Symmetries in porous flows: recursive solutions of the Brinkman equation in polygonal ducts

Density-driven instabilities between miscible fluids in a vertical Hele-Shaw cell are investigated by means of experimental measurements, as well as two- and three-dimensional numerical simulations. The experiments focus on the early stages of the instability growth, and they provide detailed information regarding the growth rates and most amplified wavenumbers as a function of the governing Rayleigh number Ra. They identify two clearly distinct parameter regimes: a low-Ra, ‘Hele-Shaw’ regime in which the dominant wavelength scales as Ra−1, and a high-Ra ‘gap’ regime in which the length scale of the instability is 5±1 times the gap width. The experiments are compared to a recent linear stability analysis based on the Brinkman equation. The analytical dispersion relationship for a step-like density profile reproduces the experimentally observed trend across the entire Ra range. Nonlinear simulations based on the two- and three-dimensional Stokes equations indicate that the high-Ra regime is characterized by an instability across the gap, wheras in the low-Ra regime a spanwise Hele-Shaw mode dominates.

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A Uniformly Stable Nonconforming FEM Based on Weighted Interior Penalties for Darcy-Stokes-Brinkman Equations

Numerical Mathematics Theory Methods and Applications ◽

10.4208/nmtma.2017.m1610 ◽

2017 ◽

Vol 10 (1) ◽

pp. 22-43 ◽

Cited By ~ 2

Author(s):

Peiqi Huang ◽

Zhilin Li

Keyword(s):

Finite Element ◽

Mixed Finite Element ◽

Discontinuous Coefficients ◽

Brinkman Equation ◽

Optimal Error Estimates ◽

Discrete Problem ◽

Brinkman Equations ◽

Approximation Errors ◽

Interior Penalties ◽

Uniformly Stable

AbstractA nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

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ESTIMATION OF HETEROGENEITY OF THE ATMOSPHERIC AIR VELOCITY FIELD IN ADSORBERS OF FRONT-END PURIFICATION UNITS FOR AIR SEPARATION PLANTS

Vestnik Tomskogo gosudarstvennogo universiteta Matematika i mekhanika ◽

10.17223/19988621/70/10 ◽

2021 ◽

pp. 117-126

Author(s):

O.N. Filimonova ◽

◽

A.A. Vorobyov ◽

A.S. Vikulin ◽

◽

...

Keyword(s):

Mathematical Model ◽

Velocity Field ◽

Granular Layer ◽

Air Separation ◽

Brinkman Equation ◽

Turbulent Regime ◽

Atmospheric Air ◽

Displacement Mode ◽

Front End ◽

Purification Unit

Assuming unidirectional motion of compressed atmospheric air through a vertical cylindrical adsorbent with a fixed granular layer of the front-end purification unit adsorbent, the mathematical model for estimating the heterogeneity of a hydrodynamic velocity field in the radial and axial directions in a turbulent regime is proposed. The model is based on the boundary layer approximation of the Darcy – Brinkman – Forchheimer phenomenological equation. The steady-state flow at low permeability of the granular layer is identified using the collocation method, and the approximate analytical solution is obtained which justifies the applicability of an ideal displacement mode when describing the carrier medium motion. Numerical integration of a boundary value problem of the model equation using the finite-difference method with Richardson extrapolation confirms the conclusion validity. The structure of an accelerated turbulent flow having constant flow velocity in the input section shows that for small Forchheimer coefficients, the Darcy – Brinkman equation is used to obtain the analytical ratio for calculating the length of the initial hydrodynamic section. The proposed mathematical model for estimating the heterogeneity of the velocity field in adsorbers with a stationary dispersed layer is applicable for a laminar flow regime. Testing of this approach by assessing velocity field uniformity for a mass-produced front-end purification unit of air separation plants has shown its efficiency.

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Analysis for Flow of an Incompressible Brinkman-Type Fluid in Thin Medium with Friction

Journal of Function Spaces ◽

10.1155/2021/5112840 ◽

2021 ◽

Vol 2021 ◽

pp. 1-8

Author(s):

Soumia Manaa ◽

Salah Boulaaras ◽

Hamid Benseridi ◽

Mourad Dilmi ◽

Sultan Alodhaibi

Keyword(s):

Fluid Flow ◽

Variational Formulation ◽

Reynolds Equation ◽

Three Dimensional ◽

Stationary Regime ◽

Limit Problem ◽

Brinkman Equation ◽

Asymptotic Convergence ◽

Thin Domain ◽

Uniqueness Of The Solution

In this paper, we consider the Brinkman equation in the three-dimensional thin domain ℚ ε ⊂ ℝ 3 . The purpose of this paper is to evaluate the asymptotic convergence of a fluid flow in a stationary regime. Firstly, we expose the variational formulation of the posed problem. Then, we presented the problem in transpose form and prove different inequalities for the solution u ε , p ε independently of the parameter ε . Finally, these estimates allow us to have the limit problem and the Reynolds equation and establish the uniqueness of the solution.

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The non-Darcy characteristics of fault water inrush in karst tunnel based on flow state conversion theory

Thermal Science ◽

10.2298/tsci2106415c ◽

2021 ◽

Vol 25 (6 Part B) ◽

pp. 4415-4421

Author(s):

Zheng-Zheng Cao ◽

Yu-Feng Xue ◽

Hao Wang ◽

Jia-Rui Chen ◽

Yu-Lou Ren

Keyword(s):

Flow Velocity ◽

Water Inrush ◽

Coupled Model ◽

Flow Fields ◽

Darcy Law ◽

Brinkman Equation ◽

Flow State ◽

Navier Stokes Equation ◽

Velocity Change ◽

Key Factor

The fault water inrush is a key factor which leads to tunnel construction in karst regions. Based on the fluid mechanics principles, the paper addresses a numer?ical coupled model for karst fault tunnel with COMSOL Multiphysics software. Besides, the Darcy law equation, Brinkman equation, and Navier-Stokes equation are inserted to stimulate the steady flow of aquifer, the non-linear seepage of fault and the free flow in tunnel excavating area in software, respectively. Then, the pres?sure and flow velocity in three flow fields are analyzed under different permeability ratios in numerical model. It is shown that the fault permeability is the key factor affecting water inrush, and that the pressure and flow velocity change visibly in adjacent domains between two flow fields.

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Preconditioning of Linear Systems Arising in Finite Element Discretizations of the Brinkman Equation

Large-Scale Scientific Computing - Lecture Notes in Computer Science ◽

10.1007/978-3-642-29843-1_43 ◽

2012 ◽

pp. 381-389

Author(s):

P. Popov

Keyword(s):

Finite Element ◽

Linear Systems ◽

Brinkman Equation ◽

Finite Element Discretizations ◽

Element Discretizations

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Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer

Mathematics ◽

10.3390/math8111961 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1961

Author(s):

Kanognudge Wuttanachamsri

Keyword(s):

Mathematical Model ◽

Exact Solution ◽

Free Boundary ◽

Horizontal Plane ◽

Numerical Solutions ◽

Numerical Results ◽

Brinkman Equation ◽

Average Height ◽

Free Interface ◽

The Mathematical Model

Cilia on the surface of ciliated cells in the respiratory system are organelles that beat forward and backward to generate metachronal waves to propel mucus out of lungs. The layer that contains the cilia, coating the interior epithelial surface of the bronchi and bronchiolesis, is called the periciliary layer (PCL). With fluid nourishment, cilia can move efficiently. The fluid in this region is named the PCL fluid and is considered to be an incompressible, viscous, Newtonian fluid. We propose there to be a free boundary at the tips of cilia underlining a gas phase while the cilia are moving forward. The Brinkman equation on a macroscopic scale, in which bundles of cilia are considered rather than individuals, with the Stefan condition was used in the PCL to determine the velocity of the PCL fluid and the height/shape of the free boundary. Regarding the numerical methods, the boundary immobilization technique was applied to immobilize the moving boundaries using coordinate transformation (working with a fixed domain). A finite element method was employed to discretize the mathematical model and a finite difference approach was applied to the Stefan problem to determine the free interface. In this study, an effective stroke is assumed to start when the cilia make a 140∘ angle to the horizontal plane and the velocitiesof cilia increase until the cilia are perpendicular to the horizontal plane. Then, the velocities of the cilia decrease until the cilia make a 40∘ angle with the horizontal plane. From the numerical results, we can see that although the velocities of the cilia increase and then decrease, the free interface at the tips of the cilia continues increasing for the full forward phase. The numerical results are verified and compared with an exact solution and experimental data from the literature. Regarding the fixed boundary, the numerical results converge to the exact solution. Regarding the free interface, the numerical solutions were compared with the average height of the PCL in non-cystic fibrosis (CF) human tissues and were in excellent agreement. This research also proposes possible values of parameters in the mathematical model in order to determine the free interface. Applications of these fluid flows include animal hair, fibers and filter pads, and rice fields.

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Dynamics of Swimmers in Fluids with Resistance

Fluids ◽

10.3390/fluids5010014 ◽

2020 ◽

Vol 5 (1) ◽

pp. 14 ◽

Cited By ~ 1

Author(s):

Cole Jeznach ◽

Sarah D. Olson

Keyword(s):

Fluid Dynamics ◽

Fluid Flow ◽

Cell Body ◽

Complex Dynamics ◽

Viscous Fluids ◽

Hydrodynamic Interactions ◽

Brinkman Equation ◽

Resistance Parameter ◽

Fluid Resistance ◽

Stationary Obstacles

Micro-swimmers such as spermatozoa are able to efficiently navigate through viscous fluids that contain a sparse network of fibers or other macromolecules. We utilize the Brinkman equation to capture the fluid dynamics of sparse and stationary obstacles that are represented via a single resistance parameter. The method of regularized Brinkmanlets is utilized to solve for the fluid flow and motion of the swimmer in 2-dimensions when assuming the flagellum (tail) propagates a curvature wave. Extending previous studies, we investigate the dynamics of swimming when varying the resistance parameter, head or cell body radius, and preferred beat form parameters. For a single swimmer, we determine that increased swimming speed occurs for a smaller cell body radius and smaller fluid resistance. Progression of swimmers exhibits complex dynamics when considering hydrodynamic interactions; attraction of two swimmers is a robust phenomenon for smaller beat amplitude of the tail and smaller fluid resistance. Wall attraction is also observed, with a longer time scale of wall attraction with a larger resistance parameter.

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