Permeable Walls Research Articles

Thermophoresis of a spherical particle in a permeable microchannel with thermal stress slip

A study of the thermophoretic motion of a spherical aerosol particle in a microchannel with permeable walls bounding a porous medium is presented. The field equations of energy and momentum for the system are solved using a semi-analytical method with the help of a boundary collocation technique. The study includes the effect of thermal stress slip at the surface of the particle in addition to temperature jump, thermal creep, and frictional slip.

TURBULENT BOUNDARY LAYER ON A PERMEABLE PLATE IN SUPERSONIC FLOW WITH POSITIVE PRESSURE GRADIENT UNDER FOREIGN GAS INJECTION

A numerical simulation of a turbulent boundary layer on a permeable plate with helium injection into a supersonic xenon flow in the presence of a positive longitudinal pressure gradient has been carried out. The injection regimes are considered, in which the temperature of the injected gas is lower than the temperature of the adiabatic impermeable wall and the stagnation temperature of the oncoming flow. The existence of a minimum temperature of the permeable wall is confirmed, at which the wall temperature is lower than the temperature of the injected gas, while there are two sections along the length of the permeable plate in which the adiabatic conditions are satisfied. The results of calculations of a supersonic flow with a longitudinal pressure gradient differ significantly from the results obtained for a gradientless flow around a plate, both for subcritical and critical injection.

MHD flow of non-Newtonian ferro nanofluid between two vertical porous walls with Cattaneo–Christov heat flux, entropy generation, and time-dependent pressure gradient

This article studies the magnetohydrodynamic flow of non-Newtonian ferro nanofluid subject to time-dependent pressure gradient between two vertical permeable walls with Cattaneo–Christov heat flux and entropy generation. In this study, blood is considered as non-Newtonian fluid (couple stress fluid). Nanoparticles’ shape factor, Joule heating, viscous dissipation, and radiative heat impacts are examined. This investigation is crucial in nanodrug delivery, pharmaceutical processes, microelectronics, biomedicines, and dynamics of physiological fluids. The flow governing partial differential equations are transformed into the system of ordinary differential equations by deploying the perturbation process and then handled with Runge–Kutta 4th-order procedure aided by the shooting approach. Hamilton–Crosser model is employed to analyze the thermal conductivity of different shapes of nanoparticles. The obtained results reveal that intensifying Eckert number leads to a higher temperature, while the reverse is true for increased thermal relaxation parameter. Heat transfer rate escalates for increasing thermal radiation. Entropy dwindles for intensifying thermal relaxation parameter.

Viscous Flow by Expanding or Shrinking the Gap with Permeable Walls Through Hermite Wavelet Method

Viscous Flow by Expanding or Shrinking the Gap with Permeable Walls Through Hermite Wavelet Method

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

Oblique wave trapping by a permeable wall breakwater connected with thick surface porous layer

ABSTRACT The reflection of surface gravity waves by a vertical wall breakwater connected with a thick surface porous layer is analysed under the framework of potential flow theory. The thick surface porous layer is placed within the confined space available between the vertical permeable wall and the impermeable seawall. The analytical investigation is progressed by creating the hidden water chamber beneath the surface porous layer. The Matched Eigenfunction Expansion Method (MEEM) is used based on the continuity of velocity and pressure along with orthogonal mode-coupling relation. The hidden water chamber will be helpful to enhance the trapping of incident wave energy and also to reduce the quantity of construction material. The present study results are well agreed with the results available in the literature when a hidden water chamber is absent. The effect of permeable wall porosity, thick surface layer porosity, angle of wave contact, depth of the surface porous layer, thickness of the permeable wall, and hidden chamber spacing on wave reflection against incident wave properties and breakwater physical properties are studied. A comparative study is performed between three different types of porous breakwaters to investigate the integrity of the proposed breakwater. The minor value of wave reflection and gradual reduction of fluid force experienced by the rigid wall is achieved due to the presence of a surface layer, which maximises the incident wave damping, as most of the wave energy propagates near the free surface.

Stokes flow solutions in infinite and semi‐infinite porous channels

AbstractWe derive a class of exact solutions for Stokes flow in infinite and semi‐infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no‐slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi‐infinite channel domain, the ground‐state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane‐based water purification, heat and mass transfer, and fuel cells.

Short-crested wave-current forces around a concentric multiple-cylinder structure

ABSTRACT The potential flow theory is used to develop a new analytical method to solve the diffraction problem of short-crested incident waves with uniform current acting on a concentric multiple-cylinder system. The influence of uniform current on the hydrodynamic performance of the concentric structure is discussed. The incident angle and speed of the currents have a significant influence on the short-crested wave force and run-up on the concentric structure, i.e. the wave force and wave run-up increase significantly when wave and uniform current directions are the same, while decreasing when in the opposite direction of the wave and uniform current. Additionally, the effects of parameters such as the current incidence angle, current speed, porous-effect parameters, number of perforated walls, and short-crestedness of regular waves on the hydrodynamic performance of the concentric structures are valuated by numerical experiments. It is observed that as the number of permeable walls increases, wave load on the impermeable internal cylinder gradually decreases and the wave surface around is more even. This study is expected to provide theoretical guidance for the design of nearshore architecture.

Mathematical model of fluid flow in a channel with reabsorption at permeable walls

Mathematical model of fluid flow in a channel with reabsorption at permeable walls

Multiscale Analysis of Permeable and Impermeable Wall Models for Seawater Reverse Osmosis Desalination

In recent years, high permeability membranes (HPMs) have attracted wide attention in seawater reverse osmosis (SWRO) desalination. However, the limitation of hydrodynamics and mass transfer characteristics for conventional spiral wound modules defeats the advantage of HPMs. Feed spacer design is one of the effective ways to improve module performance by enhancing permeation flux and mitigating membrane fouling. Herein, we propose a multiscale modeling framework that integrates a three-dimensional multi-physics model with a permeable wall and an impermeable wall, respectively, at a sub-millimeter scale and a system-level model at a meter scale. Using the proposed solution framework, a thorough quantitative analysis at different scales is conducted and it indicates that the average errors of the friction coefficient and the Sherwood number using the impermeable wall model are less than 2% and 9%, respectively, for commercial SWRO membrane (water permeability 1 L m−2 h−1 bar−1) and HPMs (3 L m−2 h−1 bar−1, 5 L m−2 h−1 bar−1 and 10 L m−2 h−1 bar−1) systems, compared to the predictions using the permeable wall model. Using both the permeable and impermeable wall models, the system-level simulations, e.g., specific energy consumption, average permeation flux, and the maximum concentration polarization factor at the system inlet are basically the same (error < 2%), while the impermeable wall model has a significant advantage in computational efficiency. The multiscale framework coupling the impermeable wall model can be used to guide the efficient and accurate optimal spacer design and system design for HPMs using, e.g., a machine learning approach.

When is the Porous, Laminar Flow Problem with Slip Condition Well Posed? And Where Does the Solution Lie?

The aim of this article is to advance the current state of knowledge for steady, isothermal, incompressible, laminar flow within a channel featuring a non-zero tangential (or slip) velocity at the permeable walls. There has been significant interest in understanding the solutions to these problems. However, a firm mathematical understanding of the solutions to the slip problem and their properties is yet to be fully developed. For example, we still do not know: if the slip problem is well-posed; where the precise solution lies; if and how approximations converge to the solution; and what the estimates on approximation errors are. Herein we formulate a new mathematical foundation that includes existence; uniqueness; location; approximation; convergence and error estimates. Our strategy involves developing insight via new and interesting connections between the boundary value problem arising from modelling the laminar flow with slip velocity, and the theory of fixed points of operators.

Buoyancy-driven flow of magnetohydrodynamic hybrid nanofluids in an open cavity with permeable horizontal walls

The buoyancy-driven flow of hybrid nanofluids in an open cavity is examined in the presence of a magnetic field with an angle to the horizontal axis. The right end of the cavity is open, and the left wall is supposed to be heated. Moreover, the horizontal walls are deemed to be permeable and maintained at ambient temperature. The successive over-relaxation (SOR) technique is applied with the finite-difference method to solve the dimensionless equations. A comparison is executed with formerly published results which provide a good agreement. The grid refinement test has also been carried out to increase the accuracy. Flow and temperature profiles are investigated for the Rayleigh number ( Ra = 104, 105, 106), Reynolds number (Re = 5, 8, 10, 20, 100), Hartmann number ( Ha = 0, 5, 10), concentration of nanoparticles ( φ1 or φ2 = 0.0, 0.05, 0.1), angle of inclination of the magnetic field ( γ = 0°, 30°, 45°), aspect ratio ( A = 1, 2, 3, 4), and suction/blow parameter ( S = −1, 0, 1). Results have been elucidated based on streamlines, isotherms, local Nusselt number ( Nu), and average Nusselt number ( Nuavg). It is noted that the changes in the variables significantly affect streamlines and isotherms. The heat transfer for Cu-Fe3O4/water hybrid nanofluid is higher about 9.98% compared to Fe3O4/water nanofluid and 26.41% compared to water. Furthermore, all other parameters noticeably augment both Nu and Nuavg.

Laminar Flow in a PEM Fuel Cell Cathode Channel

A Berman’s model for the laminar flow of incompressible fluid in a channel with permeable walls is extended for variable along the channel velocity of injection. The system of two–dimensional continuity and Navier–Stokes equations is reduced to a single ODE for the transversal velocity with coefficients depending on position along the channel. Numerical solution for the flow in the cathode channel of a PEM fuel cell is presented. The flow velocity profile across the channel is almost indistinguishable from the Poiseuille’s parabolic shape; however, the distribution of pressure gradient and longitudinal velocity differ quite significantly from the Berman’s result.

Analysis of a Steady MHD Mixed Convection Fluid Flow in a Microchannel within Permeable Walls with Suction and Injection Parameters

Analysis of a Steady MHD Mixed Convection Fluid Flow in a Microchannel within Permeable Walls with Suction and Injection Parameters

The analysis of clay soil shrinkage

The paper presents the analysis of the capillary and water-colloidal theories of clayey soil shrinkage. The method by which the volumetric shrinkage of the clay soil was determined is that the soil sample is made conical, with an angle at the top of 45-75º, and it is dried in a container geometrically similar to the sample with permeable walls. With kaolinite clays, the shrinkage process has a duration of one week, with hydromicaceous clay of around two weeks, and with montmorillonitic clays three weeks. There is a parabolic dependence between the duration of clay shrinkage and the specific surface area of clay. A series of experiments and calculations have been performed, based on Kulchitsky-Usyarov clay soil model, to determine the limit values of capillary pressure in montmorillonite, hydromicaceous and kaolinite clays using the Laplace formula. At the limit of normal shrinkage, the total porosity for Ca-montmorillonite is 0.45, for Ca-hydromica - 0.25, for Na-kaolinite - 0.385. Calculations showed that during shrinkage the capillary pressure reaches: for Ca-montmorillonite – 48.6 MPa; for Ca-hydromica 8 MPa; for Na-kaolinite - 5.2 MPa. The results obtained practically coincide with the results obtained by K. Terzaghi. Studies on the cyclic “moisture-drying” of clays selected at an experimental site in Ujar (Azerbaijan) showed that after the first “swelling-shrinkage” cycle, with secondary moistening, the swelling of the samples was 12-18 times higher than the primary value.

A PULSATILE WILLIAMSON NANOFLUID FLOW WITH MOTILE MICROORGANISMS BETWEEN TWO PERMEABLE WALLS: EFFECT OF MODIFIED DARCY'S LAW

The aim of the proposed scheme is to look at the non-Newtonian Williamson magnetohydrodynamics (MHD) pulsatile movement of an incompressible nanofluid in the existence of motile microorganisms. The flow is saturated in porous media between two permeable vertical walls. The non-Newtonian fluid is described by employing the modified Darcy's law. An unchanged normal magnetic field pervades the structure to the walls. The novelty of the ongoing work originates from a desire to highlight the role played by microbes and bacteria, as well as nanoparticles in the use of significant pulsatile pump flow. The heat transmission, effects of viscous dissipation, and thermal radiation, in addition to volumetric nanoparticles fraction and microorganism concentration, are scrutinized. Regarding the pulsatile flow, each of the distributions is divided into two steady and oscillatory parts. The nonlinear fundamental equations are analyzed by means of the homotopy perturbation method (HPM). The impact of numerous physical restrictions is examined and depicted in a series of graphs. It is found that the existence of microorganisms obstructs the flow and reduces its speed. This infection can be avoided by reducing the parameters of buoyancy and bio-convection. All heat-related parameters are found as incremental factors of heat flux, which is an important phenomenon in many medical and engineering applications.

Hall and Joule Heating Impacts on the Rabinowitsch Fluid in a tapered Channel with Permeable Walls

A mathematical model was created to study the influences of Hall current and Joule heating with wall slip conditions on peristaltic motion of Rabinowitsch fluid model through a tapered symmetric channel with Permeable Walls. The governing equations are simplified under low Reynolds number and the long-wavelength approximations. The perturbation method is used to solve the momentum equation. The physiological phenomena are studied for a certain set of pertinent parameters. The effects offered here show that the presence of the hall parameter, coefficient of pseudo-plasticity, and Hartman number impact the flow of the fluid model. Additional, study reveals that a height in the Hall parameter and the velocity slip parameter increases the trapping bolus's apparition. Furthermore, the magnitude of the trapped bolus can be reduced by enhancing the magnetic field.

Radiative and MHD Effects on Time-Dependent Thermal-Material Transfer by Micropolar Binary Mixture

Radiation is an important branch of thermal engineering which includes geophysical thermal insulation, ground water pollution, food processing, cooling of electronic components, oil recovery processes etc. An analysis of unsteady magneto-convective heat-mass transport by micropolar binary mixture of fluid passing a continuous permeable surface with thermal radiation effect has been introduced in this paper. The governing equations are transformed into coupled ordinary differential equations along with Boussinesq approximation by imposing the similarity analysis. Applying the shooting technique, the obtained non-linear coupled similarity equations are solved numerically with the help of “ODE45 MATLAB” software. The results of the numerical solutions to the problem involving velocity, temperature, concentration and micro-rotation are presented graphically for different dimensionless parameters and numbers encountered. With an increase of suction parameter, the velocity distributions very closed to the inclined permeable wall decrease slightly where 0 ≤ η ≤ 0.3 . But for the uplifting values of sunction, both micro-rotation profile and species concentration enhance through the boundary layer. The skin-friction coefficient increases about 61%, 13%, 27% for rising values of Prandtl number (0.71-7), radiation effect (0 - 1) and thermal Grashof number (5-10), respectively, but an adverse effect is observed for magnetic field (1 - 4), inclined angle 0 0 − 60 0 and Schmidt number (0.22 - 0.75). Heat transfer and mass transfer reduce about 82%, 53%, respectively, in increasing of Pr (0.71-1) and 36%, 11%, respectively, in increasing of thermal radiation (0 - 1). The surface couple stress increases about 26%, 49%, 64% and 30% with the increasing values of magnetic field (1-4), inclination angle 0 0 − 60 0 , suction (0-1) and Schmidt number (0.22-0.75), respectively. Finally, the present study has been compared with the earlier published results. It is observed that the comparison bears a good agreement.

Study of non-Newtonian synovial fluid flow by a recursive approach

This study analyzes the non-Newtonian synovial fluid flow between the joints in a synovitis, which is a diseased condition due to inflammation of synovial membrane. It is assumed in this study that the secretion of synovial fluid through the inflamed synovial membrane is a linear function of the membrane length. The mathematical modeling of synovial fluid through a synovial membrane is made by the non-Newtonian Linear Phan-Thien–Tanner (LPTT) fluid model through a thin conduit having permeable walls. The nonlinear flow of LPTT fluid gives the non-homogeneous complex boundary value problem, and the recursive approach is used to solve the problem. The flow of synovial fluid along and across the membrane is calculated under the inflamed membrane, and results are displayed through graphs. The axial pressure required for the non-Newtonian fluid flow and deformation of synovial fluid that produces the shearing forces near the synovial membrane are also calculated. The purpose of this research is to observe the shear stress on the synovial fluid and inflammation rate on the flow along the membrane at different position and pressure required for the flow of synovial fluid in diseased condition. The mathematical and graphical results for pressure, flow, volume flux, and streamline are calculated and plotted using the software MATHEMATICA. This study is very helpful for the biomedical engineers to measure the compression force and shear stress on the synovial fluid in a diseased condition and can be controlled by the viscosity of the synovial fluid.

One Case of Shock-free Deceleration of a Supersonic Flow in a Constant Cross Section Area Channel

Air flows with supersonic speeds are used in many cases, for example, as aircraft air intakes, wind tunnels, and energy separation devices. In many cases it is necessary to decelerate the flow to sonic speeds. Traditionally the deceleration realized through the shocks system, which leads to total pressure losses. The article considers the method of deceleration of supersonic flows using permeable surfaces. In this case, the deceleration process occurs without shocks and, therefore, with lower total pressure losses. We have considered the flow in a tube with permeable wall located behind a supersonic nozzle. One-dimensional and axisymmetric mathematical models of such a device are developed. The calculation results are compared with experimental data. It is shown that, depending on the ratio of pressure inside the tube and the ambient pressure, different flow regimes inside the tube are possible: pure subsonic, transitional from supersonic to subsonic, and pure supersonic. The transition from supersonic to subsonic flow occurs without shocks due to the suction and friction combined effects.