Modeling Particle Suspensions Using Lattice Boltzmann Method

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ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (88 download)

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Book Synopsis Modeling Particle Suspensions Using Lattice Boltzmann Method by : Wenbin Mao

Download or read book Modeling Particle Suspensions Using Lattice Boltzmann Method written by Wenbin Mao and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Particle suspensions are common both in nature and in various technological applications. The complex nature of hydrodynamic interactions between particles and the solvent makes such analysis difficult that often requires numerical modeling to understand the behavior of particle suspensions. In this dissertation, we employ a hybrid computational model that integrates a lattice spring model for solid mechanics and a lattice Boltzmann model for fluid dynamics. We use this model to study several practical problems in which the dynamics of spherical and spheroidal particles and deformable capsules in dilute suspensions plays an important role. The results of our studies yield new information regarding the dynamics of solid particle in pressure-driven channel flows and disclose the nonlinear effects associated with fluid inertia leading to particle cross-stream migration. This information not only give us a fundamental insight into the dynamics of dilute suspensions, but also yield engineering guidelines for designing high throughput microfluidic devices for sorting and separation of synthetic particles and biological cells. We first demonstrate that spherical particles can be size-separated in ridged microchannels. Specifically, particles with different sizes follow distinct trajectories as a result of the nonlinear inertial effects and secondary flows created by diagonal ridges in the channel. Then, separation of biological cells by their differential stiffness is studied and compared with experimental results. Cells with different stiffness squeeze through narrow gaps between solid diagonal ridges and channel wall, and migrate across the microchannel with different rates depending on their stiffness. This deformability-based microfluidic platform may be valuable for separating diseased cells from healthy cells, as a variety of cell pathologies manifest through the change in mechanical cell stiffness. Finally, the dynamics of spheroid particles in simple shear and Poiseuille flows are studied. Stable rotational motion, cross-stream migration, and equilibrium trajectories of non-spherical particles in flow are investigated. Effects of particle and fluid inertia on dynamics of particles are disclosed. The dependence of equilibrium trajectory on particle shape reveals a potential application for shape based particle separation.

Direct Simulation of Flexible Particle Suspensions Using Lattice-Boltzmann Equation with External Boundary Force

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ISBN 13 :
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Book Rating : 4.:/5 (668 download)

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Book Synopsis Direct Simulation of Flexible Particle Suspensions Using Lattice-Boltzmann Equation with External Boundary Force by : Jingshu Wu

Download or read book Direct Simulation of Flexible Particle Suspensions Using Lattice-Boltzmann Equation with External Boundary Force written by Jingshu Wu and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Determination of the relation between the bulk or rheological properties of a particle suspension and its microscopic structure is an old and important problem in physical science. In general, the rheology of particle suspension is quite complex, and the problem becomes even more complicated if the suspending particle is deformable. Despite these difficulties, a large number of theoretical and experimental investigations have been devoted to the analysis and prediction of the rheological behavior of particle suspensions. However, among these studies there are very few investigations that focus on the role of particle deformability. A novel method for full coupling of the fluid-solid phases with sub-grid accuracy for the solid phase is developed. In this method, the flow is computed on a fixed regular 'lattice' using the lattice Boltzmann method (LBM), where each solid particle, or fiber, is mapped onto a Lagrangian frame moving continuously through the domain. The motion and orientation of the particle are obtained from Newtonian dynamics equations. The deformable particle is modeled by the lattice-spring model (LSM). The fiber deformation is calculated by an efficient flexible fiber model. The no-slip boundary condition at the fluid-solid interface is based on the external boundary force (EBF) method. This method is validated by comparing with known experimental and theoretical results.

Modeling of Magnetic Particle Suspensions for Simulations

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Publisher : CRC Press
ISBN 13 : 1351679007
Total Pages : 269 pages
Book Rating : 4.3/5 (516 download)

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Book Synopsis Modeling of Magnetic Particle Suspensions for Simulations by : Akira Satoh

Download or read book Modeling of Magnetic Particle Suspensions for Simulations written by Akira Satoh and published by CRC Press. This book was released on 2017-02-03 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main objective of the book is to highlight the modeling of magnetic particles with different shapes and magnetic properties, to provide graduate students and young researchers information on the theoretical aspects and actual techniques for the treatment of magnetic particles in particle-based simulations. In simulation, we focus on the Monte Carlo, molecular dynamics, Brownian dynamics, lattice Boltzmann and stochastic rotation dynamics (multi-particle collision dynamics) methods. The latter two simulation methods can simulate both the particle motion and the ambient flow field simultaneously. In general, specialized knowledge can only be obtained in an effective manner under the supervision of an expert. The present book is written to play such a role for readers who wish to develop the skill of modeling magnetic particles and develop a computer simulation program using their own ability. This book is therefore a self-learning book for graduate students and young researchers. Armed with this knowledge, readers are expected to be able to sufficiently enhance their skill for tackling any challenging problems they may encounter in future.

Lattice Boltzmann Method And Its Application In Engineering

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Publisher : World Scientific
ISBN 13 : 9814508314
Total Pages : 419 pages
Book Rating : 4.8/5 (145 download)

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Book Synopsis Lattice Boltzmann Method And Its Application In Engineering by : Zhaoli Guo

Download or read book Lattice Boltzmann Method And Its Application In Engineering written by Zhaoli Guo and published by World Scientific. This book was released on 2013-03-25 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lattice Boltzmann method (LBM) is a relatively new simulation technique for the modeling of complex fluid systems and has attracted interest from researchers in computational physics. Unlike the traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh.This book will cover the fundamental and practical application of LBM. The first part of the book consists of three chapters starting form the theory of LBM, basic models, initial and boundary conditions, theoretical analysis, to improved models. The second part of the book consists of six chapters, address applications of LBM in various aspects of computational fluid dynamic engineering, covering areas, such as thermo-hydrodynamics, compressible flows, multicomponent/multiphase flows, microscale flows, flows in porous media, turbulent flows, and suspensions.With these coverage LBM, the book intended to promote its applications, instead of the traditional computational fluid dynamic method.

Lattice Boltzmann Simulation of Dense Rigid Spherical Particle Suspensions Using Immersed Boundary Method

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ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (15 download)

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Book Synopsis Lattice Boltzmann Simulation of Dense Rigid Spherical Particle Suspensions Using Immersed Boundary Method by :

Download or read book Lattice Boltzmann Simulation of Dense Rigid Spherical Particle Suspensions Using Immersed Boundary Method written by and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Lattice Boltzmann Modeling

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Publisher : Springer Science & Business Media
ISBN 13 : 3540279822
Total Pages : 178 pages
Book Rating : 4.5/5 (42 download)

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Book Synopsis Lattice Boltzmann Modeling by : Michael C. Sukop

Download or read book Lattice Boltzmann Modeling written by Michael C. Sukop and published by Springer Science & Business Media. This book was released on 2007-04-05 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here is a basic introduction to Lattice Boltzmann models that emphasizes intuition and simplistic conceptualization of processes, while avoiding the complex mathematics that underlies LB models. The model is viewed from a particle perspective where collisions, streaming, and particle-particle/particle-surface interactions constitute the entire conceptual framework. Beginners and those whose interest is in model application over detailed mathematics will find this a powerful 'quick start' guide. Example simulations, exercises, and computer codes are included.

The Lattice Boltzmann Method

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Publisher : Springer
ISBN 13 : 3319446495
Total Pages : 705 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis The Lattice Boltzmann Method by : Timm Krüger

Download or read book The Lattice Boltzmann Method written by Timm Krüger and published by Springer. This book was released on 2016-11-07 with total page 705 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to the theory, practice, and implementation of the Lattice Boltzmann (LB) method, a powerful computational fluid dynamics method that is steadily gaining attention due to its simplicity, scalability, extensibility, and simple handling of complex geometries. The book contains chapters on the method's background, fundamental theory, advanced extensions, and implementation. To aid beginners, the most essential paragraphs in each chapter are highlighted, and the introductory chapters on various LB topics are front-loaded with special "in a nutshell" sections that condense the chapter's most important practical results. Together, these sections can be used to quickly get up and running with the method. Exercises are integrated throughout the text, and frequently asked questions about the method are dealt with in a special section at the beginning. In the book itself and through its web page, readers can find example codes showing how the LB method can be implemented efficiently on a variety of hardware platforms, including multi-core processors, clusters, and graphics processing units. Students and scientists learning and using the LB method will appreciate the wealth of clearly presented and structured information in this volume.

Accelerated Lattice Boltzmann Model for Colloidal Suspensions

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Publisher : Springer Science & Business Media
ISBN 13 : 1489974024
Total Pages : 161 pages
Book Rating : 4.4/5 (899 download)

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Book Synopsis Accelerated Lattice Boltzmann Model for Colloidal Suspensions by : Hassan Farhat

Download or read book Accelerated Lattice Boltzmann Model for Colloidal Suspensions written by Hassan Farhat and published by Springer Science & Business Media. This book was released on 2014-04-12 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: Colloids are ubiquitous in the food, medical, cosmetics, polymers, water purification, and pharmaceutical industries. The thermal, mechanical, and storage properties of colloids are highly dependent on their interface morphology and their rheological behavior. Numerical methods provide a convenient and reliable tool for the study of colloids. Accelerated Lattice Boltzmann Model for Colloidal Suspensions introduce the main building-blocks for an improved lattice Boltzmann–based numerical tool designed for the study of colloidal rheology and interface morphology. This book also covers the migrating multi-block used to simulate single component, multi-component, multiphase, and single component multiphase flows and their validation by experimental, numerical, and analytical solutions. Among other topics discussed are the hybrid lattice Boltzmann method (LBM) for surfactant-covered droplets; biological suspensions such as blood; used in conjunction with the suppression of coalescence for investigating the rheology of colloids and microvasculature blood flow. The presented LBM model provides a flexible numerical platform consisting of various modules that could be used separately or in combination for the study of a variety of colloids and biological flow deformation problems.

The Lattice Boltzmann Equation

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Publisher : Oxford University Press
ISBN 13 : 9780198503989
Total Pages : 308 pages
Book Rating : 4.5/5 (39 download)

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Book Synopsis The Lattice Boltzmann Equation by : S. Succi

Download or read book The Lattice Boltzmann Equation written by S. Succi and published by Oxford University Press. This book was released on 2001-06-28 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: Certain forms of the Boltzmann equation, have emerged, which relinquish most mathematical complexities of the true Boltzmann equation. This text provides a detailed survey of Lattice Boltzmann equation theory and its major applications.

Solid-particles Deposition Through a Turbulent Impinging Jet Using Lattice Boltzmann Method

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ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (18 download)

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Book Synopsis Solid-particles Deposition Through a Turbulent Impinging Jet Using Lattice Boltzmann Method by : Ali Abdulkadhim

Download or read book Solid-particles Deposition Through a Turbulent Impinging Jet Using Lattice Boltzmann Method written by Ali Abdulkadhim and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Solid particle distribution on an impingement surface has been simulated utilizing a graphical processing unit (GPU). An in-house computational fluid dynamics (CFD) code has been developed to investigate a 3D turbulent impinging jet using the lattice Boltzmann method (LBM) in conjunction with large eddy simulation (LES) and the multiple relaxation time (MRT) models. This work proposed an improvement in the LBM-cellular automata (LBM-CA) probabilistic method. In the current model, the fluid flow utilizes the D3Q19 LBM lattice model, while the particles movement employs the D3Q27 one. The particle numbers are defined at the same regular LBM (fluid) nodes, and the transport of particles from one node to its neighbouring nodes are determined in accordance with the particle bulk density and velocity by considering all the external forces. The previous CA models distribute particles at each time step without considering the local particles number and velocity at each node. The present model overcomes the deficiencies of the previous LBM-CA models and, therefore, can better capture the dynamic interaction between particles and the surrounding turbulent flow field. Despite increasing popularity of the LBM-MRT model in simulating complex multiphase fluid flows, this approach is still expensive in term of memory size and computational time required to perform 3D simulations. To improve the throughput of simulations, a single GeForce GTX TITAN X GPU is used in the present work. The CUDA parallel programming platform and the CuRAND library are utilized to form an efficient LBM-MRT-CA algorithm. The LBM-MRT fluid (i.e. no particles) model results were compared with two benchmark test cases ones. The first case is a turbulent free square jet, and the second one is a circular turbulent impinging jet for L/D=2 at Reynolds number equals to 25,000, where L is the nozzle-to-surface distance and D is the jet diameter. The LBM-CA simulation methodology was first validated against a benchmark test case involving particle deposition on a square cylinder confined in a duct. The flow was unsteady and laminar at Re=200 (Re is the Reynolds number), and simulations were conducted for different Stokes numbers. The GPU code was then used to simulate the particle transport and deposition in a turbulent impinging jet at Re=10,000. The effect of changing Stokes number on the particle deposition profile was studied at different L/D ratios, i.e. L/D=2, 4, and 6. The current model was finally used to simulate the particle impaction pattern from a circular jet for L/D=0.5, where the effect of changing Stokes and Reynolds numbers on the particle transport and deposition was examined. The present LBM-CA solutions agree well with other results available in the open literature. For comparative studies, another in-house serial CPU code was also developed, coupling LBM with the classical Lagrangian particle dispersion model. Agreement between results obtained with LBM-CA and LBM-Lagrangian models and the experimental data for the impinging jet case of L/D=0.5 is generally good, and the present LBM-CA approach on GPU achieves a speedup ratio of about 150 against the serial code running on a single CPU. Another new model was proposed to incorporate the solid particle phase effect (i.e. two-way coupling) on the fluid flow. The LMB-Lagrangian approach was used in this model to track solid particles in the computational domain. The solid particle phase was considered as a porous medium moving in the computational domain. The impact of the porous medium (i.e. the solid particle phase) on the fluid flow characteristics (e.g. fluid velocity) is a function of the particle phase volume fraction and velocity in the LBM. Particle-particle collision (i.e. four-way coupling) was also considered in this model by utilizing the discrete element method (DEM). This approach can numerically capture the multi-particle collision behaviours in dense particle suspension problems. This model data were compared with the numerical study ones for a single bubble injected in a fluidized bed, and the results of the bubble diameters at different injection velocity were in good agreement.

Lattice-Gas Cellular Automata and Lattice Boltzmann Models

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Publisher : Springer
ISBN 13 : 3540465863
Total Pages : 320 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Lattice-Gas Cellular Automata and Lattice Boltzmann Models by : Dieter A. Wolf-Gladrow

Download or read book Lattice-Gas Cellular Automata and Lattice Boltzmann Models written by Dieter A. Wolf-Gladrow and published by Springer. This book was released on 2004-10-19 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) are relatively new and promising methods for the numerical solution of nonlinear partial differential equations. The book provides an introduction for graduate students and researchers. Working knowledge of calculus is required and experience in PDEs and fluid dynamics is recommended. Some peculiarities of cellular automata are outlined in Chapter 2. The properties of various LGCA and special coding techniques are discussed in Chapter 3. Concepts from statistical mechanics (Chapter 4) provide the necessary theoretical background for LGCA and LBM. The properties of lattice Boltzmann models and a method for their construction are presented in Chapter 5.

Numerical Simulation of Particle Suspension Flows

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ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (659 download)

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Book Synopsis Numerical Simulation of Particle Suspension Flows by : Byoungjin Chun

Download or read book Numerical Simulation of Particle Suspension Flows written by Byoungjin Chun and published by . This book was released on 2006 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The accuracy of solid-fluid boundary conditions for the lattice Boltzmann method has been investigated in a certain class of flows with the multiple-relaxation-time model. In general the multi-reflection rule is the most accurate. The accuracy of the linear and quadratic interpolation rules is viscosity dependent, even though they show a significant improvement at lower viscosity. Our equilibrium interpolation rule shows especially better accuracy for flows through densely packed spheres at coarser resolutions.

Computer Simulation Study of Collective Phenomena in Dense Suspensions of Red Blood Cells under Shear

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Publisher : Springer Science & Business Media
ISBN 13 : 3834823767
Total Pages : 169 pages
Book Rating : 4.8/5 (348 download)

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Book Synopsis Computer Simulation Study of Collective Phenomena in Dense Suspensions of Red Blood Cells under Shear by : Timm Krüger

Download or read book Computer Simulation Study of Collective Phenomena in Dense Suspensions of Red Blood Cells under Shear written by Timm Krüger and published by Springer Science & Business Media. This book was released on 2012-10-02 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: The rheology of dense red blood cell suspensions is investigated via computer simulations based on the lattice Boltzmann, the immersed boundary, and the finite element methods. The red blood cells are treated as extended and deformable particles immersed in the ambient fluid. In the first part of the work, the numerical model and strategies for stress evaluation are discussed. In the second part, the behavior of the suspensions in simple shear flow is studied for different volume fractions, particle deformabilities, and shear rates. Shear thinning behavior is recovered. The existence of a shear-induced transition from a tumbling to a tank-treading motion is demonstrated. The transition can be parameterized by a single quantity, namely the effective capillary number. It is the ratio of the suspension stress and the characteristic particle membrane stress. At the transition point, a strong increase in the orientational order of the red blood cells and a significant decrease of the particle diffusivity are observed. However, the average cell deformation shows no signature of the transition.

Dynamics of Colloidal and Polymeric Suspensions Via a Fluctuating Lattice-Boltzmann Model

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Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (66 download)

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Book Synopsis Dynamics of Colloidal and Polymeric Suspensions Via a Fluctuating Lattice-Boltzmann Model by : O. Berk Usta

Download or read book Dynamics of Colloidal and Polymeric Suspensions Via a Fluctuating Lattice-Boltzmann Model written by O. Berk Usta and published by . This book was released on 2007 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: ABSTRACT: We investigate the dynamic and static properties of polymeric suspensions via a numerical simulation method. This method couples Newtonian mechanics of solid phase suspended particles with a fluctuating lattice-Boltzmann fluid. This results in the solution of the Navier-Stokes equation, including thermal fluctuations and hydrodynamic interactions for the solid particles. The coupling can be achieved either through boundary conditions for a finite sized particle or through a Stokes-like frictional coupling for point particles. The frictional coupling procedure is a very efficient approach due to the use of point particles which take up much smaller volumes. This in turn results in simulations of far fewer lattice nodes, while retaining essential hydrodynamic features for dynamics of polymers. This approach has enabled us to simulate polymer chains with more than 1000 segments, including hydrodynamic interactions. This discretization is nearly an order of magnitude higher than what is feasible using traditional Brownian dynamics methods. The fluctuating lattice-Boltzmann method is presented for both colloidal and polymeric suspensions. A verification of the fluctuations for the case of colloidal finite sized particles is presented as a test case. However, the main theme of this thesis is the simulations of dilute polymer solutions.

The Lattice Boltzmann Equation: For Complex States of Flowing Matter

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Publisher : Oxford University Press
ISBN 13 : 0192538853
Total Pages : 784 pages
Book Rating : 4.1/5 (925 download)

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Book Synopsis The Lattice Boltzmann Equation: For Complex States of Flowing Matter by : Sauro Succi

Download or read book The Lattice Boltzmann Equation: For Complex States of Flowing Matter written by Sauro Succi and published by Oxford University Press. This book was released on 2018-04-13 with total page 784 pages. Available in PDF, EPUB and Kindle. Book excerpt: Flowing matter is all around us, from daily-life vital processes (breathing, blood circulation), to industrial, environmental, biological, and medical sciences. Complex states of flowing matter are equally present in fundamental physical processes, far remote from our direct senses, such as quantum-relativistic matter under ultra-high temperature conditions (quark-gluon plasmas). Capturing the complexities of such states of matter stands as one of the most prominent challenges of modern science, with multiple ramifications to physics, biology, mathematics, and computer science. As a result, mathematical and computational techniques capable of providing a quantitative account of the way that such complex states of flowing matter behave in space and time are becoming increasingly important. This book provides a unique description of a major technique, the Lattice Boltzmann method to accomplish this task. The Lattice Boltzmann method has gained a prominent role as an efficient computational tool for the numerical simulation of a wide variety of complex states of flowing matter across a broad range of scales; from fully-developed turbulence, to multiphase micro-flows, all the way down to nano-biofluidics and lately, even quantum-relativistic sub-nuclear fluids. After providing a self-contained introduction to the kinetic theory of fluids and a thorough account of its transcription to the lattice framework, this text provides a survey of the major developments which have led to the impressive growth of the Lattice Boltzmann across most walks of fluid dynamics and its interfaces with allied disciplines. Included are recent developments of Lattice Boltzmann methods for non-ideal fluids, micro- and nanofluidic flows with suspended bodies of assorted nature and extensions to strong non-equilibrium flows beyond the realm of continuum fluid mechanics. In the final part, it presents the extension of the Lattice Boltzmann method to quantum and relativistic matter, in an attempt to match the major surge of interest spurred by recent developments in the area of strongly interacting holographic fluids, such as electron flows in graphene.

Particulate Flows

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Publisher : Springer Science & Business Media
ISBN 13 : 1468471090
Total Pages : 155 pages
Book Rating : 4.4/5 (684 download)

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Book Synopsis Particulate Flows by : Donald A. Drew

Download or read book Particulate Flows written by Donald A. Drew and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: This IMA Volume in Mathematics and its Applications PARTICULATE FLOWS: PROCESSING AND RHEOLOGY is based on the proceedings of a very successful one-week workshop with the same title, which was an integral part of the 1995-1996 IMA program on "Mathematical Methods in Materials Science." We would like to thank Donald A. Drew, Daniel D. Joseph, and Stephen L. Passman for their excellent work as organizers of the meeting. We also take this opportunity to thank the National Science Foun dation (NSF), the Army Research Office (ARO) and the Office of Naval Research (ONR), whose financial support made the workshop possible. A vner Friedman Robert Gulliver v PREFACE The workshop on Particulate Flows: Processing and Rheology was held January 8-12, 1996 at the Institute for Mathematics and its Applications on the University of Minnesota Twin Cities campus as part of the 1995- 96 Program on Mathematical Methods in Materials Science. There were about forty participants, and some lively discussions, in spite of the fact that bad weather on the east coast kept some participants from attending, and caused scheduling changes throughout the workshop. Heterogeneous materials can behave strangely, even in simple flow sit uations. For example, a mixture of solid particles in a liquid can exhibit behavior that seems solid-like or fluid-like, and attempting to measure the "viscosity" of such a mixture leads to contradictions and "unrepeatable" experiments. Even so, such materials are commonly used in manufacturing and processing.

Lattice Boltzmann Method

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Publisher :
ISBN 13 : 9781447174240
Total Pages : 228 pages
Book Rating : 4.1/5 (742 download)

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Book Synopsis Lattice Boltzmann Method by : Abdulmajeed A. Mohamad

Download or read book Lattice Boltzmann Method written by Abdulmajeed A. Mohamad and published by . This book was released on 2019 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introducing the Lattice Boltzmann Method in a readable manner, this book provides detailed examples with complete computer codes. It avoids the most complicated mathematics and physics without scarifying the basic fundamentals of the method.