Stress Jump Boundary Condition Research Articles

DRAG ON A POROUS SPHERE ENCLOSED IN A SOLID CORE EMBEDDED IN COUPLE STRESS FLUID

In this paper, the problem of steady and axisymmetrical creeping flow of couple stress fluid past a permeable sphere enclosed by a solid core is analyzed. The continuous case of normal velocity and tangential velocity, stress jump boundary condition, and couple stress to be vanishing conditions are applied on the surface of the porous sphere, and the nonpenetrability boundary condition is applied for solid sphere. The problem is expressed by using the Stokes and Brinkman equations, which describe both the flow outside and inside the porous sphere, respectively. Expressions for the couple stress tensor and velocity fields are obtained in terms of Gegenbauer polynomials and Macdonald functions. Both the pressure distribution and the stream function solution for the axially symmetric motion are explicitly solved. An analytical determination for the flow field in terms of stream function is examined by wielding the method of separation of variables. The drag force felt by a permeable sphere due to the external and internal flow is calculated. The impact of the viscosity coefficients and couple stress parameter on drag is investigated numerically, and the findings are displayed in graphical form. The findings show that the uniform flow of a couple stress fluid past a porous sphere enclosed by a solid core with stress jump condition has less drag than the flow of a couple stress fluid through a porous sphere with continuous case of shear stress, and the presence of stress jump coefficients reduces the drag force, pressure, and couple stresses. With reference to earlier, well-known cases, some unique cases of flow past a porous sphere have been validated.

Exploration of peristaltic pumping of Casson fluid flow through a porous peripheral layer in a channel

Abstract This article is aimed to investigate the peristaltic pumping of a two-layered model in a two-dimensional channel. The core region occupies Casson fluid, while the porous medium occupies the peripheral region. The fluid flow in a porous medium was described with a suitable model using the Brinkman-extended Darcy equation. In the interface between fluid and porous medium, a shear stress jump boundary condition was applied. Closed-form solutions were obtained in both regions (core and peripheral). The physical quantities of peristaltic flow, such as axial velocity, pumping and change in the interface, were derived and explained. The fluid flow was analyzed by different physical parameters such as viscosity, permeability, porosity, Casson parameter and Darcy number. It is observed that the peristalsis mechanism has greater pressure in a two-layered model containing a non-Newtonian fluid in contact with a porous medium compared to a viscous fluid in the peripheral layer. It was observed that pumping decreased with the increase in Darcy number and an increase in shear stress jump constant resulted in increasing the pumping. The outcomes of the pumping phenomenon may be helpful for understanding the fluid flow aspects of blood flow in capillaries.

Permeability Characteristics of Porous Rock with Conduits under Stokes–Brinkman–Darcy Coupling Model

Conduits, the main channels within a rock mass, can significantly affect the permeability characteristics of rock masses. This paper presents a Stokes–Brinkman–Darcy coupling model to study the permeability characteristics of porous rock masses with conduits. In this model, the Navier–Stokes equation and the Brinkman-extended Darcy equation are adopted to describe the fluid motions in the conduit and porous rock matrix, respectively. The analytical solutions of both the velocity distribution and the equivalent permeability coefficient, which is the average permeability for the complete rock volume, are derived under the stress jump boundary condition for a rock mass with a conduit. The results of the sensitivity analysis show that the equivalent permeability coefficient of the rock mass with a conduit is positively correlated with the permeability of the rock mass, the relative aperture of the conduit, and the porosity and is negatively correlated with the stress jump coefficient; the aperture of the conduit plays a dominant role in the equivalent permeability coefficient. In addition, the analytical solution of the equivalent permeability coefficient is verified by a seepage test.

Effect of magnetic field on the slow motion of a porous spheroid: Brinkman’s model

The major goal of this work is to analyze the magnetic effect on the creeping viscous flow past a porous spheroidal particle, a particle of slightly deformed spherical shape. Brinkman’s model is proposed to govern the flow in the porous media. Boundary value problem considers the conditions of continuity of velocity components, continuity of normal stresses, and stress jump boundary condition for tangential stress. A transverse magnetic field of uniform nature is applied to the flow. An expression for the drag force acting on the spheroidal particle is derived analytically. The effects of the physical parameters involved in the flow like permeability, deformation, Hartmann number’s, viscosity ratio, and stress jump coefficient parameters are visualized through graphs and tables. The applied magnetic field seems to suppress the flow of fluid that leads to the increase in the drag experienced on the porous spheroid. It is also observed that the increase in the deformation, stress jump, and permeability decreases the drag coefficient. Our results without magnetic effect match with the results reported earlier in the literature.

INFLUENCE OF MAGNETIC FIELD ON THE STOKES FLOW THROUGH POROUS SPHEROID: HYDRODYNAMIC PERMEABILITY OF A MEMBRANE USING CELL MODEL TECHNIQUE

The present work concerns an analysis of the creeping flow of steady, axisymmetric Stokes flow of an electrically conducting, viscous, incompressible fluid through a swarm of porous spheroidal particles in the presence of a uniform magnetic field. Cell model technique has been used to model the problem. Four known boundary conditions, Happel's, Kuwabara's, Kvashnin's, and Cunningham's (Mehta-Morse's), are used to find the hydrodynamic permeability of the membrane built up by porous spheroidal particles by using the perturbation method. The stress jump boundary condition for tangential stresses, along with the continuity of normal stress and velocity components, is used at the fluid-porous interface. The variation of the dimensionless hydrodynamic permeability of the membrane with the stress jump coefficient, the Hartmann number, and the dimensionless permeability of the porous region and particle volume fraction are discussed. The presented model can be used for the evaluation of changing hydrodynamic permeability of a membrane under the influence of a uniform magnetic field in pressure-driven processes (nano-, reverse osmosis, and microfiltration).

Effect of varying viscosity on a two-layer model of the blood flow through porous blood vessels

The present work concerns the effect of radially varying viscosity on blood flow through blood vessels with porous walls. Blood is assumed as a two-fluid model consisting of a core region of suspension of all red cells constituted by the Herschel-Bulkley fluid and a peripheral layer of plasma free from the cells modeled as a Newtonian fluid. No slip condition has been used on the wall and the pressure gradient has been taken as constant. The wall of the blood vessel is composed of a thin porous (Brinkman) layer representing the glycocalyx layer. On the fluid interface the stress jump boundary condition as suggested by Ochoa-Tapia and Whitaker has been used. Analytical expressions for velocity profile, wall shear stress, rate of flow and resistance to flow have been obtained for different regions and the effects of plasma layer thickness, varying viscosity, yield stress, permeability and viscosity ratio parameter on the hemodynamical quantities are discussed and depicted graphically. A comparative analysis for a relative change in flow resistance between our model and the previously studied single and two-fluid models without porous walls has been done. The effects of various parameters on hematocrit and Fahraeus effect have also been analyzed and results of earlier works have been established as special limiting cases of the present study. A novel observation is that a decreasing viscosity ratio parameter ( $\lambda_{1}$ ) leads to decay in average concentration of RBCs leading to decay in hematocrit (Ht). It is concluded that a thick porous layer with high porosity at wall due to either a glycocalyx layer or the deposition of fatty plaques of cholesterol may lead to significant decay in hematocrit Ht and may lead to anemia.

Viscous flow past a porous sphere within a nonconcentric fictitious spherical cell

The quasi steady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous sphere situated at an arbitrary position within a virtual spherical cell along the line connecting their centers is analyzed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman model governs the flow inside the porous particle and the flow in the fictitious envelope medium is governed by Stokes equations. A general solution is constructed from the superposition of the fundamental solutions in the two spherical coordinate systems based on both the porous particle and virtual spherical cell. Boundary conditions on the particle’s surface and the fictitious spherical envelope for four known cell models are satisfied by a collocation method for truncated series. Numerical solutions for the drag force exerted on the porous sphere in the presence of the cell are obtained for various cases of the effective distance between the center of the porous particle and the fictitious envelope, the volume ratio of the porous particle over the surrounding sphere, the viscosity ratio, the stress jump coefficient, and a coefficient that is proportional to the permeability. Streamlines in and around porous sphere are presented for the unit cell models at different values of relevant physical parameters. In the limits of the motions of the porous particle in the concentric position with the cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature.

Analysis of Heat Transfer and Entropy Generation in a Channel Partially Filled With N-Layer Porous Media

Convective heat transfer in a channel partially filled with porous medium has received a lot of attention due to its wide engineering applications. However, most researches focused on a channel partially filled with single layer porous medium. In this paper, we will analyze the heat transfer and entropy generation inside a channel partially filled with N-layer porous media. The flow and the heat transfer in the porous region are described by the Darcy–Brinkman model and the local thermal nonequilibrium model, respectively. At the porous-free fluid interface, the momentum and the heat transfer are described by the stress jump boundary condition and the heat flux jump boundary condition, respectively; while at the interface between two different porous layers, the momentum and the heat transfer are described by the stress continuity boundary condition and the heat flux continuity boundary condition, respectively. The analytical solutions for the velocity and temperature in the channel are derived and used to calculate the overall Nusselt number, the total entropy generation rate, the Bejan number, and the friction factor. Furthermore, the performances of the flow and heat transfer of a channel partially filled with third-layer porous media are studied.

Simulating flow over and through porous media with application to erosion of particulate deposits

We simulate flows involving porous media and homogenous fluid using a single-domain finite-difference numerical method. The porous medium and unimpeded fluid are separated by a sharp interface where a stress jump boundary condition is implemented using a forcing term. The interface is constructed by connecting Lagrangian markers with cubic splines, allowing for any possible porous media geometry. This model is particularly flexible as it can easily account for a mobile interface. We apply our method to simulate erosion and suspension of particles from a fixed or erodible particulate deposit. The flux of particles entrained from the porous media is obtained from the computed velocity at the interface, in contrast to more common approaches that assume a flux proportional to the viscous stress at the interface.

Axisymmetric Stokes flow past a composite spheroidal shell of immiscible fluids

We study the flow of an incompressible Newtonian fluid past a composite spheroidal shell whose shape deviates slightly from that of a sphere. A composite particle referred to in this paper is a spheroidal liquid core covered with a porous layer. The Brinkman equation is used for the flow inside the porous medium and the Stokes equation is used for the flow in the fluid region. We assume that the external and internal viscous fluids are immiscible and the viscosity of the porous medium is different than the viscosity of pure liquid. The Ochoa-Tapia and Whitaker’s stress jump boundary condition for tangential stress is applied on the porous-fluid interface. Velocity and pressure distributions are found and the drag force acting on the spheroidal shell is evaluated. The analytical solution is obtained by dividing the flow into three regions. Both type of spheroids, oblate and prolate are considered. Numerical results of the normalized hydrodynamic drag force acting on the spheroidal shell are tabulated and represented graphically for different values of the parameters characterizing the stress jump coefficient, separation parameter, permeability, deformation parameter, and viscosity ratios. The analysis of the flow pattern is done by plotting streamlines and several renowned cases are deduced.

Effect of magnetic field on the hydrodynamic permeability of a membrane built up by porous spherical particles

In this paper, an analysis of steady, axi-symmetric Stokes flow of an electrically conducting viscous incompressible fluid through spherical particle covered by porous shell in presence of uniform magnetic field is presented. To model flow through the swarm of spherical particles, cell model technique has been used, i.e. porous spherical shell is assumed to be confined within a hypothetical cell of the same geometry. At the fluid-porous interface, the stress jump boundary condition for tangential stresses along with continuity of normal stress and velocity components are used. Four known boundary conditions on the hypothetical surface were considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta−Morse’s) condition. The effect of stress jump coefficient, Hartmann number, and dimensionless permeability of the porous region as well as particle volume fraction on the hydrodynamic permeability and streamlines were discussed. The patterns of streamlines were also obtained.

Hydrodynamic permeability of a membrane composed of porous spherical particles in the presence of uniform magnetic field

This work concerns the flow of an incompressible viscous fluid past a porous sphere in presence of transverse applied uniform magnetic field, using particle-in-cell method. The Brinkman equations are used in porous region and the Stokes equations for non-porous region. At the fluid-porous interface, the stress jump boundary condition for tangential stresses along with continuity of normal stress and velocity components are used. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). The hydrodynamic drag force experienced by a porous spherical particle in a cell and hydrodynamic permeability of membrane built up by porous spherical particles are evaluated. The patterns of streamlines are also obtained and discussed. The effect of stress jump coefficient, Hartmann number, dimensionless specific permeability of the porous particle and particle volume fraction on the hydrodynamic permeability and streamlines are discussed. Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.

Slow Motion of a Porous Sphere Translating Along the Axis of a Circular Cylindrical Pore Subject to a Stress Jump Condition

The coupled flow problem of an incompressible axisymmetrical quasisteady motion of a porous sphere translating in a viscous fluid along the axis of a circular cylindrical pore is discussed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stress along with continuity of normal stress and velocity components are employed. The flow through the porous particle is governed by the Brinkman model and the flow in the outside porous region is governed by Stokes equations. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The boundary conditions are satisfied first at the cylindrical pore wall by the Fourier transforms and then on the surface of the porous particle by a collocation method. The collocation solutions for the normalized hydrodynamic drag force exerted by the clear fluid on the porous particle is calculated with good convergence for various values of the ratio of radii of the porous sphere and pore, the stress jump coefficient, and a coefficient that is proportional to the permeability. The shape effect of the cylindrical pore on the axial translation of the porous sphere is compared with that of the particle in a spherical cavity; it found that the porous particle in a circular cylindrical pore in general attains a lower hydrodynamic drag than in a spherical envelope.

Axi-symmetric motion of a porous approximate sphere in an approximate spherical container

The creeping motion of a porous approximate sphere at the instant it passes the center of an approximate spherical container with Ochoa-Tapia and Whitaker’s stress jump boundary condition has been investigated analytically. The Brinkman’s model for the flow inside the porous approximate sphere and the Stokes equation for the flow in an approximate spherical container were used to study the motion. The stream function (and thus the velocity) and pressure (both for the flow inside the porous approximate sphere and inside an approximate spherical container) are calculated. The drag force experienced by the porous approximate spherical particle and wall correction factor are determined in closed forms. The special cases of porous sphere in a spherical container and oblate spheroid in an oblate spheroidal container are obtained from the present analysis. It is observed that drag not only changes with the permeability of the porous region, but as the stress jump coefficient increases, the rate of change in behavior of drag increases.

Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous concentric spherical shell-in-cell model is studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin and Cunningham/Mehta-Morse models are considered. At the fluid-porous interfaces, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. The hydrodynamic drag force acting on the porous shell by the external fluid in each of the four boundary conditions on the cell surface is evaluated. It is found that the normalized mobility of the particles (the hydrodynamic interaction among the porous shell particles) depends not only on the permeability of the porous shells and volume fraction of the porous shell particles, but also on the stress jump coefficient. As a limiting case, the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres with jump.

Creeping flow past a porous approximately spherical shell: stress jump boundary condition

The creeping flow of an incompressible viscous liquid past a porous approximately spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier–Stokes equations. The flow within the porous annular region of the shell is governed by Brinkman’s model. The boundary conditions used at the interface are continuity of the velocity, continuity of the pressure and Ochoa-Tapia and Whitaker’s stress jump condition. An exact solution for the problem and an expression for the drag on the porous approximately spherical shell are obtained. The drag is evaluated numerically for several values of the parameters governing the flow. doi:10.1017/S144618111100071X

Natural Convection in a Cavity Partially Filled with a Vertical Porous Medium

The shear stress jump boundary condition that must be imposed at an interface between a porous medium and a free fluid in an enclosure is investigated. Two-domain approach is founded and finite element method is used to solve the problem. Three stress jump coefficients 0, 1, -1 are analyzed for different Rayleigh number, permeability and thickness of porous layer. Variation of Maximum stream function and Nusselt number show stronger convection and heat transfer when the stress jump coefficient is positive. There is little distinctive in flow and heat transfer when the value of coefficient is equal to 0 and -1.

Creeping flow past a porous approximate sphere – Stress jump boundary condition

AbstractThe problem of flow past and through a porous approximate sphere in a uniform flow at small Reynolds number is considered. It is assumed that the Stokes equation holds outside the sphere and Brinkman's law holds inside the sphere. The stream function and the pressure distribution, both for the flow inside and outside are obtained in terms of Bessel and Gegenbauer functions of the first kind. The drag force experienced by the particle is determined and its variation with respect to permeability parameter for different values of stress jump coefficient is studied numerically. The special cases of flow past a porous sphere and spheroid are obtained from the present analysis.

Analytical Investigation of the Effect of Viscous Dissipation on Couette Flow in a Channel Partially Filled with a Porous Medium

An analytical study of viscous dissipation effect on the fully developed forced convection Couette flow through a parallel plate channel partially filled with porous medium is presented. A uniform heat flux is imposed at the moving plate while the fixed plate is insulated. In the fluid-only region the flow field is governed by Navier–Stokes equation while the Brinkman-extended Darcy law relationship is considered in the fully saturated porous medium. The interface conditions are formulated with an empirical constant β due to the stress jump boundary condition. Fluid properties are assumed to be constant and the longitudinal heat conduction is neglected. A closed-form solution for the velocity and temperature distributions and also the Nusselt number in the channel are obtained and the viscous dissipation effect on these profiles is briefly investigated.

Stokes flow of an assemblage of porous particles: stress jump condition

The present article investigates the overall bed permeability of an assemblage of porous particles. For the bed of porous particles, the fluid-particle system is represented as an assemblage of uniform porous spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Stokes equations are employed inside the fluid envelope and Brinkman equations are used inside the porous region. The stress jump boundary condition is used at the porous-liquid interface together with the continuity of normal stress and continuity of velocity components. On the surface of the fluid envelope, three different possible boundary conditions are tested. The obtained expression for the drag force is used to estimate the overall bed permeability of the assemblage of porous particles and the behavior of overall bed permeability is analyzed with various parameters like modified Darcy number (Da*), stress jump coefficient (β), volume fraction (e), and effective viscosity.