External turbulence-induced axial flow and instability in a vortex." Journal of Fluid Mechanics 793 (March 16, 2016): 353–79. http://dx.doi.org/10.1017/jfm.2016.123.
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External turbulence-induced axial flow in an incompressible, normal-mode stable Lamb–Oseen (two-dimensional) vortex column is studied via direct numerical simulations of the Navier–Stokes equations. Azimuthally oriented vorticity filaments, formed from external turbulence, advect radially towards or away from the vortex axis (depending on the filament’s swirl direction), resulting in a net induced axial flow in the vortex core; axial flow increases with increasing vortex Reynolds number ($Re=$ vortex circulation/viscosity). This contrasts the viscous mechanism for axial flow generation downstream of a lifting body, wherein an axial pressure gradient is produced by viscous diffusion of the swirl (Batchelor, J. Fluid Mech., vol. 20, 1964, pp. 645–658). Analysis of the self-induced motion of an arbitrarily curved external filament shows that any non-axisymmetric filament undergoes radial advection. We then studied the evolution of a vortex column starting with an imposed optimal transient growth perturbation. For a range of Re values, axial flow develops and initially grows as (time)$^{5/2}$ before decreasing after two turnover times; for $Re=10\,000$ – the highest computationally achievable – axial flow at late times becomes sufficiently strong to induce vortex instability. Contrary to a prior claim of a parent–offspring mechanism at the outer edge of the core, vorticity tilting within the core by axial flow is the underlying mechanism producing energy growth. Thus, external perturbations in practical flows (at $Re\sim 10^{7}$) produce destabilizing axial flow, possibly leading to the sought-after vortex breakup.
Dissertations / Theses on the topic "Equation de filaments de vorticités"
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Molitor, Mathieu. "Grassmanniennes non-linéaires, groupes de difféomorphismes unimodulaires et quelques équations hamiltoniennes en dimension infinie." Metz, 2007. http://www.theses.fr/2007METZ015S.
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L'objet de cette thèse est l'étude de l'équation d'un filament de vorticité dans ses diverses formes, l'équation d'Euler d'un fluide parfait incompressible G-invariant pour l'action d'un groupe de Lie ainsi que l'étude de la grassmannienne non-linéaire. Le corps de la thèse se découpe en trois chapitres : dans le premier chapitre, nous donnons la forme locale de l'équation d'un filament de vorticité et montrons que l'astuce d'Hasimoto s'étend aux cas des filaments plongés dans une variété riemannienne tridimensionnelle quelconque. Dans le deuxièeme chapitre, nous étudions le groupe des automorphismes unimodulaires de l'espace total d'un fibré principal. Nous déterminons les équations d'Euler associées et exhibons certaines suites exactes de groupes de Lie fréchétiques. Le troisièeme chapitre étudie en détail la grassmannienne non-linéaire, les diverses structures que l'on peut lui adjoindre et quelques équations hamiltoniennes associées. Un premier appendice traite de la notion de calcul différentiel sur un espace fréchétique et un deuxième appendice montre qu'il existe une structure de groupe de Lie sur le groupe des difféomorphismes unimodulaires d'une variété compacteIn this thesis, we study the vortex filament equation, the Euler equation of an incompressible fluid which is G-invariant with respect to a Lie group action and we also study the non-linear grassmanniann. Our study is organized in three chapter and two appendices : in the first chapter, we study the local form of the vortex filament equation and we show that Hasimoto's trisk extends to the the case of a filament embedded in a general three-dimensional riemannian manifold. In the second chapter, we study the group of unimodular automorphisms of the total space of a principal bundle. We compute the Euler equations associated to this group and derive some short exact sequences. In the third chapter, we study the non-linear grassmannian, some geometrical structures on it and we consider also some hamiltonian equations associated. The first appendix treats the notion of differentiable calculus on a frechet space and the second is devoted to the group of unimodular diffeomorphisms of a compact manifold
Book chapters on the topic "Equation de filaments de vorticités"
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"Simplified Asymptotic Equations for Slender Vortex Filaments." In Vorticity and Incompressible Flow, 256–302. Cambridge University Press, 2001. http://dx.doi.org/10.1017/cbo9780511613203.008.
Full textConference papers on the topic "Equation de filaments de vorticités"
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Voigt, Torsten, Peter Habisreuther, and Nikolaos Zarzalis. "Simulation of Vorticity Driven Flame Instability Using a Flame Surface Density Approach Including Markstein Number Effects." In ASME Turbo Expo 2009: Power for Land, Sea, and Air. ASMEDC, 2009. http://dx.doi.org/10.1115/gt2009-59331.
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The combustion induced and vorticity driven instability behaviour of a lean flame within a premixed combustion system under typical gas turbine conditions is investigated numerically. Over a wide range of operation conditions, the flame is stabilised by the well known mechanisms of premixed confined combustion systems interacting with a turbulent swirled flow field. However, the behaviour of the flame changes drastically just as a critical equivalence ratio is reached: the flame moves upstream close to the rotational axis at positions of originally highest downstream velocities. This flashback behaviour is known as combustion induced vortex breakdown (CIVB) [1] since the upstream propagation results from a local breakdown of the swirled flow field caused by the heat release. A Reynolds-Stress turbulence approach is used as closure model for the three dimensional uRANS simulations to capture the highly turbulent and exceedingly anisotropic flow field. In addition, a flame surface density (FSD) formulation is utilised for combustion modelling. Hence, the essential process of flame vortex interaction is considered in particular. This process is expressed by the important effects of flame strain and curvature and is directly taken into account by means of solving the Favre averaged area of flame surface with the aid of a transport equation. The considered stretch effects do not only influence the production and destruction of the flame surface but moreover modify the local heat release by altering the pseudo-chemical parameter of flame speed. This effect is captured through the Markstein number that is determined by preliminary flamelet calculations. The numerical results lead to the conclusion that the strained and re-orientated vorticity filaments are mainly responsible for the formation of locally negative velocities close to the rotational axis. The re-orientation and straining of vortex filaments is in turn controlled by the local heat release. Finally, the overall occurrence of the simulated flashback is in good accordance with the experiments but does also demonstrate the importance of strain effects on global flame behaviour.
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Bhargava, Aarushi, Kaiyuan Peng, Jerry Stieg, Reza Mirzaeifar, and Shima Shahab. "Ultrasound Actuation of Shape-Memory Polymer Filaments: Acoustic-Thermoelastic Modeling and Testing." In ASME 2017 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/smasis2017-3832.
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Controlled drug delivery (CDD) technology has received extensive attention in the past three decades due to numerous advantages of this technology when compared to the conventional methods. Despite recent efforts and substantial achievements, controlled drug releasing systems still face major challenges in practice, including chemical issues with synthesizing biocompatible drug containers and releasing the pharmaceutical compounds at the targeted location with a controlled time rate. In this work, we present experimentally-validated acoustic-thermoelastic mathematical modeling to show the feasibility of using shape memory polymers (SMPs) and focused ultrasound (FU) technology for designing a novel drug-delivery system. SMPs represent a new class of materials that have the ability of storing a temporary shape and returning to their permanent or original shape when subjected to external stimuli such as heat. FU is used as a trigger for noninvasively stimulating SMP-based drug capsules. FU has a superior capability to localize the heating effect, thus initiating the shape recovery process only in selected parts of the polymer. A multiphysics model is developed, which optimizes the design of a SMP-based CDD system using acoustic-thermoelastic analysis of a filament as the constituting base structure and quantifies its activation through FU. The analytical and numerical models are divided into three parts. The first part studies the acoustic behavior of SMPs using Khokhlov-Zabolotskaya-Kuznetsov (KZK) model. The equation solves for acoustic pressure field in a hybrid time-frequency domain using operator-splitting method and examines the effects of absorption, diffraction and nonlinear distortion on the propagating wave in the medium. The second part provides a numerical model based on Penne’s Bioheat equation to estimate the thermal field developed in SMPs as a result of focused acoustic pressure field. The third part provides a numerical framework to predict the mechanical stresses developed in SMPs under FU and consequent shape recovery. The mechanical model is formulated by a compressible neo-Hookean constitutive equation, which assumes the SMPs behave as a thermoelastic material and predicts the shape memory effect under FU. Experimental validation is performed using a FU transducer in a water tank. The recovery of thermally responsive SMPs under FU predicted by our model shows a good accordance with the experiments. The modeling results are used to optimize parameters such as nonlinear properties, input frequency, source power and dimensional effects to achieve maximum shape recovery.
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Nagai, Baku M., Muhammed Sohel Rana, Kazumasa Ameku, and Junji Chinen. "Flow Around a Suddenly Start Rotating Circular Cylinder and Introduce a New Paradox." In ASME/JSME 2007 5th Joint Fluids Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/fedsm2007-37214.
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There have been many misunderstanding about the flow around vortices for example a stationary and/or moving vortex pair. The authors have pointed out that no fluid dynamics textbooks have accepted the existence of stationary or arbitral speed moving vortices. About the vortex flow, recently the authors have found a new analytic solution of the Navier-Stokes equations for two-dimensional flow around a suddenly start rotating circular cylinder. This analytic solution explains the velocity distribution, vorticity distribution with change in time, and boundary layer thickness close to a vortex filament because of the action of viscosity. The resulting solutions are involved simple exponential function. Authors present a new construction for the solution of the Navier-Stokes equations for suddenly start rotating circular cylinder. New solution is based on the concept of the similarity solution approach using similarity variable, dimensional analysis, initial, & boundary conditions. A brief theoretical discussion is presented about the suddenly start rotating circular cylinder. The second part of the paper deals with the analytic solution being compared with experimental results in various Reynolds number. A typical measurement is that of relaxation of rotational velocities when the cylinder is subjected only to the viscous resistance. To measure the velocity distribution of the flow the experiments were made with the help of tracer particle (aluminum powder and 150-grain diameter meshes) for water and oil (Super Mulpus 68). The effects of the Reynolds number on the laminar asymmetric flow structure in the flow region are studied. The induced speed distribution in the rotation of cylinder (diameter 10 mm) circumference has examined about the Reynolds number from 26 to 522 for water consequent cylinder rpm 10, 25, 50, 75, 100 and 0.12 to 2.32 for Super Mulpus 68 Oil consequent cylinder rpm 5, 10, 25, 50, 75, 100. The relation between the induced speeds after the time had passed enough and the various cylinder rotational speeds for both analytical and experimental results are shown. At lower Reynolds number experimental results are closer to theoretical results for a finite time condition, at that time there is exist vorticity around the cylinder. We can also establish that more difference between experimental and theoretical results with higher Reynolds number. An interesting phenomenon has been observed in the flow patterns at various Reynolds number and is discussed. Finally, authors have explained the significant difference between experimental and theoretical results and a new paradox has been introduced.
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Zhou, Yuanxin, Mohammad Monirul Hasan, and Shaik Jeelani. "Effect of Carbon Nanofiber on Thermal and Tensile Properties of Polypropylene." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-13254.
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In the present study, effect of vapor grown carbon nanofiber on the mechanical and thermal properties of polypropylene was investigated. Firstly, nanofibers were dry-mixed with polypropylene powder and extruded into filaments by using a single screw extruder. Then the tensile tests were performed on the single filament at the strain rate range from 0.02/min to 2/min. Experiments results show that both neat and nano-phased polypropylene were strain rate strengthening material. The tensile modulus and yield strength both increased with increasing strain rate. Experimental results also show that infusing nanofiber into polypropylene can increase tensile modulus and yield strength, but decrease the failure strain. At the same time, thermal properties of neat and nano-phased polypropylene were characterized by TGA. TGA results have showed that the nanophased system is more thermally stable. At last, a nonlinear constitutive equation has been developed to describe strain rate sensitive behavior of neat and nano-phased polypropylene.
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Sharma, Sheshadri, and Richard Rodrigues Jettappa. "A Method to Determine the Exact Time Period of Oscillations of a Bifilar Pendulum." In ASME Turbo Expo 2010: Power for Land, Sea, and Air. ASMEDC, 2010. http://dx.doi.org/10.1115/gt2010-23531.
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A novel method to determine the exact time period of oscillations of a class of non-linear systems is presented. Taking the bifilar pendulum as an example, and employing the conservation of total energy concept, the free oscillations of the system is studied. The governing equation of motion of a bifilar pendulum is non-linear. The integration of this equation to obtain the time period of oscillation is highly complicated and only numerical solution is available. This is because the integral is singular at the extremities of the motion where the velocity will be zero. But, what cannot be achieved by integral calculus can be obtained easily by employing the definition of velocity taught in the high school curriculum. By employing this simple mathematical trick, this intractable equation is recast in a different but exact form. This leads to the identification of what is called the “Geometric Inertia” in bifilar pendulums. This Geometric Inertia is the additional inertia displayed by the system due to the constraint imposed by the two filaments as a result of the geometry of the pendulum. In the proposed method, the total displacement of the system is considered and divided into small equal segments. At the end points of each such segment, the corresponding velocity is calculated from the energy equation. Noting that the velocities are zero at the extremities of the system, an average velocity to each segment is calculated, and this average velocity is positive in each segment. The “delta” time spent by the system in each segment is now calculated by dividing the segment length by the average velocity of that segment. (From, time = displacement/velocity). The linear sum of such “delta” times gives the time period of oscillation. As the number of segments is increased, thereby reducing the segment length, the estimate becomes increasingly accurate. The proposed approach avoids a direct integration of complex, and often singular expressions that complicate the determination of time periods of oscillations of non-linear systems.
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Li, Yali, and N. C. Goulbourne. "Electro-Chemo-Mechanical Modeling of the Artery Myogenic Transient and Steady-State Response." In ASME 2014 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/imece2014-39237.
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Active contraction of smooth muscle results in the myogenic response and vasomotion of arteries, which adjusts the blood flow and nutrient supply of the organism. It is a multiphysic process coupled electrical and chemical kinetics with mechanical behavior of the smooth muscle. This paper presents a new constitutive model for the media layer of the artery wall to describe the myogenic response of artery wall for different transmural pressures. The model includes two major components: electrobiochemical, and chemomechanical parts. The electrochemical model is a lumped Hodgkin-Huxley-type cell membrane model for the nanoscopic ionic currents: calcium, sodium, and potassium. The calculated calcium concentration serves as input for the chemomechanical portion of the model; its molecular binding and the reactions with other enzyme cause the relative sliding of thin and thick filaments of the contractile unit. In the chemomechanical model, a new nonlinear viscoelastic model is proposed using a continuum mechanics approach to describe the time varying behavior of the smooth muscle. Specifically, this model captures the filament overlap effect, active stress evolution, initial velocity, and elastic recoil in the media layer. The artery wall is considered as a thin-walled cylindrical tube. Using the proposed constitutive model and the thin-walled equilibrium equation, the myogenic response is calculated for different transmural pressures. The integrated model is able to capture the pressure-diameter transient and steady-state relationship.