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1
Pritchard, David, Catriona R. McArdle, and Stephen K. Wilson. "The Stokes boundary layer for a power-law fluid." Journal of Non-Newtonian Fluid Mechanics 166, no. 12-13 (2011): 745–53. http://dx.doi.org/10.1016/j.jnnfm.2011.04.011.
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Saeger, R. B., L. E. Scriven, and H. T. Davis. "Transport processes in periodic porous media." Journal of Fluid Mechanics 299 (September 25, 1995): 1–15. http://dx.doi.org/10.1017/s0022112095003399.
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The Stokes equation system and Ohm's law were solved numerically for fluid in periodic bicontinuous porous media of simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC) symmetry. The Stokes equation system was also solved for fluid in porous media of SC arrays of disjoint spheres. The equations were solved by Galerkin's method with finite element basis functions and with elliptic grid generation. The Darcy permeability k computed for flow through SC arrays of spheres is in excellent agreement with predictions made by other authors. Prominent recirculation patterns are found for Stokes flow in bicontinuous porous media. The results of the analysis of Stokes flow and Ohmic conduction through bicontinuous porous media were used to test the permeability scaling law proposed by Johnson, Koplik & Schwartz (1986), which introduces a length parameter Λ to relate Darcy permeability k and the formation factor F. As reported in our earlier work on the SC bicontinuous porous media, the scaling law holds approximately for the BCC and FCC families except when the porespace becomes nearly spherical pores connected by small orifice-like passages. We also found that, except when the porespace was connected by the small orifice-like passages, the permeability versus porosity curve of the bicontinuous media agrees very well with that of arrays of disjoint and fused spheres of the same crystallographic symmetry.
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Marušić-Paloka, E. "On the Stokes Paradox for Power-Law Fluids." ZAMM 81, no. 1 (2001): 31–36. http://dx.doi.org/10.1002/1521-4001(200101)81:1<31::aid-zamm31>3.0.co;2-g.
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Rana, Anirudh Singh, Vinay Kumar Gupta, and Henning Struchtrup. "Coupled constitutive relations: a second law based higher-order closure for hydrodynamics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2218 (2018): 20180323. http://dx.doi.org/10.1098/rspa.2018.0323.
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In the classical framework, the Navier–Stokes–Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier–Stokes–Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier–Stokes–Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.
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Alam, M., S. Saha, and R. Gupta. "Unified theory for a sheared gas–solid suspension: from rapid granular suspension to its small-Stokes-number limit." Journal of Fluid Mechanics 870 (May 15, 2019): 1175–93. http://dx.doi.org/10.1017/jfm.2019.304.
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A non-perturbative nonlinear theory for moderately dense gas–solid suspensions is outlined within the framework of the Boltzmann–Enskog equation by extending the work of Saha & Alam (J. Fluid Mech., vol. 833, 2017, pp. 206–246). A linear Stokes’ drag law is adopted for gas–particle interactions, and the viscous dissipation due to hydrodynamic interactions is incorporated in the second-moment equation via a density-corrected Stokes number. For the homogeneous shear flow, the present theory provides a unified treatment of dilute to dense suspensions of highly inelastic particles, encompassing the high-Stokes-number rapid granular regime ($St\rightarrow \infty$) and its small-Stokes-number counterpart, with quantitative agreement for all transport coefficients. It is shown that the predictions of the shear viscosity and normal-stress differences based on existing theories deteriorate markedly with increasing density as well as with decreasing Stokes number and restitution coefficient.
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GURTIN, MORTON E., DEBRA POLIGNONE, and JORGE VIÑALS. "TWO-PHASE BINARY FLUIDS AND IMMISCIBLE FLUIDS DESCRIBED BY AN ORDER PARAMETER." Mathematical Models and Methods in Applied Sciences 06, no. 06 (1996): 815–31. http://dx.doi.org/10.1142/s0218202596000341.
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A unified framework for coupled Navier-Stokes/Cahn-Hilliard equations is developed using, as a basis, a balance law for microforces in conjunction with constitutive equations consistent with a mechanical version of the second law. As a numerical application of the theory, we consider the kinetics of coarsening for a binary fluid in two space dimensions.
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Chen, Jie, Shuyu Sun, and Zhangxin Chen. "Coupling Two-Phase Fluid Flow with Two-Phase Darcy Flow in Anisotropic Porous Media." Advances in Mechanical Engineering 6 (January 1, 2014): 871021. http://dx.doi.org/10.1155/2014/871021.
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This paper reports a numerical study of coupling two-phase fluid flow in a free fluid region with two-phase Darcy flow in a homogeneous and anisotropic porous medium region. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the anisotropic porous medium region. A Robin-Robin domain decomposition method is used for the coupled Navier-Stokes and Darcy system with the generalized Beavers-Joseph-Saffman condition on the interface between the free flow and the porous media regions. Obtained results have shown the anisotropic properties effect on the velocity and pressure of the two-phase flow.
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Boukrouche, Mahdi, Imane Boussetouan, and Laetitia Paoli. "Existence and approximation for Navier–Stokes system with Tresca’s friction at the boundary and heat transfer governed by Cattaneo’s law." Mathematics and Mechanics of Solids 23, no. 3 (2017): 519–40. http://dx.doi.org/10.1177/1081286517722587.
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We consider an unsteady non-isothermal incompressible fluid flow. We model heat conduction with Cattaneo’s law instead of the commonly used Fourier’s law, in order to overcome the physical paradox of infinite propagation speed. We assume that the fluid viscosity depends on the temperature, while the thermal capacity depends on the velocity field. The problem is thus described by a Navier–Stokes system coupled with the hyperbolic heat equation. Furthermore, we consider non-standard boundary conditions with Tresca’s friction law on a part of the boundary. By using a time-splitting technique, we construct a sequence of decoupled approximate problems and we prove the convergence of the corresponding approximate solutions, leading to an existence theorem for the coupled fluid flow/heat transfer problem. Finally, we present some numerical results.
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FELDERHOF, B. U. "Transient flow of a viscous compressible fluid in a circular tube after a sudden point impulse." Journal of Fluid Mechanics 644 (February 10, 2010): 97–106. http://dx.doi.org/10.1017/s0022112009992874.
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The flow of a viscous compressible fluid in a circular tube generated by a sudden impulse at a point on the axis is studied on the basis of the linearized Navier–Stokes equations. A no-slip boundary condition is assumed to hold on the wall of the tube. Owing to the finite velocity of sound the flow behaviour differs qualitatively from that of an incompressible fluid. The flow velocity and the pressure disturbance at any fixed point different from the source point vanish at short time and decay at long times with a t−3/2 power law.
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Okamura, Makoto. "Closure model for homogeneous isotropic turbulence in the Lagrangian specification of the flow field." Journal of Fluid Mechanics 841 (February 23, 2018): 521–51. http://dx.doi.org/10.1017/jfm.2018.98.
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This paper proposes a new two-point closure model that is compatible with the Kolmogorov$-5/3$power law for homogeneous isotropic turbulence in an incompressible fluid using the Lagrangian specification of the flow field. A closed set of three equations was derived from the Navier–Stokes equation with no adjustable free parameters. The Kolmogorov constant and the skewness of the longitudinal velocity derivative were evaluated to be 1.779 and$-0.49$, respectively, using the proposed model. The bottleneck effect was also reproduced in the near-dissipation range.
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Roustaei, A., T. Chevalier, L. Talon, and I. A. Frigaard. "Non-Darcy effects in fracture flows of a yield stress fluid." Journal of Fluid Mechanics 805 (September 16, 2016): 222–61. http://dx.doi.org/10.1017/jfm.2016.491.
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We study non-inertial flows of single-phase yield stress fluids along uneven/rough-walled channels, e.g. approximating a fracture, with two main objectives. First, we re-examine the usual approaches to providing a (nonlinear) Darcy-type flow law and show that significant errors arise due to self-selection of the flowing region/fouling of the walls. This is a new type of non-Darcy effect not previously explored in depth. Second, we study the details of flow as the limiting pressure gradient is approached, deriving approximate expressions for the limiting pressure gradient valid over a range of different geometries. Our approach is computational, solving the two-dimensional Stokes problem along the fracture, then upscaling. The computations also reveal interesting features of the flow for more complex fracture geometries, providing hints about how to extend Darcy-type approaches effectively.
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YAN, MIN, JIANYE YIN, BAOPING SUN, and XIAO MA. "GRANULE HYDRODYNAMICS METHOD: A DISCRETE ELEMENT METHOD ON FLUID MOTIONS." International Journal of Computational Methods 09, no. 01 (2012): 1240023. http://dx.doi.org/10.1142/s0219876212400233.
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A granule hydrodynamics method is proposed in studying complex dynamic behavior of wind-blown movement system. Fluid in the wind-blown movement system is discretized into elastic fluid granules in a certain scale of time and space. It is considered that fluid motion is caused by fluid granules collision and fluid density difference (or pressure difference). Based on the essential properties of fluid, the constitutive relations of fluid granules are well established in the proposed granule hydrodynamics method, fluid state equation, and fluid sound velocity derivative state equation are adopted to study motion law of fluid or fluid–solid coupling, rather than the Navier–Stokes equation in traditional fluid mechanics. Results on classical shear flow in a cavity based on the proposed granule hydrodynamics method agree well with those based on finite difference method and smoothed particle hydrodynamics method, which verifies the validity and feasibility of the granule hydrodynamics method. Two more numerical examples of solid–fluid coupling of a multiple moving solid system are also provided, which proves the feasibility, convenience, and uniqueness of the proposed method.
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HAUGEN, NILS ERLAND L., and STEINAR KRAGSET. "Particle impaction on a cylinder in a crossflow as function of Stokes and Reynolds numbers." Journal of Fluid Mechanics 661 (July 27, 2010): 239–61. http://dx.doi.org/10.1017/s0022112010002946.
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A high-order direct numerical simulation code (The Pencil Code) has been used together with the immersed boundary method on a Cartesian grid to simulate particle impaction on a cylinder in a crossflow. The direct numerical scheme concerns only the fluid flow, into which the particles are subsequently coupled through a one-way drag-coefficient law. The immersed boundary method is extended to work with high-order discretization, and the particle impaction efficiency has been measured for Stokes numbers ranging from 0.001 to 40 for a range of different Reynolds numbers. Three modes of impaction on the front side of the cylinder are identified, where, for the large-Stokes-number mode (St > 0.3), an alternative to the traditional Stokes number is presented that provides better scaling. The intermediate impaction mode has a very steep decrease in impaction efficiency as the Stokes number is decreased, and this is identified as the range of Stokes numbers where the viscous boundary layer starts to take effect. The third mode of front-side impaction is for the very small particles with St < 0.1 exactly following the flow but impacting on the cylinder due to their finite radii. There will not be any capture on the front side of the cylinder for impact angles larger than ~56° for this mode. Finally, it is found that the particle impaction on the back side of the cylinder is strongly dependent on the flow Reynolds number, where large Reynolds numbers lead to larger impaction efficiencies. The upper limiting Stokes number of back-side impaction is around 0.13, apparently irrespective of the Reynolds number.
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Bandulasena, H. C. H., W. B. Zimmerman, and J. M. Rees. "An inverse methodology for the rheology of a power-law non-Newtonian fluid." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 222, no. 5 (2008): 761–68. http://dx.doi.org/10.1243/09544062jmes747.
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The current paper presents a novel methodology for calculating the rheological para-meters for dilute aqueous solutions of a power-law non-Newtonian fluid, xanthan gum (XG). Previous studies have verified the fidelity of finite-element modelling of the Navier—Stokes equations for reproducing the velocity fields of XG solutions in a microfluidic T-junction with experimental observations obtained using micron resolution particle image velocimetry (μ-PIV). As the pressure-driven fluid is forced to turn the corner of the T-junction, a range of shear rates, and therefore viscosities, are produced within the flow system. Thus, a setup that potentially establishes the rheological profile of XG from a single experiment is selected. An inverse method based on finding the mapping between the statistical moments of the velocity field and the constitutive parameters of the viscosity profile demonstrated that such a system could potentially be used for the design of an efficient microfluidic rheometer. However, μ-PIV technology is expensive and the equipment is bulky. The current paper investigates whether different flow features could be used to establish the rheological profile.
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Jin, Yan, and Kang Ping Chen. "Fundamental equations for primary fluid recovery from porous media." Journal of Fluid Mechanics 860 (December 4, 2018): 300–317. http://dx.doi.org/10.1017/jfm.2018.874.
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Primary fluid recovery from a porous medium is driven by the volumetric expansion of thein situfluid. For production from a petroleum reservoir, primary recovery accounts for more than half of the total amount of recovered hydrocarbon. The primary recovery process is studied here at the pore scale and the macroscopic scale. The pore-scale flow is first analysed using the compressible Navier–Stokes equations and the mathematical theory for low-Mach-number flow developed by Klainerman & Majda (Commun. Pure Appl. Maths, vol. 34 (4), 1981, pp. 481–524; vol. 35 (5), 1982, pp. 629–651). An asymptotic analysis shows that the pore-scale flow is governed by the self-diffusion of the fluid and it exhibits a slip-like mass flow rate, even though the velocity satisfies the no-slip condition on the pore wall. The pore-scale density equation is then upscaled to a macroscopic diffusion equation for the density which possesses a diffusion coefficient proportional to the fluid’s kinematic viscosity. Darcy’s law is shown to be inapplicable to primary fluid recovery and it should be replaced by a new mass flux equation which depends on the porosity but not on the permeability. This is in stark contrast to the classical result and it can have important implications for hydrocarbon recovery as well as other applications.
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Box, F., K. Singh, and T. Mullin. "The interaction between rotationally oscillating spheres and solid boundaries in a Stokes flow." Journal of Fluid Mechanics 849 (June 26, 2018): 834–59. http://dx.doi.org/10.1017/jfm.2018.354.
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We present the results of an experimental and theoretical investigation into the influence of proximate boundaries on the motion of an rotationally oscillating sphere in a viscous fluid. The angular oscillations of the sphere are controlled using the magnetic torque generated by a spatially uniform, oscillatory magnetic field which interacts with a small magnet embedded within the sphere. We study the motion of the sphere in the vicinity of stationary walls that are parallel and perpendicular to the rotational axis of the sphere, and near a second passive sphere that is non-magnetic and free to move. We find that rigid boundaries introduce viscous resistance to motion that acts to suppress the oscillations of the driven sphere. The amount of viscous resistance depends on the orientation of the wall with respect to the axis of rotation of the oscillating sphere. A passive sphere also introduces viscous resistance to motion, but for this case the rotational oscillations of the active sphere establish a standing wave that imparts vorticity to the fluid and induces oscillations of the passive sphere. The standing wave is analogous to the case of an oscillating plate in a viscous fluid; the amplitude of the wave decays exponentially with radial distance from the surface of the oscillating sphere. The standing wave introduces a phase lag between the motion of the active sphere and the response of the passive sphere which increases linearly with separation distance.
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Hayat, Tasawar, Javaria Akram, Hina Zahir, and Ahmad Alsaedi. "Numerical investigation for endoscopic and Soret-Dufour effects on MHD peristaltic activity of Carreau fluid." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 12 (2018): 2960–78. http://dx.doi.org/10.1108/hff-02-2018-0050.
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Purpose The purpose of this paper is to emphasize on the impact of endoscope in MHD peristaltic flow of Carreau fluid. Heat and mass transfer phenomena are comprised of Soret and Dufour effects. Influences of mixed convection and viscous dissipation are also accounted. Wall properties and convective boundary conditions are used. Design/methodology/approach The Navier–Stokes and energy equations used the lubrication approach. The reduced system of equations is executed numerically. The graphical illustration of velocity, temperature, concentration and heat transfer coefficient for various emerging parameters is discussed. Findings The response of Weissenberg number and power law index is decaying toward velocity and temperature. Moreover impression of Soret and Dufour number on temperature is quite reverse to that of concentration. Originality/value The titled problem with the various considered effects has not been solved before, and it is of special importance in various industries. The problem is original.
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Nazeer, Mubbashar, Farooq Hussain, Fayyaz Ahmad, Sadia Iftikhar, and Gener S. Subia. "Theoretical study of an unsteady ciliary hemodynamic fluid flow subject to the Newton’s boundary conditions." Advances in Mechanical Engineering 13, no. 8 (2021): 168781402110404. http://dx.doi.org/10.1177/16878140211040462.
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This article addresses the hemodynamic flow of biological fluid through a symmetric channel. Methachronal waves induced by the ciliary motion of motile structures are the main source of Couple stress nanofluid flow. Darcy’s law is incorporated in Navier-Stokes equations to highlight the influence of the porous medium. Thermal transport by the microscopic collision of particles is governed by Fourier’s law while a separate expression is obtained for net diffusion of nanoparticles by using Fick’s law. A closed-form solution is achieved of nonlinear differential equations subject to Newton’s boundary conditions. Moreover, the current findings are compared with previous outcomes for the limiting case and found a complete coherence. Parametric study reveals that nanoflow is resisted by employing Newton’s boundary conditions. Thermal profile enhancement is contributed by the viscous dissipation parameter. Finally, one infers that hemodynamic flow of non-Newtonian fluid is an effective mode of heat and mass transfer especially, in medical sciences for the rapid transport of medicines in drug therapy.
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Bergemann, Nico, Anne Juel, and Matthias Heil. "Viscous drops on a layer of the same fluid: from sinking, wedging and spreading to their long-time evolution." Journal of Fluid Mechanics 843 (March 16, 2018): 1–28. http://dx.doi.org/10.1017/jfm.2018.127.
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We study the axisymmetric spreading of drops deposited on a pre-existing horizontal layer of the same viscous fluid. Using a combination of experiments, numerical modelling based on the axisymmetric free-surface Navier–Stokes equations and scaling analyses, we explore the drops’ behaviour in a regime where the flow is driven by gravitational and/or capillary forces while inertial effects are small. We find that during the early stages of the drops’ evolution there are three distinct spreading behaviours depending on the thickness of the liquid layer. For thin layers the fluid ahead of a clearly defined spreading front is at rest and the overall behaviour resembles that of a drop spreading on a dry substrate. For thicker films, the spreading is characterised by an advancing wedge which is sustained by fluid flow from the drop into the layer. Finally, for thick layers the drop sinks into the layer, accompanied by significant flow within the layer. As the drop keeps spreading, the evolution of its shape becomes self-similar, with a power-law behaviour for its radius and its excess height above the undisturbed fluid layer. We employ lubrication theory to analyse the drop’s ultimate long-term behaviour and show that all drops ultimately enter an asymptotic regime which is reached when their excess height falls below the thickness of the undisturbed layer.
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Chen, Jie, Shuyu Sun, and Zhengkang He. "HOMOGENIZE COUPLED STOKES–CAHN–HILLIARD SYSTEM TO DARCY'S LAW FOR TWO-PHASE FLUID FLOW IN POROUS MEDIUM BY VOLUME AVERAGING." Journal of Porous Media 22, no. 1 (2019): 1–19. http://dx.doi.org/10.1615/jpormedia.2018028699.
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Liu, Xin, and Yuan Yuan. "The self-similar solutions to full compressible Navier–Stokes equations without heat conductivity." Mathematical Models and Methods in Applied Sciences 29, no. 12 (2019): 2271–320. http://dx.doi.org/10.1142/s0218202519500465.
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In this work, we establish a class of globally defined large solutions to the free boundary problem of compressible full Navier–Stokes equations with constant shear viscosity, vanishing bulk viscosity and heat conductivity. We establish such solutions with initial data perturbed around the self-similar solutions when [Formula: see text]. In the case when [Formula: see text], solutions with bounded entropy can be constructed. It should be pointed out that the solutions we obtain in this fashion do not in general keep being a small perturbation of the self-similar solution due to the second law of thermodynamics, i.e. the growth of entropy. If, in addition, in the case when [Formula: see text], we can construct a solution as a global-in-time small perturbation of the self-similar solution and the entropy is uniformly bounded in time. Our result extends the one of Hadžić and Jang [Expanding large global solutions of the equations of compressible fluid mechanics, J. Invent. Math. 214 (2018) 1205.] from the isentropic inviscid case to the non-isentropic viscous case.
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Vianna, Rafael S., Alexsander M. Cunha, Rodrigo B. V. Azeredo, Ricardo Leiderman, and Andre Pereira. "Computing Effective Permeability of Porous Media with FEM and Micro-CT: An Educational Approach." Fluids 5, no. 1 (2020): 16. http://dx.doi.org/10.3390/fluids5010016.
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Permeability is a parameter that measures the resistance that fluid faces when flowing through a porous medium. Usually, this parameter is determined in routine laboratory tests by applying Darcy’s law. Those tests can be complex and time-demanding, and they do not offer a deep understanding of the material internal microstructure. Currently, with the development of new computational technologies, it is possible to simulate fluid flow experiments in computational labs. Determining permeability with this strategy implies solving a homogenization problem, where the determination of the macro parameter relies on the simulation of a fluid flowing through channels created by connected pores present in the material’s internal microstructure. This is a powerful example of the application of fluid mechanics to solve important industrial problems (e.g., material characterization), in which the students can learn basic concepts of fluid flow while practicing the implementation of computer simulations. In addition, it gives the students a concrete opportunity to work with a problem that associates two different scales. In this work, we present an educational code to compute absolute permeability of heterogeneous materials. The program simulates a Stokes flow in the porous media modeled with periodic boundary conditions using finite elements. Lastly, the permeability of a real sample of sandstone, modeled by microcomputed tomography (micro-CT), is obtained.
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Damak, Kamel, Abdelmoneim Ayadi, Belkacem Zeghmati, and Philippe Schmitz. "A new Navier-Stokes and Darcy's law combined model for fluid flow in crossflow filtration tubular membranes." Desalination 161, no. 1 (2004): 67–77. http://dx.doi.org/10.1016/s0011-9164(04)90041-0.
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Dong, Yuan. "Thermal rectification based on phonon hydrodynamics and thermomass theory." Communications in Applied and Industrial Mathematics 7, no. 2 (2016): 26–38. http://dx.doi.org/10.1515/caim-2016-0004.
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AbstractThe thermal diode is the fundamental device for phononics. There are various mechanisms for thermal rectification, e.g. different temperature dependent thermal conductivity of two ends, asymmetric interfacial resistance, and nonlocal behavior of phonon transport in asymmetric structures. The phonon hydrodynamics and thermomass theory treat the heat conduction in a fluidic viewpoint. The phonon gas flowing through the media is characterized by the balance equation of momentum, like the Navier-Stokes equation for fluid mechanics. Generalized heat conduction law thereby contains the spatial acceleration (convection) term and the viscous (Laplacian) term. The viscous term predicts the size dependent thermal conductivity. Rectification appears due to the MFP supersession of phonons. The convection term also predicts rectification because of the inertia effect, like a gas passing through a nozzle or diffuser.
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Hughes, T. J. R., L. P. Franca, and M. Mallet. "A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics." Computer Methods in Applied Mechanics and Engineering 54, no. 2 (1986): 223–34. http://dx.doi.org/10.1016/0045-7825(86)90127-1.
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Saqr, Khalid M. "Computational fluid dynamics simulations of cerebral aneurysm using Newtonian, power-law and quasi-mechanistic blood viscosity models." Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 234, no. 7 (2020): 711–19. http://dx.doi.org/10.1177/0954411920917531.
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Cerebral aneurysm is a fatal neurovascular disorder. Computational fluid dynamics simulation of aneurysm haemodynamics is one of the most important research tools which provide increasing potential for clinical applications. However, computational fluid dynamics modelling of such delicate neurovascular disorder involves physical complexities that cannot be easily simplified. Recently, it was shown that the Newtonian simplification used to close the shear stress tensor of the Navier–Stokes equation is not sufficient to explore aneurysm haemodynamics. This article explores the differences between the latter simplification, non-Newtonian power-law model and a newly proposed quasi-mechanistic model. The modified Krieger model, which treats blood as a suspension of plasma and particles, was implemented in computational fluid dynamics context here for the first time and is made available to the readers in a C# code in the supplementary material of this article. Two middle-cerebral artery and two anterior-communicating artery aneurysms, all ruptured, were utilized here as case studies. It was shown that the modified Krieger model had higher sensitivity for wall shear stress calculations in comparison with the other two models. The modified Krieger model yielded lower wall shear stress values consistently in comparison with the other two models. Moreover, the modified Krieger model has generally predicted higher pressure in the aneurysm models. Based on published aneurysm rupture studies, it is believed that ruptured aneurysms are usually correlated with lower wall shear stress values than unruptured ones. Therefore, this work concludes that the modified Krieger model is a potential candidate for providing better clinical relevance to aneurysm computational fluid dynamics simulations.
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Seshasayanan, Kannabiran, and Basile Gallet. "Dynamo saturation down to vanishing viscosity: strong-field and inertial scaling regimes." Journal of Fluid Mechanics 864 (February 13, 2019): 971–94. http://dx.doi.org/10.1017/jfm.2019.12.
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We present analytical examples of fluid dynamos that saturate through the action of the Coriolis and inertial terms of the Navier–Stokes equation. The flow is driven by a body force and is subject to global rotation and uniform sweeping velocity. The model can be studied down to arbitrarily low viscosity and naturally leads to the strong-field scaling regime for the magnetic energy produced above threshold: the magnetic energy is proportional to the global rotation rate and independent of the viscosity $\unicode[STIX]{x1D708}$. Depending on the relative orientations of global rotation and large-scale sweeping, the dynamo bifurcation is either supercritical or subcritical. In the supercritical case, the magnetic energy follows the scaling law for supercritical strong-field dynamos predicted on dimensional grounds by Pétrélis & Fauve (Eur. Phys. J. B, vol. 22, 2001, pp. 271–276). In the subcritical case, the system jumps to a finite-amplitude dynamo branch. The magnetic energy obeys a magneto-geostrophic scaling law (Roberts & Soward, Annu. Rev. Fluid Mech., vol. 4, 1972, pp. 117–154), with a turbulent Elsasser number of the order of unity, where the magnetic diffusivity of the standard Elsasser number appears to be replaced by an eddy diffusivity. In the absence of global rotation, the dynamo bifurcation is subcritical and the saturated magnetic energy obeys the equipartition scaling regime. We consider both the vicinity of the dynamo threshold and the limit of large distance from threshold to put these various scaling behaviours on firm analytical ground.
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Coulaud, O., P. Morel, and J. P. Caltagirone. "Numerical modelling of nonlinear effects in laminar flow through a porous medium." Journal of Fluid Mechanics 190 (May 1988): 393–407. http://dx.doi.org/10.1017/s0022112088001375.
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This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium. The method chosen is the numerical resolution of flow equations at a pore scale. The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base. For a given flow through this medium the pressure drop is evaluated numerically.The Navier-Stokes equations are discretized by the mixed finite-element method. The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation. The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & Pironneau.For Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law. For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's law.On the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media. In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied.
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Hussong, Jeanette, Wim-Paul Breugem, and Jerry Westerweel. "A continuum model for flow induced by metachronal coordination between beating cilia." Journal of Fluid Mechanics 684 (August 30, 2011): 137–62. http://dx.doi.org/10.1017/jfm.2011.282.
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AbstractIn this numerical study we investigate the flow induced by metachronal coordination between beating cilia arranged in a densely packed layer by means of a continuum model. The continuum approach allows us to treat the problem as two-dimensional as well as stationary, in a reference frame moving with the speed of the metachronal wave. The model is used as a computationally efficient design tool to investigate cilia-induced transport of a Newtonian fluid in a plane channel. Contrary to prior continuum models, the present approach accounts for spatial variations in the porosity along the metachronal wave and thus ensures conservation of mass within the cilia layer. Using porous-media theory the governing volume-averaged Navier–Stokes (VANS) equations are derived and closure formulations are given explicitly for the model. This makes it possible to investigate cilia-induced flow with a continuum model in both the viscous regime and the inertial regime. The results show that metachronal coordination can act as a transport mechanism in both regimes. Porosity variations appear to be the key mechanism for correct prediction of the fluid transport in the viscous flow regime. The reason is that spatial variations in the porosity break the symmetry of the drag distribution along the metachronal wave. A new insight that has been gained is that the fluid transport reverses, thus switches from plectic to antiplectic metachronism, for the same cilia beat cycle when the wavespeed is increased such that inertial effects occur. Based on a parameter study, the net transport in the channel is described by a power-law relation of the amplitude, length and speed of the metachronal wave. It is found that the wavelength has the strongest effect on the viscosity-dominated fluid transport.
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Djenidi, L., R. A. Antonia, and S. L. Tang. "Scale invariance in finite Reynolds number homogeneous isotropic turbulence." Journal of Fluid Mechanics 864 (February 7, 2019): 244–72. http://dx.doi.org/10.1017/jfm.2019.28.
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The problem of homogeneous isotropic turbulence (HIT) is revisited within the analytical framework of the Navier–Stokes equations, with a view to assessing rigorously the consequences of the scale invariance (an exact property of the Navier–Stokes equations) for any Reynolds number. The analytical development, which is independent of the 1941 (K41) and 1962 (K62) theories of Kolmogorov for HIT for infinitely large Reynolds number, is applied to the transport equations for the second- and third-order moments of the longitudinal velocity increment, $(\unicode[STIX]{x1D6FF}u)$. Once the normalised equations and the constraints required for complying with the scale-invariance property of the equations are presented, results derived from these equations and constraints are discussed and compared with measurements. It is found that the fluid viscosity, $\unicode[STIX]{x1D708}$, and the mean kinetic energy dissipation rate, $\overline{\unicode[STIX]{x1D716}}$ (the overbar denotes spatial and/or temporal averaging), are the only scaling parameters that make the equations scale-invariant. The analysis further leads to expressions for the distributions of the skewness and the flatness factor of $(\unicode[STIX]{x1D6FF}u)$ and shows that these distributions must exhibit plateaus (of different magnitudes) in the dissipative and inertial ranges, as the Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ increases indefinitely. Also, the skewness and flatness factor of the longitudinal velocity derivative become constant as $Re_{\unicode[STIX]{x1D706}}$ increases; this is supported by experimental data. Further, the analysis, backed up by experimental evidence, shows that, beyond the dissipative range, the behaviour of $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}$ with $n=2$, 3 and 4 cannot be represented by a power law of the form $r^{\unicode[STIX]{x1D701}_{n}}$ when the Reynolds number is finite. It is shown that only when $Re_{\unicode[STIX]{x1D706}}\rightarrow \infty$ can an $n$-thirds law (i.e. $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}\sim r^{\unicode[STIX]{x1D701}_{n}}$, with $\unicode[STIX]{x1D701}_{n}=n/3$) emerge, which is consistent with the onset of a scaling range.
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Zamansky, R., F. Coletti, M. Massot, and A. Mani. "Turbulent thermal convection driven by heated inertial particles." Journal of Fluid Mechanics 809 (November 10, 2016): 390–437. http://dx.doi.org/10.1017/jfm.2016.630.
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The heating of particles in a dilute suspension, for instance by radiation, chemical reactions or radioactivity, leads to local temperature fluctuations in the fluid due to the non-uniformity of the disperse phase. In the presence of a gravity field, the fluid is set in motion by the resulting buoyancy forces. When the particle density is different than that of the fluid, the fluid motion alters the spatial distribution of the particles and possibly strengthens their concentration inhomogeneities. This in turn causes more intense local heating. Direct numerical simulations in the Boussinesq limit show this feedback loop. Various regimes are identified depending on the particle inertia. For very small particle inertia, the macroscopic behaviour of the system is the result of many thermal plumes that are generated independently of each other. For significant particle inertia, clusters of particles are observed and their dynamics controls the flow. The emergence of very intermittent turbulent fluctuations shows that the flow is influenced by the larger structures (turbulent convection) as well as by the small-scale dynamics that affect particle segregation and thus the flow forcing. Assuming thermal equilibrium between the particles and the fluid (i.e. infinitely fast thermal relaxation of the particle), we investigate the evolution of statistical observables with the change of the main control parameters (namely the particle number density, the particle inertia and the domain size), and propose a scaling argument for these trends. Concerning the energy density in the spectral space, it is observed that the turbulent energy and temperature spectra follow a power law, the exponent of which varies continuously with the Stokes number. Furthermore, the study of the spectra of the temperature and momentum forcing (and thus of the concentration/temperature and velocity/temperature correlations) gives strong support to the proposed feedback loop mechanism. We then discuss the intermittency of the flow, and analyse the effect of relaxing some of the simplifying assumptions, thus assessing the relevance of the original studied configuration.
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Lin, Tsing-Fa, Tsai-Shou Chang, and Yu-Feng Chen. "Development of Oscillatory Asymmetric Recirculating Flow in Transient Laminar Opposing Mixed Convection in a Symmetrically Heated Vertical Channel." Journal of Heat Transfer 115, no. 2 (1993): 342–52. http://dx.doi.org/10.1115/1.2910685.
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Detailed flow and thermal characteristics in transient laminar opposing mixed convection in a vertical plane channel subject to a symmetric heat input are numerically investigated. First, a linear stability analysis was employed to evidence the occurrence of flow bifurcation. Then, the unsteady Navier–Stokes equations along with the continuity and energy equations were respectively integrated by a third-order upwind and power-law finite-difference scheme with the resulting matrices inverted by the Fast Fourier Transform and conjugated gradient methods. Reverse flow in the form of symmetric, elongated recirculating cells is initiated earlier and is stronger in a lower Prandtl number fluid with higher opposing buoyancy and Reynolds number and longer heated section length. At a high opposing buoyancy, sudden flow asymmetry and oscillation occur simultaneously in a nearly steady flow after the initial transient. Periodic flow and thermal evolution are noted in space and time. An empirical equation for the condition for inducing flow oscillation is proposed.
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Chang, Fon-Chieh, and John R. Hull. "Computer Modeling of Electromagnetic Fields and Fluid Flows for Edge Containment in Continuous Casting." Journal of Manufacturing Science and Engineering 127, no. 4 (2004): 724–30. http://dx.doi.org/10.1115/1.2039101.
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A computer model was developed to predict eddy currents and fluid flows in molten steel. The model was verified by comparing predictions with experimental results of liquid-metal containment and fluid flow in electromagnetic (EM) edge dams (EMDs) designed at Inland Steel (Ispat Industries Ltd.) for twin-roll casting. This mathematical model can greatly shorten casting research on the use of EM fields for liquid metal containment and control. It can also optimize the existing casting processes and minimize expensive, time-consuming full-scale testing. The model was verified by comparing predictions with experimental results of liquid metal containment and fluid flow in EM edge dams designed at Inland Steel (Ispat Industries Ltd.) for twin-roll casting. Numerical simulation was performed by coupling a three-dimensional (3D) finite-element EM code (ELEKTRA) and a 3D finite-difference fluids code (CaPS-EM) to solve Maxwell’s equations, Ohm’s law, Navier-Stokes equations, and transport equations of turbulence flow in a casting process that uses EM fields. ELEKTRA is able to predict the eddy-current distribution and EM forces in complex geometry. CaPS-EM is capable of modeling fluid flows with free surfaces and dynamic rollers. The computed 3D magnetic fields and induced eddy currents in ELEKTRA are used as input to flow-field computations in CaPS-EM. Results of the numerical simulation compared well with measurements obtained from both static and dynamic tests.
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Alkhedher, Mohammad, Pouyan Talebizadehsardari, Arameh Eyvazian, Afrasyab Khan, and Naeim Farouk. "Wave Dispersion Analysis of Fluid Conveying Nanocomposite Shell Reinforced by MWCNTs Considering the Effect of Waviness and Agglomeration Efficiency." Polymers 13, no. 1 (2021): 153. http://dx.doi.org/10.3390/polym13010153.
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The current paper is aimed to investigate the effects of waviness, random orientation, and agglomeration factor of nanoreinforcements on wave propagation in fluid-conveying multi-walled carbon nanotubes (MWCNTs)-reinforced nanocomposite cylindrical shell based on first-order shear deformable theory (FSDT). The effective mechanical properties of the nanocomposite cylindrical shell are estimated employing a combination of a novel form of Halpin-Tsai homogenization model and rule of mixture. Utilized fluid flow obeys Newtonian fluid law and it is axially symmetric and laminar flow and it is considered to be fully developed. The effect of flow velocity is explored by implementing Navier-Stokes equation. The kinetic relations of nanocomposite shell are calculated via FSDT. Moreover, the governing equations are derived using the Hamiltonian approach. Afterward, a method which solves problems analytically is applied to solve the obtained governing equations. Effects of a wide range of variants such as volume fraction of MWCNTs, radius to thickness ratio, flow velocity, waviness factor, random orientation factor, and agglomeration factor on the phase velocity and wave frequency of a fluid conveying MWCNTs-reinforced nanocomposite cylindrical shell were comparatively illustrated and the results were discussed in detail.
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HALDENWANG, P. "Laminar flow in a two-dimensional plane channel with local pressure-dependent crossflow." Journal of Fluid Mechanics 593 (November 23, 2007): 463–73. http://dx.doi.org/10.1017/s0022112007008622.
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Long ducts (or pipes) composed of transpiring (e.g. porous) walls are at the root of numerous industrial devices for species separation, as tangential filtration or membrane desalination. Similar configurations can also be involved in fluid supply systems, as irrigation or biological fluids in capillaries. A transverse leakage (or permeate flux), the strength of which is assumed to depend linearly on local pressure (as in Starling's law for capillary), takes place through permeable walls. All other dependences, as osmotic pressure or partial fouling due to polarization of species concentration, are neglected. To analyse this open problem we consider the simplest situation: the steady laminar flow in a two-dimensional channel composed of two symmetrical porous walls.First, dimensional analysis helps us to determine the relevant parameters. We then revisit the Berman problem that considers a uniform crossflow (i.e. pressure-independent leakage). We expand the solution in a series of Rt, the transverse Reynolds number. We note this series has a rapid convergence in the considered range of Rt (i.e. Rt ≤ O(1)). A particular method of variable separation then allows us to derive from the Navier–Stokes equations two new ordinary differential equations (ODE), which correspond to first and second orders in the development in Rt, whereas the zero order recovers the Regirer linear theory. Finally, both new ODEs are used to study the occurrence of two undesirable events in the filtration process: axial flow exhaustion (AFE) and crossflow reversal (CFR). This study is compared with a numerical approach.
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Moffatt, H. K., and Yoshifumi Kimura. "Towards a finite-time singularity of the Navier–Stokes equations Part 1. Derivation and analysis of dynamical system." Journal of Fluid Mechanics 861 (December 31, 2018): 930–67. http://dx.doi.org/10.1017/jfm.2018.882.
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The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity$\unicode[STIX]{x1D708}$is studied, starting from a finite-energy configuration of two vortex rings of circulation$\pm \unicode[STIX]{x1D6E4}$and radius$R$, symmetrically placed on two planes at angles$\pm \unicode[STIX]{x1D6FC}$to a plane of symmetry$x=0$. The minimum separation of the vortices,$2s$, and the scale of the core cross-section,$\unicode[STIX]{x1D6FF}$, are supposed to satisfy the initial inequalities$\unicode[STIX]{x1D6FF}\ll s\ll R$, and the vortex Reynolds number$R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the ‘tipping points’) is determined solely by the curvature$\unicode[STIX]{x1D705}(\unicode[STIX]{x1D70F})$at the tipping points and by$s(\unicode[STIX]{x1D70F})$and$\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$, where$\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D6E4}/R^{2})t$is a dimensionless time variable. The Biot–Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot–Savart law breaks down just before this singularity is realised, when$\unicode[STIX]{x1D705}s$and$\unicode[STIX]{x1D6FF}/\!s$become of order unity. The dynamical system admits ‘partial Leray scaling’ of just$s$and$\unicode[STIX]{x1D705}$, and ultimately full Leray scaling of$s,\unicode[STIX]{x1D705}$and$\unicode[STIX]{x1D6FF}$, conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter$\unicode[STIX]{x1D705}$; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot–Savart approach just before the incipient singularity is realised. The Euler flow situation ($\unicode[STIX]{x1D708}=0$) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.
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Mei, Lanju, Defu Cui, Jiayue Shen, et al. "Electroosmotic Mixing of Non-Newtonian Fluid in a Microchannel with Obstacles and Zeta Potential Heterogeneity." Micromachines 12, no. 4 (2021): 431. http://dx.doi.org/10.3390/mi12040431.
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This paper investigates the electroosmotic micromixing of non-Newtonian fluid in a microchannel with wall-mounted obstacles and surface potential heterogeneity on the obstacle surface. In the numerical simulation, the full model consisting of the Navier–Stokes equations and the Poisson–Nernst–Plank equations are solved for the electroosmotic fluid field, ion transport, and electric field, and the power law model is used to characterize the rheological behavior of the aqueous solution. The mixing performance is investigated under different parameters, such as electric double layer thickness, flow behavior index, obstacle surface zeta potential, obstacle dimension. Due to the zeta potential heterogeneity at the obstacle surface, vortical flow is formed near the obstacle surface, which can significantly improve the mixing efficiency. The results show that, the mixing efficiency can be improved by increasing the obstacle surface zeta potential, the flow behavior index, the obstacle height, the EDL thickness.
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Issakhov, Alibek, and Medina Imanberdiyeva. "Numerical Study of the Movement of Water Surface of Dam Break Flow by VOF Methods for Various Obstacles." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 5 (2020): 475–500. http://dx.doi.org/10.1515/ijnsns-2018-0278.
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AbstractIn this paper, the movement of the water surface is numerically simulated when a dam is broken by the volume of fluid (VOF) method. The mathematical model is based on the Navier–Stokes equations and uses the large eddy simulation turbulent model, describing the flow of an incompressible viscous fluid and the equation for the phase. These equations are discretized by the finite-volume method. Numerical PISO (Pressure-Implicit with Splitting of Operators) algorithm was chosen for numerical solution of this equation system. The movement of the water surface is captured by using the VOF method, which leads to a strict mass conservation law. The accuracy of the three-dimensional model and the chosen numerical algorithm were tested using several laboratory experiments on dam break problem. In each of the problems, the obtained results were compared with the experimental data and several calculations by other authors and in each of the test problems, the developed model showed results close to the experimental data. Comparison of simulation results with experimental data for various turbulent models was also performed. And also two combined problems were performed which are more close to real conditions; with the help of these problems, flooding zones and flooding time were identified that would help in evacuating people from dangerous zones.
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Ray, Baidurja, and Lance R. Collins. "Investigation of sub-Kolmogorov inertial particle pair dynamics in turbulence using novel satellite particle simulations." Journal of Fluid Mechanics 720 (February 27, 2013): 192–211. http://dx.doi.org/10.1017/jfm.2013.24.
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AbstractClustering (or preferential concentration) of weakly inertial particles suspended in a homogeneous isotropic turbulent flow is driven primarily by the smallest eddies at the so-called Kolmogorov scale. In particle-laden large-eddy simulations (LES), these small scales are not resolved by the grid and hence their effect on both the resolved flow scales and the particle motion have to be modelled. In order to predict clustering in a particle-laden LES, it is crucial that the subgrid model for the particles captures the mechanism by which the subgrid scales affect the particle motion (Ray & Collins, J. Fluid Mech., vol. 680, 2011, pp. 488–510). In this paper, we describe novel satellite particle simulations (SPS), in which we study the clustering and relative velocity statistics of inertial particles at separation distances well below the Kolmogorov length scale. SPS is designed to isolate pairwise interactions of particles, and is therefore well suited for developing two-particle models. We show that the power-law dependence of the radial distribution function (RDF), a statistical measure of clustering, is predicted by the SPS in excellent agreement with direct numerical simulations (DNS) for Stokes numbers up to 3, implying that no explicit information from the inertial range is required to accurately describe particle clustering. This result further explains our successful prediction of the RDF power using the drift-diffusion model of Chun et al. (J. Fluid Mech., vol. 536, 2005, pp. 219–251) for $\mathit{St}\leq 0. 4$. We also consider the second-order longitudinal relative velocity structure function for the particles; we show that the SPS is able to capture its power-law exponent for $\mathit{St}\leq 0. 5$ and attribute the disagreement at larger $\mathit{St}$ to the effect of the larger scales of motion not captured by the SPS. Further, the SPS is able to capture the ‘caustic activation’ of the structure function at zero separation and predict the critical $\mathit{St}$ and rate of activation in agreement with the DNS (Salazar & Collins, J. Fluid. Mech., vol. 696, 2012, pp. 45–66). We show comparisons between filtered DNS and equivalently filtered SPS, and the findings are similar to the unfiltered case. Overall, SPS is an efficient and accurate computational tool for investigating particle pair dynamics at small separations, as well as an interesting platform for developing LES subgrid models designed to accurately reproduce particle clustering.
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Zambrano, Miller, Alan D. Pitts, Ali Salama, Tiziano Volatili, Maurizio Giorgioni, and Emanuele Tondi. "Analysis of Fracture Roughness Control on Permeability Using SfM and Fluid Flow Simulations: Implications for Carbonate Reservoir Characterization." Geofluids 2019 (April 23, 2019): 1–19. http://dx.doi.org/10.1155/2019/4132386.
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Fluid flow through a single fracture is traditionally described by the cubic law, which is derived from the Navier-Stokes equation for the flow of an incompressible fluid between two smooth-parallel plates. Thus, the permeability of a single fracture depends only on the so-called hydraulic aperture which differs from the mechanical aperture (separation between the two fracture wall surfaces). This difference is mainly related to the roughness of the fracture walls, which has been evaluated in previous works by including a friction factor in the permeability equation or directly deriving the hydraulic aperture. However, these methodologies may lack adequate precision to provide valid results. This work presents a complete protocol for fracture surface mapping, roughness evaluation, fracture modeling, fluid flow simulation, and permeability estimation of individual fracture (open or sheared joint/pressure solution seam). The methodology includes laboratory-based high-resolution structure from motion (SfM) photogrammetry of fracture surfaces, power spectral density (PSD) surface evaluation, synthetic fracture modeling, and fluid flow simulation using the Lattice-Boltzmann method. This work evaluates the respective controls on permeability exerted by the fracture displacement (perpendicular and parallel to the fracture walls), surface roughness, and surface pair mismatch. The results may contribute to defining a more accurate equation of hydraulic aperture and permeability of single fractures, which represents a pillar for the modeling and upscaling of the hydraulic properties of a geofluid reservoir.
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Sadeghi, H., M. Oberlack, and M. Gauding. "On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation." Journal of Fluid Mechanics 854 (September 6, 2018): 233–60. http://dx.doi.org/10.1017/jfm.2018.625.
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A temporally evolving turbulent plane jet is studied both by direct numerical simulation (DNS) and Lie symmetry analysis. The DNS is based on a high-order scheme to solve the Navier–Stokes equations for an incompressible fluid. Computations were conducted at Reynolds number $\mathit{Re}_{0}=8000$, where $\mathit{Re}_{0}$ is defined based on the initial jet thickness, $\unicode[STIX]{x1D6FF}_{0.5}(0)$, and the initial centreline velocity, $\overline{U}_{1}(0)$. A symmetry approach, known as the Lie group, is used to find symmetry transformations, and, in turn, group invariant solutions, which are also denoted as scaling laws in turbulence. This approach, which has been extensively developed to create analytical solutions of differential equations, is presently applied to the mean momentum and two-point correlation equations in a temporally evolving turbulent plane jet. The symmetry analysis of these equations allows us to derive new invariant (self-similar) solutions for the mean flow and higher moments of the velocities in the jet flow. The current DNS validates the consequence of Lie symmetry analysis and therefore confirms the establishment of novel scaling laws in turbulence. It is shown that the classical scaling law for the mean velocity is a specific form of the current scaling (which has a more general form); however, the scaling for the second and higher moments (such as Reynolds stresses) has a completely different structure compared to the classical scaling. While the failure of the classical scaling for the second moments of the fluctuating velocities has been noted from the jet data for many years, the DNS results nicely match with the present self-similar relations derived from Lie symmetry analysis. Key ingredients for the present results, in particular for the scaling laws of the higher moments, are symmetries, which are of a purely statistical nature. i.e. these symmetries are admitted by the moment equations, however, they are not observed by the original Navier–Stokes equations.
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Boussaha, Bilal, Mustapha Lahmar, Benyebka Bou-Said, and Hamid Boucherit. "Non-Newtonian couple-stress squeeze film behaviour between oscillating anisotropic porous circular discs with sealed boundary." Mechanics & Industry 21, no. 3 (2020): 311. http://dx.doi.org/10.1051/meca/2020004.
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The thrust of this paper is to investigate theoretically the non-Newtonian couple stress squeeze film behaviour between oscillating circular discs based on V. K. Stokes micro-continuum theory. The lubricant squeezed out between parallel porous and rigid facings is supposed to be a concentrated suspension which consists of small particles dispersed in a Newtonian base fluid (solvent). The effective viscosity of the suspension is determined by using the Krieger-Dougherty viscosity model for a given volume fraction of particles in the base fluid. For low frequency and amplitude of sinusoidal squeezing where cavitation as well as turbulence are unlikely, the governing equations including the modified Reynolds equation coupled with the modified Darcy's equation are derived and solved numerically using the finite difference method and a sub-relaxed iterative procedure. The slip velocity at the porous-fluid interface is directly evaluated by means of the modified Darcy's law considering laminar and isothermal squeezing flow. For a given volume fraction, the couple stress effects on the squeeze film characteristics are analyzed through the dimensionless couple stress parameter ℓ˜ considering sealed and unsealed boundary of the porous disc. The obtained relevant results reveal that the use of couple stress suspending fluids as lubricants and the effect of sealing the boundary of the porous matrix improves substantially the squeeze film behaviour by increasing the squeeze film force. On the other hand, side leakage flow calculated in the sealed case remains constant in comparison to that of open end (unsealed) porous disc for all values of couple stress parameter and volume fraction of particle.
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Singh, Tej Pratap, Amitesh Kumar, and Ashok Kumar Satapathy. "Fluid flow analysis of a turbulent offset jet impinging on a wavy wall surface." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 234, no. 2 (2019): 544–63. http://dx.doi.org/10.1177/0954406219880209.
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The fluid flow characteristics of a turbulent offset jet impinging on a wavy wall surface has been investigated numerically. Two-dimensional Reynolds-averaged Navier–Stokes (RANS) equations are solved by the finite volume method. In the governing differential equations, the convective and diffusive terms are discretized by the power law upwind scheme and second-order central difference, respectively. The semi-implicit method for pressure linked equation algorithm is utilized to link the pressure to the velocity. The offset ratio is set to 7.0 and the Reynolds number is fixed to 15,000. The width of the jet is taken as the characteristic length. The amplitude of the wavy wall surface is varied from 0.1 to 0.7 with an interval of 0.1 and the number of cycle is fixed to 10. The results of fluid flow and turbulent characteristics of the offset jet are presented in the form of contours of streamline, velocity vector, turbulent kinetic energy, dissipation rate, pressure, and Reynolds shear stress. The variation in integral constant of momentum flux, wall shear stress, and pressure along the wall is presented and also compared. The decay in the maximum streamwise velocity in the downstream direction and jet half-width along the streamwise direction are also presented and discussed. The wavy surface introduces some remarkable features, which are not present in a normal plane wall case. These features have been discussed in detail.
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VOLFSON, DMITRI, and JORGE VIÑALS. "Flow induced by a randomly vibrating boundary." Journal of Fluid Mechanics 432 (April 10, 2001): 387–408. http://dx.doi.org/10.1017/s0022112001003585.
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We study the flow induced by random vibration of a solid boundary in an otherwise quiescent fluid. The analysis is motivated by experiments conducted under the low level and random effective acceleration field that is typical of a microgravity environment. When the boundary is planar and is being vibrated along its own plane, the variance of the velocity field decays as a power law of distance away from the boundary. If a low-frequency cut-off is introduced in the power spectrum of the boundary velocity, the variance decays exponentially for distances larger than a Stokes layer thickness based on the cut-off frequency. Vibration of a gently curved boundary results in steady streaming in the ensemble average of the tangential velocity. Its amplitude diverges logarithmically with distance away from the boundary, but asymptotes to a constant value instead if a low-frequency cut-off is considered. This steady component of the velocity is shown to depend logarithmically on the cut-off frequency. Finally, we consider the case of a periodically modulated solid boundary that is being randomly vibrated. We find steady streaming in the ensemble average of the first-order velocity, with flow extending up to a characteristic distance of the order of the boundary wavelength. The structure of the flow in the vicinity of the boundary depends strongly on the correlation time of the boundary velocity.
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FIGUEROA-ESPINOZA, BERNARDO, and JEAN FABRE. "Taylor bubble moving in a flowing liquid in vertical channel: transition from symmetric to asymmetric shape." Journal of Fluid Mechanics 679 (May 19, 2011): 432–54. http://dx.doi.org/10.1017/jfm.2011.159.
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The velocity and shape of Taylor bubbles moving in a vertical channel in a Poiseuille liquid flow were studied for the inertial regime, characterized by large Reynolds numbers. Numerical experiments were carried out for positive (upward) and negative (downward) liquid mean velocity. Previous investigations in tube have reported that for upward flow the bubble is symmetric and its velocity follows the law of Nicklin, whereas for certain downward flow conditions the symmetry is broken and the bubble rises appreciably faster. To study the bubble motion and to identify the existence of a transition, a two-dimensional numerical code that solves the Navier–Stokes equations (through a volume of fluid implementation) was used to obtain the bubble shape and the rise velocity for different liquid mean velocities. A reference frame located at the bubble tip and an irregular grid were implemented to allow long simulation times without an excessively large numerical domain. It was observed that whenever the mean liquid velocity exceeded some critical value, bubbles adopted a symmetric final shape even though their initial shape was asymmetric. Conversely, if the mean liquid velocity was smaller than the critical value, a transition to a non-symmetric shape occurred, along with a correspondingly faster velocity. It was also found that surface tension has a stabilizing effect on the transition.
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Jánosdeák, Egon. "The limit of validity of the Newtonian fluid friction law in flows around vehicles and the limit of validity of the Navier-Stokes equations." Periodica Polytechnica Transportation Engineering 40, no. 2 (2012): 67. http://dx.doi.org/10.3311/pp.tr.2012-2.04.
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Zare, A., H. Emdad, and E. Goshtasbirad. "Feedback control of laminar flow behind backward-facing step by POD analysis and using perturbed Navier–Stokes equations." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 226, no. 3 (2011): 648–59. http://dx.doi.org/10.1177/0954406211416051.
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The purpose of this article is to design a reduced order model based on the proper orthogonal decomposition/Galerkin projection and perturbation method to develop a non-autonomous model. The resulting model can be used in optimal control of flow over backward-facing step. The main disadvantage of the proper orthogonal decomposition approach for control purposes is that, controlling parameters or inputs do not show up explicitly in the resulting reduced order system. The perturbation method can solve this problem and insert control inputs in the resulting system. The resulting system captures the time-varying influence of the controlling parameters and precisely predicts the Navier–Stokes response to external excitations. At last, optimal control theory is introduced to design a control law for a non-linear forced reduced model, which attempts to minimize the vorticity content in the fluid domain. The test bed is laminar flow behind backward-facing step [Formula: see text] actuated by a pair of blowing/suction jets. Results show that the wall jet can significantly influence the flow field and delay separation, while the perturbation method can predict the flow field in an accurate manner. The method is also found to be fast and efficient in computational time.
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Sahu, Krishnkant, and Satish C. Sharma. "Influence of bearing surface irregularities on hybrid slot-entry journal bearing with electrically conducting lubricant." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 234, no. 8 (2019): 1185–207. http://dx.doi.org/10.1177/1350650119896195.
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This study concerns with the numerical simulation of a hybrid slot entry journal bearing lubricated with electrically conducting lubricant under the influence of magnetic field for both thermal and isothermal conditions. The Navier–Stokes equation has been used to formulate the flow of electrically conducting lubricant through slot restrictor and combining the Lorentz force in the equations of motion, together with the Ohm’s law and Maxwell equations. Further, the effect of surface irregularities on bearing surface is considered to analyse the performance of the slot-entry bearing. The surface irregularities asperity profile has been modelled in both axial as well as circumferential directions. Finite element method is used to solve the Modified MHD Reynolds equation. To compute the bearing performance characteristic parameters, a MATLAB source code based on Gauss–Seidel iteration method has been developed. A comparative numerical analysis has been carried out for an electrically conducting lubricant, Newtonian lubricant, bearing surface having irregularities and bearing with smooth surface. The numerically simulated results indicate that considering the bearing surface irregularities and MHD effects enhances the value of fluid film damping coefficients [Formula: see text] and the value of minimum fluid film thickness [Formula: see text].
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Harding, Brendan. "Convergence analysis of inertial lift force estimates using the finite element method." ANZIAM Journal 60 (July 4, 2019): C65—C78. http://dx.doi.org/10.21914/anziamj.v60i0.14094.
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We conduct a convergence analysis for the estimation of inertial lift force on a spherical particle suspended in flow through a straight square duct using the finite element method. Specifically, we consider the convergence of an inertial lift force approximation with respect to a range of factors including the truncation of the domain, the resolution of the tetrahedral mesh and the boundary conditions imposed at the (truncated) ends of the domain. Additionally, we compare estimates obtained via the Lorentz reciprocal theorem with those obtained via a direct integration of fluid stress over the particle surface. 
 
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Rani, Sarma L., Vijay K. Gupta, and Donald L. Koch. "Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 1. Drift and diffusion flux closures." Journal of Fluid Mechanics 871 (May 22, 2019): 450–76. http://dx.doi.org/10.1017/jfm.2019.204.
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In this two-part study, we present the development and analysis of a stochastic theory for characterizing the relative positions of monodisperse, low-inertia particle pairs that are settling rapidly in homogeneous isotropic turbulence. In the limits of small Stokes number and Froude number such that $Fr\ll St_{\unicode[STIX]{x1D702}}\ll 1$, closures are developed for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions. The theory focuses on the relative motion of particle pairs in the dissipation regime of turbulence, i.e. for pair separations smaller than the Kolmogorov length scale. In this regime, the theory approximates the fluid velocity field in a reference frame following the primary particle as locally linear. In this part 1 paper, we present the derivation of closure approximations for the drift and diffusion fluxes in the p.d.f. equation for pair relative positions $\boldsymbol{r}$. The drift flux contains the time integral of the third and fourth moments of the ‘seen’ fluid velocity gradients along the trajectories of primary particles. These moments may be analytically resolved by making approximations regarding the ‘seen’ velocity gradient. Accordingly, two closure forms are derived specifically for the drift flux. The first invokes the assumption that the fluid velocity gradient along particle trajectories has a Gaussian distribution. In the second drift closure, we account for the correlation time scales of dissipation rate and enstrophy by decomposing the velocity gradient into the strain-rate and rotation-rate tensors scaled by the turbulent dissipation rate and enstrophy, respectively. An analytical solution to the p.d.f. $\langle P\rangle (r,\unicode[STIX]{x1D703})$ is then derived, where $\unicode[STIX]{x1D703}$ is the spherical polar angle. It is seen that the p.d.f. has a power-law dependence on separation $r$ of the form $\langle P\rangle (r,\unicode[STIX]{x1D703})\sim r^{\unicode[STIX]{x1D6FD}}$ with $\unicode[STIX]{x1D6FD}\sim St_{\unicode[STIX]{x1D702}}^{2}$ and $\unicode[STIX]{x1D6FD}<0$, analogous to that for the radial distribution function of non-settling pairs. An explicit expression is derived for $\unicode[STIX]{x1D6FD}$ in terms of the drift and diffusion closures. The $\langle P\rangle (r,\unicode[STIX]{x1D703})$ solution also shows that, for a given $r$, the clustering of $St_{\unicode[STIX]{x1D702}}\ll 1$ particles is only weakly anisotropic, which is in conformity with prior observations from direct numerical simulations of isotropic turbulence containing settling particles.