### Citations

1458 |
Quantum Mechanics
- Landau, Lifshitz
- 1965
(Show Context)
Citation Context ... 4.2. Cylindrical Couette Flow The next test was conducted for cylindrical Couette flow, which is a flow between two coaxial rotating cylinders. There is a well-known stationary solution for this flow=-=[17]-=-: �=ar� b r (35) where a= � 2 R 2 2 R2−R1 2 , b=−� 2 R 2 1 R 2 2 −R1 R 2 2 (36) Here � 2 is the velocity of the outer cylinder (if the inner one is not rotating), R 1, R 2 are inner and outer cylinder... |

108 |
Theory of the Lattice Boltzmann Method : From the Boltzmann equation to the Lattice Boltzmann equation
- He, Luo
- 1997
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Citation Context ...s: �=∑ i f i , � �=∑ i f i c i (3) Note that lattice vectors and their weights are constructed to satisfy symmetry conditions [1] and to recover density and velocity for continuous Boltzmann dynamics =-=[12]-=-. Typical lattice stencil for a two-dimensional space is the D2Q9 one, which contains of 9 lattice vectors with the following coordinates [2] � 0 0 ∣�1∣ 0 0 �1 ∣−1∣ 0 0 −1 ∣�1 �1 ∣−1 �1 ∣−1 −1 ∣�1 −1�... |

90 |
On Pressure and Velocity Boundary Conditions for the Lattice Boltzmann Bgk Model,”
- Zou, He
- 1997
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Citation Context ...s numerically stable, accurate and local; it has a good physical background. Also, we provide theoretical justifications of a popular Zou-He boundaries technique and an isothermal boundaries algorithm=-=[10]-=-. These 4justifications are based on the maximum entropy method derived. Finally, while conducting numerical benchmarks, typical straight boundaries algorithm (Zou-He) was compared to a typical curve... |

43 |
On boundary conditions in lattice Boltzmann methods,”
- Chen, Martinez
- 1996
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Citation Context ...just plane boundary conditions), Chen-Martinez boundary nodes (which support curved boundary conditions, but need specific treatment for curved boundaries, so they were used just for plane walls) [5],=-=[8]-=-,[10]. 4.1. Poiseuille Flow We consider an academic two-dimensional Poiseuille flow within the following setting. Flow is force-driven to prevent pressure and density changes and to make the simulatio... |

38 | An accurate curved boundary treatment in the lattice Boltzmann method - Mei, Luo, et al. - 1999 |

34 | Multireflection boundary conditions for lattice Boltzmann models - Ginzburg, d’Humières - 2003 |

28 | The lattice Boltzmann equation method: Theoretical interpretation, numerics and implications
- Nourgaliev, Dinh, et al.
- 1998
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Citation Context ...nt to specify either velocity or density (not both of them) due to the Navier-Stokes equation properties. A ubiquitous approach for no-slip boundaries implementation is the usage of bounce-back nodes =-=[1]-=-. Also, several specific methods exist for constant density and constant velocity boundaries modeling on straight walls [4]. An accurate method for constant velocity boundaries simulation on curved wa... |

27 | An extrapolation method for boundary conditions in lattice Boltzmann method - Guo, Zheng, et al. - 2002 |

19 |
Simplified Thermal Lattice Boltzmann Model For Incompressible Thermal Flows," Physical Review E,
- Peng, Shu, et al.
- 2003
(Show Context)
Citation Context ...entioned above, are measured in lattice units [13]. LBM method analysis and linking it to the Navier-Stokes equation are usually performed through one of the following techniques: 1. Taylor expansion =-=[14]-=- 2. asymptotic analysis [15] 3. Chapman-Enskog expansion [1]. 83. Maximum Entropy Boundaries In this section maximum entropy boundaries method will be explained. 3.1. Method Idea F B E C 6 C 2 C 5 D ... |

13 |
Straight velocity boundaries in the lattice Boltzmann method.
- Latt, Chopard, et al.
- 2008
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Citation Context ...sk of implementing boundary conditions to make them correspond to the macroscopic constraints. The most wide-spread boundaries are solid no-slip ones, constant velocity or constant density boundaries =-=[4]-=-. Note that it is always sufficient to specify either velocity or density (not both of them) due to the Navier-Stokes equation properties. A ubiquitous approach for no-slip boundaries implementation i... |

11 |
On laminar flow through a uniformly porous
- Terrill, Thomas
- 1969
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Citation Context ...ial velocity is u , pressure is p , membrane velocity, directed outside, is V , cylinder radius is a , non-dimensional radial coordinate �=r 2 /a 2 . A general solution for such a problem has the form=-=[20]-=- �=V f �� �/���� u=u 0 f ' ����1−2V z/a� p= p 0 ��� A z 2 �Bz���V 2 � 2 f ' / R− f 2 /� � (37) where u 0 is a characteristic velocity, R=V a /� is suction Reynolds number (don't mix it with the main R... |

11 |
Laminar pipe flow with injection and suction through a porous wall
- Yuan, Finkelstein
- 1956
(Show Context)
Citation Context ...rmined during the solution of the boundary-value problem (38) due to sufficient boundary conditions. 23The approximate solution to this problem, obtained with the perturbation theory, can be found in=-=[22]-=-,[23] f ���=2�−� 2 � R 36 � 4� −9�2 �6� 3 −� 4 �, K =−2� 3 2 R This solution is applicable, when wall Reynolds number R is in the range [−2 ;2] . A D3Q19 scheme was utilized in the simulation. Cylinde... |

9 |
The lattice Boltzmann equation for natural convection in a two-dimensional cavity with a partially heated wall
- Barrios, Rechtman, et al.
- 2005
(Show Context)
Citation Context ... time t . These probabilities are normalized by mass of the fluid. � is the mass relaxation time, 6which is related to the kinematic viscosity as 2 �=c s ��−1/2� , where cs=1/��3� is the sound speed =-=[11]-=-. Equilibrium particle population in direction i is eq f i =wi �[1�3�⋅c i� 9 2 ��⋅c i� 2 − 3 2 � 2 ] , (2) where macroscopic density in the given node is � , macroscopic velocity is � . Lattice vector... |

4 |
Lattice-Boltzmann algorithm for simulating thermal two-phase flow
- Palmer, Rector
(Show Context)
Citation Context ... in lattice units [13]. LBM method analysis and linking it to the Navier-Stokes equation are usually performed through one of the following techniques: 1. Taylor expansion [14] 2. asymptotic analysis =-=[15]-=- 3. Chapman-Enskog expansion [1]. 83. Maximum Entropy Boundaries In this section maximum entropy boundaries method will be explained. 3.1. Method Idea F B E C 6 C 2 C 5 D C 3 O C 0 C 1 A C 7 C 4 C 8 ... |

4 |
Laminar flow in channels with wall suction or injection: a new model to study multi-channel filtration systems
- Oxarango, Schmitz
- 2004
(Show Context)
Citation Context ...d during the solution of the boundary-value problem (38) due to sufficient boundary conditions. 23The approximate solution to this problem, obtained with the perturbation theory, can be found in[22],=-=[23]-=- f ���=2�−� 2 � R 36 � 4� −9�2 �6� 3 −� 4 �, K =−2� 3 2 R This solution is applicable, when wall Reynolds number R is in the range [−2 ;2] . A D3Q19 scheme was utilized in the simulation. Cylinder len... |

3 |
Bachmann O: Application of the multi distribution function lattice Boltzmann approach to thermal flows
- Parmigiani, Huber, et al.
(Show Context)
Citation Context ...vectors ∣c i∣ 2 =1 , w i =1/36 for vectors ∣c i∣ 2 =2 . 7An important point in the LBM framework is specific units conversion: all the physical values, mentioned above, are measured in lattice units =-=[13]-=-. LBM method analysis and linking it to the Navier-Stokes equation are usually performed through one of the following techniques: 1. Taylor expansion [14] 2. asymptotic analysis [15] 3. Chapman-Enskog... |

3 |
Lattice-Boltzmann accuracy in pore-scale flow simulation
- Maier, Bernard
- 2010
(Show Context)
Citation Context ...stem widths (Y axis) were obtained in the first set of experiments and errors between these solutions and expected solutions were measured. Solutions error was computed according to the Maier L 2 norm=-=[16]-=- in the space of possible system states by the following formula: 17�= � ∑ ∣u nodes theor −u exp ∣ 2 � ∑ ∣u nodes theor ∣ 2 (34) Numerical simulations were stopped when the difference by the given no... |

2 | Flow of a Newtonian fluid through a permeable tube: The application to the proximal renal tubule - Marshall, Trowbridge |

2 |
Exact solution for flow in a porous pipe with unsteady wall suction and/or injection
- Tsangaris, Vlachakis, et al.
- 2007
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Citation Context ...e operating according to the Darcy's law[18-22]. Fluid velocity through membranes is assumed to be constant and predefined for simplicity. Despite there is an exact analytical solution for such a case=-=[21]-=-, it turns out that an approximate solution is simpler and easier for analysis and implementation. The following variables are introduced: axial velocity is � , radial velocity is u , pressure is p , ... |

1 |
Parallel Lattice Boltzmann Methods for
- Körner, Pohl, et al.
(Show Context)
Citation Context ...er density and velocity for continuous Boltzmann dynamics [12]. Typical lattice stencil for a two-dimensional space is the D2Q9 one, which contains of 9 lattice vectors with the following coordinates =-=[2]-=- � 0 0 ∣�1∣ 0 0 �1 ∣−1∣ 0 0 −1 ∣�1 �1 ∣−1 �1 ∣−1 −1 ∣�1 −1� (4) Lattice weights for this scheme are the following: w i =4/9 for a vector with ∣c i∣ 2 =0 , w i =1/9 for vectors ∣c i∣ 2 =1 , w i =1/36 f... |

1 | Impact of conduit geometry and bed porosity on flow and dispersion in noncylindrical sphere packings - Khirevich, Höltzel, et al. - 2007 |