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Academic literature on the topic 'Mixing equations'

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Published: 25 May 2024

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Journal articles on the topic "Mixing equations"

1

Brigham, W. E. "Mixing Equations in Various Geometries." SPE Reservoir Engineering 1, no. 02 (March 1, 1986): 203–8. http://dx.doi.org/10.2118/4585-pa.

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Blasone, M., P. Jizba, and L. Smaldone. "Schwinger-Dyson equations and flavor mixing." Journal of Physics: Conference Series 1071 (August 2018): 012003. http://dx.doi.org/10.1088/1742-6596/1071/1/012003.

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Łoskot, K., and R. Rudnicki. "Mixing and some integro-functional equations." Aequationes Mathematicae 40, no. 1 (December 1990): 78–82. http://dx.doi.org/10.1007/bf02112282.

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Kostić, Marko. "Hypercyclic and topologically mixing properties of abstract timefractional equations with discrete shifts." Sarajevo Journal of Mathematics 9, no. 2 (November 2013): 257–69. http://dx.doi.org/10.5644/sjm.09.2.10.

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Eggl, Maximilian F., and Peter J. Schmid. "Shape optimization of stirring rods for mixing binary fluids." IMA Journal of Applied Mathematics 85, no. 5 (May 14, 2020): 762–89. http://dx.doi.org/10.1093/imamat/hxaa012.

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Abstract Mixing is an omnipresent process in a wide range of industrial applications, which supports scientific efforts to devise techniques for optimizing mixing processes under time and energy constraints. In this endeavour, we present a computational framework based on nonlinear direct-adjoint looping for the enhancement of mixing efficiency in a binary fluid system. The governing equations consist of the nonlinear Navier–Stokes equations, complemented by an evolution equation for a passive scalar. Immersed and moving stirrers are treated by a Brinkman penalization technique, and the full system of equations is solved using a Fourier-based pseudospectral approach. The adjoint equations provide gradient and sensitivity information which is in turn used to improve an initial mixing strategy, based on shape, rotational and path modifications. We utilize a Fourier-based approach for parameterizing and optimizing the embedded stirrers and consider a variety of geometries to achieve enhanced mixing efficiency. We consider a restricted optimization space by limiting the time for mixing and the rotational velocities of all stirrers. In all cases, non-intuitive shapes are found which produce significantly enhanced mixing efficiency.

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XUE, SHE-SHENG. "NEUTRINO MASSES AND MIXINGS." Modern Physics Letters A 14, no. 39 (December 21, 1999): 2701–8. http://dx.doi.org/10.1142/s0217732399002844.

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We propose a novel theoretical understanding of neutrino masses and mixings, which is attributed to the left–right symmetric feature of the regularized standard model at short distances. We try to explain the smallness of Dirac neutrino masses and the decoupling of the right-handed neutrino as a free particle. Neutrino masses and mixing angles are completely related to each other in the Schwinger–Dyson equations for their self-energy functions. The solutions to these equations and a possible pattern of masses and mixings are discussed.

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Hall, Kenneth R., Gustavo A. Iglesias-Silva, and G. Ali Mansoori. "Quadratic mixing rules for equations of state." Fluid Phase Equilibria 91, no. 1 (November 1993): 67–76. http://dx.doi.org/10.1016/0378-3812(93)85079-2.

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Kirkup, S. M., M. Wadsworth, D. G. Armour, R. Badheka, and J. A. Van Den Berg. "Computational solution of the atomic mixing equations." International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 11, no. 4 (July 1998): 189–205. http://dx.doi.org/10.1002/(sici)1099-1204(199807/08)11:4<189::aid-jnm301>3.0.co;2-a.

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Chitsaz, Mahzad, and Mani Fathali. "Effect of External Magnetic Field on Dynamics of Two-dimensional Isotropic Conducting Flow." E3S Web of Conferences 95 (2019): 02003. http://dx.doi.org/10.1051/e3sconf/20199502003.

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In this article, the impact of external uniform magnetic field on the dynamic characteristics and mixing parameters of two-dimensional isotropic magnetohydrodynamic (MHD) flow is investigated. For this purpose, the direct numerical simulation (DNS) is applied to two-dimensional incompressible Navier-Stokes and magnetic induction equations by pseudo-spectral method. Governing equations are considered in the N-S vorticity equations to guarantee the incompressibility conditions and remove the pressure term from equations. The Results of the calculations show that the deformation of vortexes by external magnetic field reduces the mixing efficiency. It is also demonstrated that in MHD flow the energy is exchanged by Lorentz force between the flow and the magnetic field in such a way that the kinetic energy decreases and consequently mixing of the fluid is reduced. This energy transfer causes reduction of viscous dissipation of energy and mixing efficiency, despite increasing the total dissipated energy rate.

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Lozovatsky, I. D., and H. J. S. Fernando. "Mixing efficiency in natural flows." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1982 (January 13, 2013): 20120213. http://dx.doi.org/10.1098/rsta.2012.0213.

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It is argued that the mixing efficiency of naturally occurring stratified shear flows, γ = Rf /(1− Rf ), where Rf is the flux Richardson number, is dependent on at least two governing parameters: the gradient Richardson number Ri and the buoyancy Reynolds number Re b = ε / vN 2 . It is found that, in the range approximately 0.03< Ri <0.4, which spans 10 4 < Re b <10 6 , the mixing efficiency obtained via direct measurements of fluxes and property gradients in the stable atmospheric boundary layer and homogeneous/stationary balance equations of turbulent kinetic energy (TKE) is nominally similar to that evaluated using the scalar balance equations. Outside these Ri and Re b ranges, the commonly used flux-estimation methodology based on homogeneity and stationarity of TKE equations breaks down (e.g. buoyancy effects are unimportant, energy flux divergence is significant or flow is non-stationary). In a wide range, 0.002< Ri <1, the mixing efficiency increases with Ri , but decreases with Re b . When Ri is in the proximity of Ri cr ∼0.1–0.25, γ can be considered a constant γ ≈0.16–0.2. The results shed light on the wide variability of γ noted in previous studies.

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Dissertations / Theses on the topic "Mixing equations"

1

Johnson, A. E. "The effects of curvature and divergence on turbulent mixing layers." Thesis, University of Surrey, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.279304.

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Pibal, Douglas J. "Development and validation of MM5 MOS-based forecast equations for mixing height." abstract and full text PDF (free order & download UNR users only), 2007. http://0-gateway.proquest.com.innopac.library.unr.edu/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:1446430.

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McCabe, Ryan Matthew. "Small-scale coastal dynamics and mixing from a Lagrangian perspective /." Thesis, Connect to this title online; UW restricted, 2008. http://hdl.handle.net/1773/10963.

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Akhtar, Kareem. "Numerical Investigation using RANS Equations of Two-dimensional Turbulent Jets and Bubbly Mixing layers." Thesis, Virginia Tech, 2010. http://hdl.handle.net/10919/34512.

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This thesis presents numerical investigations of two-dimensional single-phase turbulent jets and bubbly mixing layers using Reynolds-Averaged Navier-Stokes (RANS) equations. The behavior of a turbulent jet confined in a channel depends on the Reynolds number and geometry of the channel which is given by the expansion ratio (channel width to jet thickness) and offset ratio (eccentricity of the jet entrance). Steady solutions to the RANS equations for a two-dimensional turbulent jet injected in the middle of a channel have been obtained. When no entrainment from the channel base is allowed, the flow is asymmetric for a wide range of expansion ratio at high Reynolds number. The jet attaches to one of the channel side walls. The attachment length increases linearly with the channel width for fixed value of Reynolds number. The attachment length is also found to be independent of the (turbulent) jet Reynolds number for fixed expansion ratio. By simulating half of the channel and imposing symmetry, we can construct a steady symmetric solution to the RANS equations. This implies that there are possibly two solutions to the steady RANS equations, one is symmetric but unstable, and the other solution is asymmetric (the jet attaches to one of the side walls) but stable. A symmetric solution is also obtained if entrainment from jet exit plane is permitted. Fearn et al. (Journal of Fluid Mechanics, vol. 121, 1990) studied the laminar problem, and showed that the flow asymmetry of a symmetric expansion arises at a symmetry-breaking bifurcation as the jet Reynolds number is increased from zero. In the present study the Reynolds number is high and the jet is turbulent. Therefore, a symmetry-breaking bifurcation parameter might be the level of entrainment or expansion ratio. The two-dimensional turbulent bubbly mixing layer, which is a multiphase problem, is investigated using RANS based models. Available experimental data show that the spreading rate of turbulent bubbly mixing layers is greater than that of the corresponding single phase flow. The presence of bubbles also increases the turbulence level. The global structure of the flow proved to be sensitive to the void fraction. The present RANS simulations predict this behavior, but different turbulence models give different spreading rates. There is a significant difference in turbulence kinetic energy between numerical predictions and experimental data. The models tested include k-Îµ, shear-stress transport (SST), and Reynolds stress transport (SSG) models. All tested turbulence models under predict the spreading rate of the bubbly mixing layer, even though they accurately predict the spreading rate for single phase flow. The best predictions are obtained by using SST model.
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Morlock, Merlin B. "Nonlinear mixing of two collinear Rayleigh waves." Thesis, Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/50280.

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Nonlinear mixing of two collinear, initially monochromatic, Rayleigh waves propagating in the same direction in an isotropic, nonlinear elastic solid is investigated: analytically, by finite element method simulations and experimentally. In the analytical part, it is shown that only collinear mixing in the same direction fulfills the phase matching condition based on Jones and Kobett 1963 for the resonant generation of the second harmonics, as well as the sum and difference frequency components caused by the interaction of the two fundamental waves. Next, a coupled system of ordinary differential equations is derived based on the Lagrange equations of the second kind for the varying amplitudes of the higher harmonic and combination frequency components of the fundamentals waves. Numerical results of the evolution of the amplitudes of these frequency components over the propagation distance are provided for different ratios of the fundamental wave frequencies. It is shown that the energy transfer is larger for higher frequencies, and that the oscillation of the energy between the different frequency components depends on the amplitudes and frequencies of the fundamental waves. Furthermore, it is illustrated that the horizontal velocity component forms a shock wave while the vertical velocity component forms a pulse in the case of low attenuation. This behavior is independent of the two fundamental frequencies and amplitudes that are mixed. The analytical model is then extended by implementing diffraction effects in the parabolic approximation. To be able to quantify the acoustic nonlinearity parameter, β, general relations based on the plane wave assumption are derived. With these relations a β is expressed, that is analog to the β for longitudinal waves, in terms of the second harmonics and the sum and the difference frequencies. As a next step, frequency and amplitude ratios of the fundamental frequencies are identified, which provide a maximum amplitude of one of the second harmonics as well as the sum or difference frequency components to enhance experimental results. Subsequently, the results of the analytical model are compared to those of finite element method simulations. Two dimensional simulations for small propagation distances gave similar results for analytical and finite element simulations. Consquently. this shows the validity of the analytical model for this setup. In order to demonstrate the feasibility of the mixing technique and of the models, experiments were conducted using a wedge transducer to excite mixed Rayleigh waves and an air-coupled transducer to detect the fundamentals, second harmonics and the sum frequency. Thus, these experiments yield more physical information compared to the case of using a single fundamental wave. Further experiments were conducted that confirm the modeled dependence on the amplitudes of the generated waves. In conclusion, the results of this research show that it is possible to measure the acoustic nonlinearity parameter β to quantify material damage by mixing Rayleigh waves on up to four ways.

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Sommerville, Lesley Laverne. "A Parabolized navier-stokes model for static mixers." Thesis, Georgia Institute of Technology, 1993. http://hdl.handle.net/1853/19535.

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Helgesen, James Karl. "Particle mixing and diffusion in the turbulent wake of cylinder arrays." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/11227.

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Aroza, Benlloch Javier. "Dynamics of strongly continuous semigroups associated to certain differential equations." Doctoral thesis, Universitat Politècnica de València, 2015. http://hdl.handle.net/10251/57186.

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[EN] The purpose of the Ph.D. Thesis "Dynamics of strongly continuous semigroups associated to certain differential equations'' is to analyse, from the point of view of functional analysis, the dynamics of solutions of linear evolution equations. These solutions can be represented by a strongly continuous semigroup on an infinite-dimensional Banach space. The aim of our research is to provide global conditions for chaos, in the sense of Devaney, and stability properties of strongly continuous semigroups which are solutions of linear evolution equations. This work is composed of three principal chapters. Chapter 0 is introductory and defines basic terminology and notation used, besides presenting the basic results that we will use throughout this thesis. Chapters 1 and 2 describe, in general way, a strongly continuous semigroup induced by a semiflow in Lebesgue and Sobolev spaces which is a solution of a linear first order partial differential equation. Moreover, some characterizations of the main dynamical properties, including hypercyclicity, mixing, weakly mixing, chaos and stability are given along these chapters. Chapter 3 describes the dynamical properties of a difference equation based on the so-called birth-and-death model and analyses the conditions previously proven for this model improving them by employing a different strategy. The goal of this thesis is to characterize dynamical properties of these kind of strongly continuous semigroups in a general way, whenever possible, and to extend these results to another spaces. Along this memory, these findings are compared with the previous ones given by many authors in recent years.
[ES] La presente memoria "Dinámica de semigrupos fuertemente continuos asociadas a ciertas ecuaciones diferenciales'' es analizar, desde el punto de vista del análisis funcional, la dinámica de las soluciones de ecuaciones de evolución lineales. Estas soluciones pueden ser representadas por semigrupos fuertemente continuos en espacios de Banach de dimensión infinita. El objetivo de nuestra investigación es proporcionar condiciones globales para obtener caos, en el sentido de Devaney, y propiedades de estabilidad de semigrupos fuertemente continuos, los cuales son soluciones de ecuaciones de evolución lineales. Este trabajo está compuesto de tres capítulos principales. El Capítulo 0 es introductorio y define la terminología básica y notación usada, además de presentar los resultados básicos que usaremos a lo largo de esta tesis. Los Capítulos 1 y 2 describen, de forma general, un semigrupo fuertemente continuo inducido por un semiflujo en espacios de Lebesgue y en espacios de Sobolev, los cuales son solución de una ecuación diferencial lineal en derivadas parciales de primer orden. Además, algunas caracterizaciones de las principales propiedades dinámicas, incluyendo hiperciclicidad, mezclante, débil mezclante, caos y estabilidad, se obtienen a lo largo de estos capítulos. El Capítulo 3 describe las propiedades dinámicas de una ecuación en diferencias basada en el llamado modelo de nacimiento-muerte y analiza las condiciones previamente probadas para este modelo, mejorándolas empleando una estrategia diferente. La finalidad de esta tesis es caracterizar propiedades dinámicas para este tipo de semigrupos fuertemente continuos de forma general, cuando sea posible, y extender estos resultados a otros espacios. A lo largo de esta memoria, estos resultados son comparados con los resultados previos dados por varios autores en años recientes.
[CAT] La present memòria "Dinàmica de semigrups fortament continus associats a certes equacions diferencials'' és analitzar, des del punt de vista de l'anàlisi funcional, la dinàmica de les solucions d'equacions d'evolució lineals. Aquestes solucions poden ser representades per semigrups fortament continus en espais de Banach de dimensió infinita. L'objectiu de la nostra investigació es proporcionar condicions globals per obtenir caos, en el sentit de Devaney, i propietats d'estabilitat de semigrups fortament continus, els quals són solucions d'equacions d'evolució lineals. Aquest treball està compost de tres capítols principals. El Capítol 0 és introductori i defineix la terminologia bàsica i notació utilitzada, a més de presentar els resultats bàsics que utilitzarem al llarg d'aquesta tesi. Els Capítols 1 i 2 descriuen, de forma general, un semigrup fortament continu induït per un semiflux en espais de Lebesgue i en espais de Sobolev, els quals són solució d'una equació diferencial lineal en derivades parcials de primer ordre. A més, algunes caracteritzacions de les principals propietats dinàmiques, incloent-hi hiperciclicitat, mesclant, dèbil mesclant, caos i estabilitat, s'obtenen al llarg d'aquests capítols. El Capítol 3 descrivís les propietats dinàmiques d'una equació en diferències basada en el model de naixement-mort i analitza les condicions prèviament provades per aquest model, millorant-les utilitzant una estratègia diferent. La finalitat d'aquesta tesi és caracteritzar propietats dinàmiques d'aquest tipus de semigrups fortament continus de forma general, quan siga possible, i estendre aquests resultats a altres espais. Al llarg d'aquesta memòria, aquests resultats són comparats amb els resultats previs obtinguts per diversos autors en anys recents.
Aroza Benlloch, J. (2015). Dynamics of strongly continuous semigroups associated to certain differential equations [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/57186
TESIS

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Ostatnicky, Tomas. "Model calculation of four-wave mixing polarization and dynamics in bulk and confined semiconductors." Université Louis Pasteur (Strasbourg) (1971-2008), 2005. https://publication-theses.unistra.fr/public/theses_doctorat/2005/OSTATNICKY_Tomas_2005.pdf.

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Lee, Wei-Koon. "Chaotic mixing in wavy-type channels and two-layer shallow flows." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:f5fcbe34-babb-4fae-9204-28de8774eb98.

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This thesis examines chaotic mixing in wavy-type channels and two-layer shallow water flow. For wavy-type channels, the equations of motion for vortices and fluid particles are derived assuming two-dimensional irrotational, incompressible flow. Instantaneous positions of the vortices and particles are determined using Lagrangian tracking, and are conformally mapped to the physical domain. Unsteady vortex motion is analysed, and vortex-induced chaotic mixing in the channels studied. The dynamics of mixing associated with the evolution of the separation bubble, and the invariant manifolds are examined. Mixing efficiencies of the different channel conﬁgurations are compared statistically. Fractal enhancement of productivity is identified in the study of auto-catalytic reaction in the wavy channel. For the two-layer shallow water model, an entropy-correction free Roe type two-layer shallow water solver is developed for a hyperbolic system with non-conservative products and source terms. The scheme is well balanced and satisfies the C-property such that smooth steady solutions are second order accurate. Numerical treatment of the wet-dry front of both layers and the loss of hyperbolicity are incorporated. The solver is tested rigorously on a number of 1D and 2D benchmark test cases. For 2D implementation, a dynamically adaptive quadtree grid generation system is adopted, giving results which are in excellent agreement with those on regular grids at a much lower cost. It is also shown that algebraic balancing cannot be applied directly to a two-layer shallow water ﬂow due to the lack of simultaneous referencing for the still water position for both layers. The adaptive two-layer shallow water solver is applied successfully to ﬂow in an idealised tidal channel and to tidal-driven ﬂow in Tampa Bay, Florida. Finally, chaotic advection and particle mixing is studied for wind-induced recirculation in two-layer shallow water basins, as well as Tampa Bay, Florida.

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Books on the topic "Mixing equations"

1

Center, Langley Research, ed. Simulations of diffusion-reaction equations with implications to turbulent combustion modeling. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1993.

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Girimaji, Sharath S. Simulations of diffusion-reaction equations with implications to turbulent combustion modeling. Hampton, Va: Institute for Computer Applications in Science and Engineering, 1993.

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Center, Langley Research, ed. Simulations of diffusion-reaction equations with implications to turbulent combustion modeling. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1993.

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Center, Langley Research, ed. Simulations of diffusion-reaction equations with implications to turbulent combustion modeling. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1993.

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Orbey, Hasan. Modeling vapor-liquid equilibria: Cubic equations of state and their mixing rules. New York: Cambridge University Press, 1998.

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G, Ostrovskii Alexander, ed. Advection and diffusion in random media: Implications for sea surface temperature anomalies. Dordrecht: Kluwer Academic, 1997.

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Miroslav, Krstić, ed. Flow control by feedback: Stabilization and mixing. London: Springer, 2003.

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United States. National Aeronautics and Space Administration., ed. Numerical solutions of the complete Navier-Stokes equations: Progress report no. 16 for the period July 1, 1998 to December 31, 1989. Raleigh, N.C: Dept. of Mechanical and Aerospace Engineering, North Carolina State University, 1989.

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United States. National Aeronautics and Space Administration., ed. Numerical solutions of the complete Navier-Stokes equations: Progress report no. 27 for the period October 1, 1995 to September 30, 1996. Raleigh, N.C: Dept. of Mechanical and Aerospace Engineering, North Carolina State University, 1996.

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J, Morris Philip, and United States. National Aeronautics and Space Administration., eds. Supersonic coaxial jet noise predictions. [Washington, D.C.]: National Aeronautics and Space Administration, 1995.

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Book chapters on the topic "Mixing equations"

1

Mansoori, G. Ali. "Mixing Rules for Cubic Equations of State." In Equations of State, 314–30. Washington, DC: American Chemical Society, 1986. http://dx.doi.org/10.1021/bk-1986-0300.ch015.

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Copeman, Thomas W., and Paul M. Mathias. "Recent Mixing Rules for Equations of State." In Equations of State, 352–70. Washington, DC: American Chemical Society, 1986. http://dx.doi.org/10.1021/bk-1986-0300.ch017.

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Ely, James F. "Improved Mixing Rules for One-Fluid Conformal Solution Calculations." In Equations of State, 331–50. Washington, DC: American Chemical Society, 1986. http://dx.doi.org/10.1021/bk-1986-0300.ch016.

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Constantin, Peter, Laura Gioia Andrea Keller, and Camilla Nobili. "Existence, uniqueness, regularity and long time behavior of hydrodynamic evolution equations." In Transport, Fluids, and Mixing, 63–94. Warsaw, Poland: De Gruyter Open, 2017. http://dx.doi.org/10.1515/9783110571240-003.

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Panagiotopoulos, A. Z., and R. C. Reid. "New Mixing Rule for Cubic Equations of State for Highly Polar, Asymmetric Systems." In Equations of State, 571–82. Washington, DC: American Chemical Society, 1986. http://dx.doi.org/10.1021/bk-1986-0300.ch028.

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Maasoumi, Esfandiar. "Mixing Forecasts in Linear Simultaneous Equations Under Quadratic Loss." In Contributions to Consumer Demand and Econometrics, 176–88. London: Palgrave Macmillan UK, 1992. http://dx.doi.org/10.1007/978-1-349-12221-9_10.

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Lipp, Tobias, Grégoire Loeper, and Olivier Pironneau. "Mixing Monte-Carlo and Partial Differential Equations for Pricing Options." In Partial Differential Equations: Theory, Control and Approximation, 323–47. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-41401-5_13.

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Castillo-Chavez, C., and S. Busenberg. "On the Solution of the Two-Sex Mixing Problem." In Differential Equations Models in Biology, Epidemiology and Ecology, 80–98. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/978-3-642-45692-3_7.

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Benmekki, E. H., T. Y. Kwak, and G. A. Mansoori. "Van der Waals Mixing Rules for Cubic Equations of State." In ACS Symposium Series, 101–14. Was,hington, DC: American Chemical Society, 1987. http://dx.doi.org/10.1021/bk-1987-0329.ch009.

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Kostić, Marko. "Hypercyclic and Topologically Mixing Properties of Certain Classes of Abstract Time-Fractional Equations." In Difference Equations, Discrete Dynamical Systems and Applications, 155–70. Berlin, Heidelberg: Springer Berlin Heidelberg, 2016. http://dx.doi.org/10.1007/978-3-662-52927-0_12.

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Conference papers on the topic "Mixing equations"

1

Fernandes, Sara, and Sumitha Jayachandran. "Conductance and Mixing Time in Discrete Dynamical Systems." In Proceedings of the Twelfth International Conference on Difference Equations and Applications. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814287654_0018.

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Kepros, John G. "Raman scattering or four-wave mixing." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1985. http://dx.doi.org/10.1364/oam.1985.tuq6.

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An attempt was made to prove the identity of the set of equations describing the Raman phenomenon with the set describing four-wave mixing. The result is a classic proof by contradiction: for the two phenomena to be the same, 1 would have to equal 3/2, 1 would have to equal 5/2. The Raman phenomenon is described by Hertzberg as the vibronic coupling of monochromatic light with the quantized vibrational states of a molecule, producing sidebands from integral quanta. This process is considered one of the initial proofs of quantum mechanics. Four-wave mixing is the optical application of a radar model: two signals of frequency f1 and f2 are transmitted through a radar amplifier withfour (or more) waves emitted, of the form f1, f2, jf1, + kf2, and jf1−kf2. When the Raman equations are equated with the CARS, HORSES, SCIS-SORS…equations2 where j = 1,2,3 … and f2 is greater than f1, the result is inconsistent with logic. Thus, one is led to the conclusion that the Raman effect and the collection of acronyms do not describe the same effect. It is recommended that the Optical Society of America take the editorial position that submissions which label their topic contrary to the proposed definitions be returned to the author for revision.

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Johansen, Per Michael, and Arne Skov Jensen. "Dynamics of Magnetophotorefractive Wave Mixing." In Photorefractive Materials, Effects, and Devices II. Washington, D.C.: Optica Publishing Group, 1993. http://dx.doi.org/10.1364/pmed.1993.fre.1.

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In resonant photorefractive experiments the charge-density distribution in the conduction band is moving and an intrinsic magnetic field proportional to the velocity of the distribution is produced. In such cases the free electron plasma can interact with an externally applied magnetic field. On top of that materials exposed to a magnetic field exhibit Faraday rotation (Voigt birefringence) of the state of polarization of the optical bean.*. The purpose of the present summary is to describe the effect of both phenomena in a standard wave mixing setup. The photorefractive band transport model in the case v here magnetic fields are present is only modified via a term in the current density equation and reads (1) where e is the numeric elementary charge, µ the free electron mobility, n the number density' of free electrons, E the total electric field which may consist of an externally applied term and the norihnear generated spacecharge field, D the diffusion constant, and B the magnetic field term which may consist of an external part and an internal part (stemming from electronic motion). The remaining equations describing the photorefractive effect are the conventional ones. Now, by inferring the small modulation approximation, a perturbational relation for the physical quantities and, by Fourie- transforming in space-time, a steady-state response function for the space-charge electric field are found. ThL photorefractive frequency response function, expressed as a function of the conventional photorefractive parameters, is given by where the function P is given by (3) The actual definition of the parameters in Eqs. (2) and (3) can be found in Ref. [1]. Equation (2) is taken in the case where an externally magnetic field is applied perpendicular to the grating wave vector K. If it is assumed that the constituents of the incident intensity are monochromatic plane waves, that may be frequency shifted a small amount, the Fourier transformation of the perturbed intensity is a sum of δ-functions peaking at k = ±K and at ω = ±Ω. The space-charge field is now given by (4) A typical response function for GaAs:Cr is shown in Fig. 1. The characteristic parameters used in the calculation are taken from Refs. [2, 3, 4], The externally applied electric field is Eθ = 10 V/m, the applied magnetic field is Bθ = 2 T, the angle between the applied electric field and the grating wave vector is 20°, and the angle between the applied magnetic field and the z-axis is 45°. Recently, experiments on photorefr?ctive wave mixing with Faraday rotation in a diluted magnetic semiconductor of Cd] MnTe have been demonstrated. The gain showed an oscillatory behaviour as a function of the' magnetic field, and it was demonstrated that the magnetic field controls the direction and magnitude of the energy transfer. In what follows the coupled wave equations including uie effect of optical activity, the Faraday effect, and the effect of linear absorption will be given. Mcreover, the magnetic effects in the space-charge field given above will be taken into account. Numeric solutions to the coupled differential equations will be given. The equations are given in the following form (5) (6) (7) (8) where α is the absorption constant, Γθ the birefringence coupling constant, p the rotatory power, V the Verdet constant, <5 the angle between the z-axis and the magnetic field vector, and Γ (K,Ω) the photorefractive coupling parameter. A(. and are the polarization components perpendicular and^ parallel to the plane of incidence, respectively. In Fig. 2 a typical numerical solution to the above equations is given for photorefractive GaAs:Cr. The applied magnetic field is Bθ = 2 T, The electric field is Eθ - 106 V/m, the grating spacing is A - 34 µm, and the frequency shift is Ω = 94 rad/s. In such a material the optical activity is zero so the oscillations are due to the Faraday effect. Figure 1. The space-charge response function for GaAs:Cr. Figure 2. Intensity of the probe beam in GaAs:Cr as a function of distance in the crystal for a 10 mm crystal.

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4

Farzad, M. Hosseini. "Exact solution of coupled wave equations in degenerate four-wave mixing." In 17th Congress of the International Commission for Optics: Optics for Science and New Technology. SPIE, 1996. http://dx.doi.org/10.1117/12.2316105.

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5

Sargent, Murray, and Stephan W. Koch. "Multiwave mixing in semiconductor media." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1988. http://dx.doi.org/10.1364/oam.1988.fb3.

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We consider an intrinsic semiconductor medium subject to an injection current, an arbitrarily intense electromagnetic wave, and one or two non-saturating waves. We assume that the saturating wave intensity is constant throughout the interaction region and ignore transverse variations. We further assume that the carrier–carrier scattering is sufficiently fast that the carriers are described approximately by Fermi-Dirac distributions. We find that the probe-wave propagation is described by Beer’s law or coupled-mode equations. These equations are determined by a Fourier expansion technique highly developed for two-level atoms. The absorption and coupling coefficients experience asymmetric dips generated by pump scattering off carrier-density pulsations induced by the pump–probe interference. These effects are reminiscent of coherent-dip phenomena well-known in two-level systems but differ significantly in form.

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6

Wa¨lter, Bettina, and Peter Ehrhard. "Numerical Simulation of Fluid Flows and Mixing in Microchannels Induced by Internal Electrodes." In ASME 2009 7th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2009. http://dx.doi.org/10.1115/icnmm2009-82016.

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We investigate the influence of internal electrodes onto the flow field, governed by electroosmosis and electrophoresis in a modular rectangular microchannel. As internal electrodes can be positioned at lower distances, they can be operated at lower voltages and still ensure strong electrical field strength. Even at lower voltages, electrode reactions influence the species concentration fields, and the crucial question arises, whether at the electrodes all species can be kept in dissolution or whether some species are released in gaseous form. The position and charge of multiple internal electrodes is a further focus of our investigations: wall-tangential electrical field components are responsible for pumping, wall-normal electrical field components are responsible for mixing. Hence, an optimized position and charge of all electrodes will lead to an optimized electrical field, designed to fulfill the desired tasks of the modular microchannel. The mathematical model for the numerical treatment relies on a first-principle description of the EDL and the electrical forces caused by the electrical field between the internal electrodes. Hence, the so-called Debye-Hu¨ckel approximation is avoided. The governing system of equations consists of a Poisson equation for the electrical potential, the continuity and Navier-Stokes equations for the flow field, species transport equations, based on the Nernst-Planck equation, and a charge transport equation. Further, a model for the electrode reactions, based on the Butler-Volmer equation, is in place. The simulations are time-dependent and two-dimensional in nature and employ a FVM.

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Van-Le, Thuy Hong, Sangmo Kang, Yong Kweon Suh, and Yangyang Wang. "Chaotic Mixing in Three-Dimensional Micro Channel." In ASME 2008 First International Conference on Micro/Nanoscale Heat Transfer. ASMEDC, 2008. http://dx.doi.org/10.1115/mnht2008-52193.

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The quality of chaotic mixing in three-dimensional micro channel flow has been numerically studied using Fractional-step method (FSM) and particle tracking techniques such as Poincare´ section and Lyapunov exponents. The flow was driven by pressure distribution and the chaotic mixing was generated by applying alternating current to electrodes embedded on the bottom wall at a first half period and on the top wall at a second half period. The equations governing the velocity and concentration distributions were solved using FSM based on Finite Volume approach. Results showed that the mixing quality depended significantly on the modulation period. The modulation period for the best mixing performance was determined based on the mixing index for various initial conditions of concentration distribution. The optimal values of modulation period obtained by the particle tracking techniques were compared with those from the solution of concentration distribution equation using FSM and CFX software and the comparison showed their good match.

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Tang, G. H., X. J. Gu, R. W. Barber, D. R. Emerson, Y. H. Zhang, and J. M. Reese. "Pulsating Electroosmotic Flow and Wall Block Mixing in Microchannels." In ASME 2008 First International Conference on Micro/Nanoscale Heat Transfer. ASMEDC, 2008. http://dx.doi.org/10.1115/mnht2008-52207.

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Understanding electroosmotic flow in microchannels is of both fundamental and practical significance for the design and optimization of various microfluidic devices to control fluid motion. Electroosmotic flows in microfluidic systems are restricted to the low Reynolds number regime, and mixing in these systems becomes problematic due to negligible inertial effects. To enhance the species mixing effect, the current study presents a numerical investigation of steady-state electroosmotic flow mixing in smooth microchannels, channels patterned with surface blocks, channels patterned with heterogeneous surfaces, as well as pulsating electroosmotic flow. The lattice Boltzmann equations, which recover the nonlinear Poisson-Boltzmann equation, the Navier-Stokes equation including the external force term, and the diffusion equation, were solved to obtain the electric potential distribution in the electrolyte, the velocity field, and the species concentration distribution, respectively. The simulation results confirm that wall blocks, heterogeneous surfaces, and electroosmotic pulsating flow can all change the flow pattern and enhance mixing in microfluidic systems. In addition, it is shown that pulsating flow provides the most promising method for enhancing the mixing efficiency.

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Deb, P., and Pradip Majumdar. "Direct Numerical Simulation of Mixing of a Passive in Decaying Turbulence." In ASME 1999 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/imece1999-1086.

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Abstract Research on turbulent mixing processes is of great interest to those working on turbulent-reactive flows. In this paper, a detailed study has been performed for the evolution of scalar fields of different initial integral scales in decaying, homogeneous and isotropic turbulence using DNS technique. Passive scalar mixing in a cubical decaying, homogeneous, isotropic turbulence field is considered. The three-dimensional incompressible Navier-Stokes equations together with scalar equation are solved using Fractional Step Method. The convective and diffusive terms in governing equations are discretised by Compact Finite Difference Scheme. The 32 × 32 × 32 uniform staggered grids are used. The present simulation is performed at Taylor Reynolds number of 28.83. In this paper, the evolution of scalar RMS and scalar dissipation rate for different integral length scales has been presented. The initial velocity vector and Probability Density Function (PDF) of scalar at different eddy turn over time have also been presented.

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Svensson, Erik D. "Computational Characterization of Passive Fluid Mixing in Microfluidics." In ASME 7th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2004. http://dx.doi.org/10.1115/esda2004-58422.

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In this work we computationally characterize fluid mixing in a number of passive microfluidic mixers. Generally, in order to systematically study and characterize mixing in realistic fluid systems we (1) compute the fluid flow in the systems by solving the stationary three-dimensional Navier-Stokes equations or Stokes equations with a finite element method, and (2) compute various measures indicating the degree of mixing based on concepts from dynamical systems theory, i.e., the sensitive dependence on initial conditions and mixing variance.

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Reports on the topic "Mixing equations"

1

Peters, Hartmut. Development of a Two-Equation Turbulence Model for Mean Shear- and Internal Wave-Driven Mixing. Fort Belvoir, VA: Defense Technical Information Center, September 2010. http://dx.doi.org/10.21236/ada542572.

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Mikkelsen, D. R. Solution of the Fokker-Planck equation with mixing of angular harmonics by beam-beam charge exchange. Office of Scientific and Technical Information (OSTI), September 1989. http://dx.doi.org/10.2172/5577657.

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Rahai, Hamid, and Jeremy Bonifacio. Virus Control Aboard a Commuter Bus. Mineta Transporation Institute, October 2023. http://dx.doi.org/10.31979/mti.2023.2248.

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A major health concern for public transit users is exposure to viruses from other passengers. This numerical study examines virus containment aboard a public bus with changes to the bus ventilation system. The virus was modeled as a 2.5 µm round solid particle released from the mouth of the infectious passenger at a rate of 21 particles per second at a mouth velocity of 0.278 m/sec. The air delivery to the cabin was two linear ceiling slots spanning the length of the bus delivering 59.38 m3/min (2,097 CFM) of air at a mean velocity of 1 m/sec. Two different axial and vertical linear exhaust slots placed on the side walls were investigated to examine how they affected virus containment and spread to the other parts of the cabin. Simulations were performed for both cases of the bus in transit and at the bus stop when the drop-off door was opened. Results indicate during transit that virus spread was contained to passengers sitting immediately in front of and behind the infectious passenger and the level of virus concentration could merit an increased risk of infection with increased virus residence time. However, augmented air mixing was observed between inside and outside air during the passenger drop-off with viruses spread to the front and back of the bus with reduced concentration and risk of infection. Analytical analyses of the risk of infection using the Wells-Riley equation were performed for the bus ventilation using 100% recirculating air without filtration, and 50% and 100% fresh air ventilation. Results indicate a high risk of infection when recirculating air is used, but the risk is reduced significantly with 50% and 100% fresh air ventilation. These results are critical to informing bus manufacturers, transit agencies, planners, and public transportation users about the potential of virus containment using a new ventilation system.

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Ground-water flow simulation and chemical and isotopic mixing equation analysis to determine source contributions to the Missouri River alluvial aquifer in the vicinity of the Independence, Missouri, well field. US Geological Survey, 2002. http://dx.doi.org/10.3133/wri024208.

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