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On a hydrodynamic permeability of a system of coaxial partly porous cylinders with superhydrophobic surfaces
... Vasin et al. [21], Tiwari et al. [22] used Brinkman and Darcy formulations respectively to analyze flow past swarm of solid cylindrical particles with porous layers having varied specific permeability. Filippov and Koroleva [23] provided a study of the Stokes-Brinkman system with varying viscosity that describes the fluid flow along an ensemble of partially porous cylindrical particles with superhydrophobic surface. Recently Filippov and Ryzhikh [24] employed the cell method to calculate the hydrodynamic permeability of a porous medium composed of a set of partially porous spherical particles with varying viscosity and solid impermeable cores taking tangential stress jump condition into account. ...
... As the considered BVPs statements have the only difference on the cell surface and this difference involves spin vector, the biggest discrepancy in curves is observed for the angular velocities at r → m. Dashed curve in Fig. 2b tends to zero with r → m in accordance with condition (24), while solid curve in Fig. 2b demonstrates behavior analogous to the processes with free boundary for r → m, which is characterized by no couple stress condition (23). Negative values of angular velocity in Fig. 2b point out that the direction of the spin vector is opposite to that shown in Fig. 1. ...
... Negative values of angular velocity in Fig. 2b point out that the direction of the spin vector is opposite to that shown in Fig. 1. Linear velocity profiles, shown in Fig. 2a, increase from solid core towards cell surface under both boundary conditions (23) and (24). They demonstrate very close to each other and almost linear growth in the porous region and deformed parabolic growth in the liquid region. ...
The present paper considers the flow of micropolar fluid through a membrane modeled as a swarm of solid cylindrical particles with porous layer using the cell model technique. Traditional boundary conditions on hypothetical cell surface were added with an additional condition: the no spin condition / no couple stress condition. Expressions for velocity and microrotation vector components have been obtained analytically. Effect of various parameters such as particle volume fraction, permeability parameter, micropolarity number etc. on hydrodynamic permeability of membrane has been discussed.
... Fillipov and Koroleva [21] solved the governing equations of porous and non-porous regions with varying viscosity using the cell method and studied the existence of superhydrophobic sur-Accepted to Phys. Fluids 10.1063/5.0143317 4 faces. ...
The present study is an attempt to deal with hydrodynamic and thermal aspects of the incompressible Carreau fluid flow past a membrane consisting of uniformly distributed aggregates of porous cylindrical particles enclosing a solid core which aims to provide a comprehensive study of the impact of non-Newtonian nature of Carreau fluid in the filtration process through membranes. The non-Newtonian characteristic of Carreau fluid is adopted to describe the mechanism of the pseudoplastic flow through membranes. The layout of the fluid flow pattern is separated into two distinct areas in which the area adjacent to the solid core of the cylindrical particle is considered as porous. However, the region surrounding the porous cylindrical particle is taken as non-porous (clear fluid region). The Brinkman equation governs the porous region, whereas the non-porous region is regulated by the Stokes equation. The nonlinear governing equations of the Carreau fluid flow in the different regions are solved using an asymptotic series expansion in terms of the small parameters such as Weissenberg number $(\mathrm{We}\ll 1)$ and a non-dimensional parameter $(S\ll 1)$ for the higher permeability of the porous material. The notable determination of the present study is that the Carreau fluid parameters, such as the Weissenberg number, Power-law index, and viscosity ratio parameter have a significant impact on the velocity, and hence the membrane permeability, Kozeny constant and temperature profile. The findings of the proposed work may be instrumented in analyzing various processes, including wastewater treatment filtration processes, and blood flow through smooth muscle cells.
... Filippov and Koroleva (2017) and Koroleva (2017) discussed the qualitative properties of the viscous fluid flow through membranes describing a partially filled porous material using varying viscosity approach and deduced the existence and uniqueness of the solutions as well as uniform estimates for boundary value problems. Further, Filippov and Koroleva (2018) investigated the dependencies of the hydrodynamical permeability of the membranes on the varying nature of viscosity using the polynomial and exponential viscosity models. Recently, Tiwari and Chauhan (2019b,a,c,d) examined the impact of hematocrit-dependent viscosity on the circulation of blood flow through microvessels and observed that the variable nature of viscosity play an important role to compute the correct measurement of the hemodynamical quantities which is more important for medical treatment. ...
The majority of the previous studies analyzed the flow of fluids with constant viscosity through membranes composed of porous cylindrical particles using the particle-in-cell approach with Brinkman equation governing the flow through porous media. However a slight variation in temperature affects the viscosity of the fluids and hence affects the filtration process of fluids through membranes. The motivation of this problem came from the fact that viscosity is concentration dependent due to presence of impurities and contaminants in the fluids and hence can be taken as function of position or temperature. The present work is a theoretical attempt to investigate the impact of temperature-dependent viscosity on the creeping flow of Jeffrey fluid through membrane consisting the aggregates of the porous cylindrical particles. The flow pattern of the Jeffrey fluid is taken along the axial direction of the cylindrical particles and the cell model approach is utilized to formulate the governing equations driven by a constant pressure gradient. The flow regime is divided into two-layer form, one is inside the porous cylindrical particle enclosing a solid core, which is governed by the Brinkman-Forccheimer equation, and another one is outside of the porous cylindrical particle, which is governed by the Stokes equation. Being a nonlinear equation, an analytical solution of the Brinkman-Forchheimer equation is intractable. To overcome this difficulty, the regular and singular perturbation method is employed to solve the Brinkman-Forchheimer equation under the assumption of temperature-dependent viscosity for small and large permeability of the porous medium, respectively; however, an analytical approach is utilized to solve the Stokes equation. The analytical expressions for velocity in different regions, hydrodynamic permeability of the membrane, and Kozeny constant are derived. The impact of various control parameters such as viscosity parameter, Forchheimer number, permeability of the porous medium, and Jeffrey fluid parameter on the above quantities are discussed and validated with previously published works on the Newtonian fluid in the limiting cases. The present work is in good agreement with the previously published work on Newtonian fluid under constant viscosity assumptions where the porous media flow was governed by the Brinkman equation. The remarkable observation of the present study is that higher viscosity and Jeffrey fluid parameters lead to enhanced velocity profile and hence the hydrodynamical permeability of the membranes. However, a decay in the Kozeny constant is observed with the increasing viscosity and Jeffrey fluid parameters. The coating of porous layer can be attributed to adsorption of polymers on the solid particles and further makes the present model to be more relevant in understanding the membrane filtration process.
... Using this model for such fluid flow, they also mentioned the streamlines and velocity profiles of various diameter ratios. Filippov and Koroleva (2018) explained the incompressible viscous fluid flow along an ensemble of partially porous cylindrical particles using the cell method on the Stokes-Brinkman model when viscosity has secondpower polynomial growth. They evaluated the hydrodynamic permeability of a porous medium modelled as a group of co-axial porous cylinders with superhydrophobic surfaces, as well as a set of partly porous particles immersed into liquid concentric cells, plus analytical formulas for the velocity of the fluid flow and the coefficient of hydrodynamic permeability. ...
This article discusses the extended expression of the stream function solution of the Brinkman equation in the cylindrical polar coordinates. Analytical solutions of this equation are available in the coordinates (Cartesian, cylindrical polar, spherical polar, and prolate spheroidal coordinates). Using a relation connecting the hypergeometric function
and modified Bessel function, stream function solutions of the Brinkman equation reported earlier by Pop and Cheng (1992) and Deo et al. (2016) are also deduced.
The present article reveals the study of an electrohydrodynamic flow through a membrane composed of a swarm of porous layered cylindrical particles adopting a heat transfer approach. The configuration of the proposed theoretical model is segregated into two regions in which the region proximate to the solid core of the cylindrical particle is a porous region. However, a region surrounded by a porous region is a non-porous (clear fluid) region. The thermal equations are employed under steady-state conditions to establish the temperature distribution when heat conduction prevails over heat convection. The Brinkman and Stokes equations regulate fluid flow through a swarm of porous layered cylindrical particles in porous and non-porous regions, respectively. With the purpose of addressing an electric field in the fluid flow process through a swarm of porous layered cylindrical particles to understand the role of a Hartmann electric number, the momentum equation and the charge density are coupled and nonlinear. The nonlinear second-order differential equation governs the momentum equation and regulates fluid flow through a swarm of porous cylindrical particles. The solutions of the energy equations for both regions are analytically obtained. The asymptotic expansions of velocities for porous and non-porous regions have been derived using the perturbation technique for the small and large values of the nonlinearity parameter α. The effects of various parameters like Hartmann electric number, Grashof number, radiation parameter, viscosity ratio parameter, and porosity of the porous material on the hydrodynamical permeability, Kozeny constant of the membrane, and temperature are analyzed graphically. A noteworthy observation is that a rising Hartmann electric number, the ratio of electric force to the viscous force, enhances the velocity, which is relatively more significant for higher permeability and hence enhances the membrane permeability; however, decay in Kozeny constant is reported with a rising Hartmann electric number. Significant velocity and membrane permeability growth are described with a rising Grashof number, a ratio of thermal buoyancy and viscous forces. The observations from the present study hold promise for advancing our understanding of critical physical and biological applications, including wastewater treatment filtration processes, petroleum reservoir rocks, and blood flow through smooth muscle cells.
The present work is an effort to investigate the dispersion process of reactive mass transport in human blood vessels for varying viscosity and permeability. Considering a four-layer model of Luo and Kuang (KL)-Newtonian fluids flowing through a biporous layered microvessel, the current research focuses on unsteady mass transport under the first-order chemical reaction. A two-fluid model is adopted where the core region contains the KL fluid, depicting the flow of blood cells, and the coaxial peripheral region represents the plasma region. The outer plasma layer containing Newtonian fluid is segregated into three sublayers, where adjacent to the KL fluid is the non-porous plasma region. The outer two regions, the Brinkman and the Brinkman-Forchheimer regions exhibit radially varying permeability and viscosity characteristics. Analyzing the impact of the Froude number on the solute dispersion process necessitates incorporating an additional body force into the analysis. The porous medium equations are solved using regular and singular perturbation techniques to obtain closed-form solutions. Nevertheless, analytical solutions for the KL fluid and non-porous plasma layer have been derived. The analytical solution of mass distribution due to advection and diffusion is obtained through the Gill and Sankarasubramanian (1970) approach with the aid of Hankel transformation. The effect of various parameters such as Darcy’s number, Forchheimer number, Reynolds number, Froude number, permeability parameters (𝜀1, 𝜀2), and viscosity
parameters (𝛼1, 𝛼2) on the transport coefficients and mean concentration are discussed graphically. Higher Froude numbers caused weaker dispersion, while the parameters of the Brinkman-Forchheimer region have a significant effect on mass transport compared to the parameters of the Brinkman region. The findings of the current study may assist physiologists in developing a more nuanced understanding of these complex processes, ultimately leading to improved clinical outcomes.
This study examines the flow of a Newtonian fluid enclosed between two non-Newtonian Jeffreys fluids with viscosity that varies with temperature within a composite vertical channel. Including a corotational Jeffreys liquid allows for considering stress dependence on the present deformation rate and its history. The proposed study's framework comprises three distinct regions, wherein the intermediate region governs Newtonian fluid flow under temperature-dependent viscosity. However, the outer layers oversee the flow of Jeffreys fluids within the porous medium, demonstrating temperature-dependent viscosity. The Brinkman–Forchheimer equation is employed to establish the governing equations applicable to both low and high permeabilities of the porous medium. This equation is nonlinear, making it challenging to find an analytical solution. Therefore, the regular and singular perturbation methods with matched asymptotic expansions are applied to derive asymptotic expressions for velocity profiles in various regions. The hydrodynamic quantities, such as flow rate, flow resistance, and wall shear stresses, are determined by deriving their expressions using velocities from three distinct regions. The graphical analysis explores the relationships between these hydrodynamic quantities and various parameters, including the Grashof number, Forchheimer number, viscosity parameter, Jeffreys parameter, conductivity ratio, effective viscosity ratio, absorption ratio, and the presence of varying thicknesses of different layers. An interesting finding is that a more pronounced velocity profile is noticed when the permeability is high and the viscosity parameter of the Newtonian region, denoted as α2, is lower than that of the surrounding area. This heightened effect can be linked to a relatively more significant decrease in the viscosity of the Jeffreys fluid, represented by μ1, as compared to the viscosity of the Newtonian fluid, μ2, as the temperature increases. The outcomes of this research hold special significance in situations like the extraction of oil from petroleum reserves, where the oil moves through porous layers with varying viscosities, including sand, rock, shale, and limestone.
In the limit of low Reynolds number, the effect of variable viscosity engendered by the rotation of a porous sphere in a closed spherical boundary is filled up with a viscous fluid. The fluid motion is steady and axisymmetric. Stokes equation holds in the liquid region and porous region obeys the Brinkman equation. The slip boundary condition is used on the surface of cavity. The continuity of velocity components and stress jump boundary condition on the porous-liquid surface are used. The couple and wall correction factor exerted by the surrounding viscous fluid on a rotating porous sphere are obtained. The effect of apparent viscosity, permeability parameter, slip parameter, and stress jump coefficient on the wall correction factor are discussed. Special cases are recovered.
The paper concerns numerical study of a Stokes-Brinkman system with varying liquid viscosity that describes the fluid flow along a set of partially porous parallel cylindrical particles, which form a fibrous
membrane, using the cell modeling. We have applied two different approaches to varying viscosity inside a porous layer: exponential and power decaying laws and compared the results. We proposed new numerical scheme based on Variation Iteration Method in order to solve the boundary value problems under consideration and compared results for velocity profiles near a porous cylindrical particle with an impermeable core placed into concentric liquid shell. We calculated the hydrodynamic permeability of a membrane, regarded as a swarm of such parallelly packed particles (fibers), being perpendicular to the membrane surface, in case of liquid flow normal to the membrane surface for both cell models mentioned above. The existence and
uniqueness of the solutions obtained as well as some uniform estimates are also discussed.
This paper concerns the evaluation of the hydrodynamic permeability of a membrane built up by non-homogeneous porous cylindrical particles using four known boundary conditions, such as Happel’s, Kuwabara’s, Kvashnin’s, Mehta-Morse/Cunningham’s. For different values of flow parameters, the variations of the hydrodynamic permeability are presented graphically and discussed. Some earlier results reported for the drag force and the hydrodynamic permeability, have been verified.
The Happel-Brenner cell method has been employed to calculate the hydrodynamic permeability of a porous medium (membrane) composed of a set of partially porous spherical particles with solid impermeable cores. This representation is used to describe the globular structure of membranes containing soluble grains. The apparent viscosity of a liquid is suggested to increase as a power function of the depth of the porous shell from the viscosity of the pure liquid at the porous medium-liquid shell interface to some larger value at the boundary with the impermeable core. All known boundary conditions used for the cell surface, i.e., those proposed by Happel, Kuwabara, Kvashnin, and Cunningham, have been considered. Important limiting cases have been analyzed.
The paper deals with study of a Stokes-Brinkman system with varying viscosity that describes the fluid flow along an ensemble of partially porous cylindrica particles using the cell approach. We have proved the existence and uniqueness of the solutions as well as derived some uniform estimates.
This book provides a comprehensive and concise description of most important aspects of experimental and theoretical investigations of porous materials and powders, with the use and application of these materials in different fields of science, technology, national economy and environment. It allows the reader to understand the basic regularities of heat and mass transfer and adsorption occurring in qualitatively different porous materials and products, and allows the reader to optimize the functional properties of porous and powdered products and materials. Written in an straightforward and transparent manner, this book is accessible to both experts and those without specialist knowledge, and it is further elucidated by drawings, schemes and photographs.
Porous materials and powders with different pore sizes are used in many areas of industry, geology, agriculture and science. These areas include (i) a variety of devices and supplies; (ii) thermal insulation and building materials; (iii) oil-bearing geological, gas-bearing and water-bearing rocks; and (iv) biological objects. Structural Properties of Porous Materials and Powders Used in Different Fields of Science and Technology is intended for a wide-ranging audience specializing in different fields of science and engineering including engineers, geologists, geophysicists, oil and gas producers, agronomists, physiologists, pharmacists, researchers, teachers and students.
In Part I of this paper a stress jump condition was developed based on the non-local form of the volume averaged Stokes' equations. The excess stress terms that appeared in the jump condition were represented in a manner that led to a tangential stress boundary condition containing a single adjustable coefficient of order one. In this paper we compare the theory with the experimental studies of Beavers and Joseph [J. Fluid Mech. 30, 197–207 (1967)], and we explore the use of a variable porosity model as a substitute for the jump condition. The latter approach does not lead to a successful representation of all the experimental data, but it does provide some insight into the complexities of the boundary region between a porous medium and a homogenous fluid.
The liquid flow through a set of nonuniform porous cylinders with a impenetrable core are studied by the Happel—Brenner cell
method. Different orientations of cylinders relative to the liquid flow, such as transverse, longitudinal, and random, are
considered. Brinkman equations are used to describe the flow of liquid in the porous layer. All known boundary conditions
on the cell surface (Happel, Kuwabara, Kvashnin, and Cunningham conditions) are considered. The models proposed can be used
to describe the processes of reverse osmosis, as well as nano-and ultrafiltration.
The hydrodynamic permeability of a membrane simulated by a set of identical impenetrable cylinders covered with a porous layer is calculated by the Happel-Brenner cell method. Both transverse and longitudinal flows of filtering liquid with respect to the cylindrical fibers that compose the membrane are studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin, and Cunningham models are considered. Brinkman equations are used to describe the flow of liquid in the porous layer. Results that correspond to previously published data are obtained for the limiting cases. Theoretical values of Kozeny constants are calculated. The models proposed can be used to describe the processes of reverse osmosis, as well as nano- and ultrafiltration.
The flow of viscous liquid in the porous medium formed by cylindrical fibers coated with a fractal porous adlayer is considered. Based on the Happel-Brenner cell method, the hydrodynamic permeability of a medium is calculated using the Brinkman equations. The analysis is performed for boundary conditions on cell surfaces of four types corresponding to the Happel, Kuwabara, Kvashnin, and Cunningham models. Different (transversal, longitudinal, and random) orientations of fibers with respect to liquid flow are considered.
Superhydrophobic surfaces combine high aspect ratio micro- or nano-topography and hydrophobic surface chemistry to create super water-repellent surfaces. Most studies consider their effect on droplets, which ball-up and roll-off. However, their properties are not restricted to modification of the behaviour of droplets, but potentially influence any process occurring at the solid-liquid interface. Here, we highlight three recent developments focused on the theme of immersed superhydrophobic surfaces. The first illustrates the ability of a superhydrophobic surface to act as a gas exchange membrane, the second demonstrates a reduction in drag during flow through small tubes and the third considers a macroscopic experiment demonstrating an increase in the terminal velocity of settling spheres.
The momentum transfer condition that applies at the boundary between a porous medium and a homogeneous fluid is developed as a jump condition based on the non-local form of the volume averaged momentum equation. Outside the boundary region this non-local form reduces to the classic transport equations, i.e. Darcy's law and Stokes' equations. The structure of the theory is comparable to that used to develop jump conditions at phase interfaces, thus experimental measurements are required to determine the coefficient that appears in the jump condition. The development presented in this work differs from previous studies in that the jump condition is constructed to join Darcy's law with the Brinkman correction to Stokos' equations. This approach produces a jump in the stress but not in the velocity, and this has important consequences for heat transfer processes since it allows the convective transport to be continuous at the boundary between a porous medium and a homogeneous fluid.
The Happel—Brenner cell model is employed to calculate the hydrodynamic permeability of membranes composed of impenetrable
spherical particles coated with nonuniform porous layers. All of the boundary conditions corresponding to the Happel, Kuwabara,
Kvashnin, and Cunningham models are considered at the cell surface. Liquid flows in the porous layers are described by the
Brinkman equation. The force applied to a composite particle flowed around by a uniform stream is calculated.
A theoretical analysis of transport through charged membranes is presented using a cell model. In an attempt to improve the capillary tube model, we have modeled the membrane by an array of charged spheres. Three sets of partial differential equations describe this system: the generalized Nernst–Planck flux equations, the Navier–Stokes equation, and the Poisson–Boltzmann equation. These equations are averaged over a cell volume to yield a set of ordinary differential equations on the gross scale, that is, a length scale of the order of the membrane thickness. It is shown for hyperfiltration that the cell model will reject salt more efficiently than the tube model, and in the limit of Stokes’ flow the present analysis reproduces the rejection coefficient previously obtained for the capillary tube model. Results for the electrodialysis mode of membrane operation are also presented. Comparison is made with the capillary tube model for the same total charge and equal volume to surface ratio. The effect of a spatially varying wall charge density on membrane performance is discussed.
The forces experienced by parallel circular cylinders or spheres which
are distributed at random and homogeneously in a viscous flow are
investigated on the basis of Stokes’ approximation. The results
for the case of spheres are applied to the problem of sedimentation and
compared with several experimental data.
The free-surface model, successfully employed to predict sedimentation, resistance to flow, and viscosity in assemblages of spherical particles, has been extended to the case of flow relative to cylinders. It is shown to be in good agreement with existing data on beds of fibers of various types and flow through bundles of heat-exchanger tubes for cases where it can reasonably be expected to apply. Close agreement in the dilute range with the only theoretical treatment for flow parallel to a square array of cylinders provides interesting validation of the model.
A calculation is given of the viscous force, exerted by a flowing fluid on a dense swarm of particles. The model underlying
these calculations is that of a spherical particle embedded in a porous mass. The flow through this porous mass is decribed
by a modification of Darcy's equation. Such a modification was necessary in order to obtain consistent boundary conditions.
A relation between permeability and particle size and density is obtained. Our results are compared with an experimental relation
due to Carman.
A variant of the cell model for describing a suspension of spherical particles in a viscous fluid is proposed. In contrast to the existing models, the requirement that the tangential component of the velocity reach a minimum with respect to the radial coordinate is imposed as additional condition on the cell surface. It is shown that this requirement corresponds to the physical pattern of flow around the system of particles. As a result, an expression is obtained for the drag of a particle in the system, and the rate of precipitation of suspensions and emulsions is calculated.
Hydrodynamic properties of fractal aggregates with radially varying permeability are investigated in terms of two parameters: radius of equivalent solid sphere experiencing the same drag as the aggregate (Ω*radius of aggregate) and fluid collection efficiency (η) of the aggregate. Resistance to the fluid flow through the aggregate is predicted to increase with increasing fractal dimension, while the fluid collection efficiency is expected to decrease. It is shown that a reasonable estimate of the fluid drag force on a fractal aggregate may be obtained by assigning a constant volume-averaged porosity to the aggregate and using any of the expressions available in the literature for aggregates with uniform permeability. The two hydrodynamic parameters, Ω and η, are used to modify the existing expressions for interactions between solid spheres to account for the porous nature of aggregates and thus calculate the collision rate kernels for interacting aggregates. The ratio of hydrodynamic radius of an aggregate to its radius of gyration predicted by the proposed model was in reasonable agreement with an experimental value reported in the literature.
A hydrodynamic permeability of membranes built up by porous cylindrical or spherical particles with impermeable core is investigated. Different versions of a cell method are used to calculate the hydrodynamic permeability of the membranes. Four known boundary conditions, namely, Happel's, Kuwabara's, Kvashnin's and Cunningham/Mehta-Morse's, are considered on the outer surface of the cell. Comparison of the resulting hydrodynamic permeability is undertaken. A possible jump of a shear stress at the fluid-membrane interface, its impact on the hydrodynamic permeability is also investigated. New results related to the calculated hydrodynamic permeability and the theoretical values of Kozeny constant are reported. Both transversal and normal flows of liquid with respect to the cylindrical fibers that compose the membrane are studied. The deduced theoretical results can be applied for the investigation of the hydrodynamic permeability of colloidal cake layers on the membrane surface, the hydrodynamic permeability of woven materials.