Governing equations

The history of Computational Fluid Dynamics, or CFD for short,, started in the early 1970’s. Around that time, it became an acronym for a combination of physics, numerical mathematics, and, to some extent, computer sciences employed to simulate fluid flows. The beginning of CFD was triggered by the availability of increasingly more powerful mainframes and the advances in CFD are still tightly coupled to the evolution of computer technology.
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Với phương pháp phần tử hữu hạn, miền tính toán được xem như là tập hợp nhiều miền con hữu hạn (finite element) có dạng hình học đơn giản (simple shapeelement). Trên mỗi miền con này, phương trình chủ đạo (governing equation) được thiết lập với sử dụng một phương pháp biến phân nào đó.
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....CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 334872742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acidfree paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 9781420053982 (Hardback) This book contains information obtained from authentic and highly regarded sources.
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The next chapter presents an analytical solution for a nanoplate with Levy boundary conditions. The free vibration analysis is based on a first order shear deformation theory which includes the small scale effect. The governing equations of motion, reformulated as two new equations called the edgezone and interior equations, are based on the nonlocal constitutive equations of Eringen.
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In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the threedimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics. 1. Introduction Large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the NavierStokes equations, with rotation, coupled to thermodynamics and salinity diﬀusiontransport equations, which account for the buoyancy forces and stratiﬁcation eﬀects under the Boussinesq approximation. ...
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Allahyari et al. Nanoscale Research Letters 2011, 6:360 http://www.nanoscalereslett.com/content/6/1/360 NANO EXPRESS Open Access Conjugate heat transfer of laminar mixed convection of a nanofluid through an inclined tube with circumferentially nonuniform heating Shahriar Allahyari1, Amin Behzadmehr1* and Seyed Masoud Hosseini Sarvari2 Abstract Laminar mixed convection of a nanofluid consisting of water and Al2O3 in an inclined tube with heating at the top half surface of a copper tube has been studied numerically.
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Computational Fluid Dynamics (CFD) is the branch of fluid dynamics providing a costeffective means of simulating real flows by the numerical solution of the governing equations. The governing equations for Newtonian fluid dynamics, namely the NavierStokes equations, have been known for over 150 years. However, the development of reduced forms of these equations is still an active area of research, in particular, the turbulent closure problem of the Reynoldsaveraged NavierStokes equations.
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PERIODIC SOLUTIONS OF A DISCRETETIME DIFFUSIVE SYSTEM GOVERNED BY BACKWARD DIFFERENCE EQUATIONS BINXIANG DAI AND JIEZHONG ZOU Received 22 November 2004 and in revised form 16 January 2005 A discretetime delayed diﬀusion model governed by backward diﬀerence equations is investigated. By using the coincidence degree and the related continuation theorem as well as some priori estimates, easily veriﬁable suﬃcient criteria are established for the existence of positive periodic solutions. 1.
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Education vouchers and charter schools are two of the most prominent and farreaching forms of familychoice policies currently in evidence in the nation’s elementary and secondary schools. As such, they present important challenges to the traditional provision of public education in schools that are created, governed, funded, and operated by state and local authorities
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In the present paper the governing equations for dynamical analysis of corrugated crossply laminated composite plates in the form of a sin wave are developed based on the KirchoffLove’s theory and the extension of Seydel’s technique. The problems of natural vibration and forced vibration of a plate with various boundary conditions are studied. Effects of factors as geometry dimensions, order of laminate as well as wavedamplitude on frequency of natural vibration, amplitude of forced vibration of the corrugated crossply laminated composite plates are also analysed. ...
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The purpose of this book is to provide in a single volume a comprehensive reference work covering the broad spectrum of mathematics for circuits and filters; circuits elements, devices, and their models; and linear circuit analysis. This book is written and developed for the practicing electrical engineers in industry, government, and academia. The goal is to provide the most uptodate information in the field. Over the years, the fundamentals of the field have evolved to include a wide range of topics and a broad range of practice.
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While the term chemical substance is a precise technical term that is synonymous with "chemical" for professional chemists, the meaning of the word chemical varies for nonchemists within the English speaking world or those using English. For industries, government and society in general in some countries,[10] the word chemical includes a wider class of substances that contain many mixtures of such chemical substances, often finding application in many vocations.
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In this chapter, we consider another application of the extended Kalman ﬁlter recurrent multilayer perceptron (EKFRMLP) scheme: the modeling of a chaotic time series or one that could be potentially chaotic. The generation of a chaotic process is governed by a coupled set of nonlinear differential or difference equations.
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CHAOTIC DYNAMICS Gaurav S. Patel Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada Simon Haykin Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada (haykin@mcmaster.ca) 4.1 INTRODUCTION In this chapter, we consider another application of the extended Kalman ﬁlter recurrent multilayer perceptron (EKFRMLP) scheme: the modeling of a chaotic time series or one that could be potentially chaotic. The generation of a chaotic process is governed by a coupled set of nonlinear differential or difference equations.
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Introduction: An ionselective electrode (ISE) is an example of an electrochemical sensor utilizing the principle of potentiometry, or measurement of the cell potential (i.e., ISE against a standard reference electrode) at nearzero current. Under these conditions, the boundary potential at the ISE–solution interface is governed by the laws of electrochemical thermodynamics or is compliant with the famous Nernst equation.
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The most important fact related with fluid motion is to understand the fluid patterns, and the flow structure ‐ vortices, recirculation zones, high mix regions, poor mix regions, calm regions, to name a few. Moreover, most of the flows have turbulent characteristics and turbulence remains one of the unsolved problems in physics. No one knows how to obtain stochastic solutions to the well‐posed set of partial differential equations that govern turbulent flows.
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Using graph representation, mechanism structures can be conveniently represented by graphs. The classiﬁcation problem can be transformed into an enumeration of nonisomorphic graphs for a prescribed number of degrees of freedom, number of loops, number of vertices, and number of edges. The degrees of freedom of a mechanism are governed by Equation (4.3).
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Classiﬁcation of Mechanisms Using graph representation, mechanism structures can be conveniently represented by graphs. The classiﬁcation problem can be transformed into an enumeration of nonisomorphic graphs for a prescribed number of degrees of freedom, number of loops, number of vertices, and number of edges. The degrees of freedom of a mechanism are governed by Equation (4.3). The number of loops, number of links, and number of joints in a mechanism are related by Euler’s equation, Equation (4.5). The loop mobility criterion is given by Equation (4.7).
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The last two explanatory variables are control variables, while the remaining variables are the corporate governance variables that we discussed in Section VA. As discussed in Section II, the signs of most of these variables are empirical issues, so we use the observed signs to interpret our results. The top section of Table 7 shows estimates of four variants of equation (6) for the full sample.
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The notion of Ecosystem Services (ES: (Daily 1997; Carpenter 2003; Kremen and Ostfeld 2005) provides a cohesive scientific view of the many mechanisms through which nature contributes to human wellbeing. Focusing on both the biophysical mechanisms of ES provision and the economic implications of ES use can allow our societies to balance the sides of the “nature vs. the economy” equation, leading to better management and governance (Millennium Ecosystem Assessment 2002).
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