A general formula for the drag on a sphere placed in a creeping unsteady micropolar fluid flow

In the present work, we investigate the creeping unsteady motion of an infinite micropolar fluid flow past a fixed sphere. The technique of Laplace transform is used. The drag formula is obtained in the physical domain analytically by using the complex inversion formula of the Laplace transform. The well known formula of Basset for the drag on a sphere placed in an unsteady viscous fluid flow and that of Ramkissoon and Majumdar for steady motion in the case of micropolar fluids are recovered as special cases. The obtained formula is employed to calculate the drag force for some micropolar fluid flows. Numerical results are obtained and represented graphically.     

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.     

MHD non-Newtonian micropolar fluid flow and heat transfer in channel with stretching walls

A study is presented for magnetohydrodynamics （MHD） flow and heat transfer characteristics of a viscous incompressible electrically conducting micropolar fluid in a channel with stretching walls. The micropolar model introduced by Eringen is used to describe the working fluid. The transformed self similar ordinary differential equations together with the associated boundary conditions are solved numerically by an algorithm based on quasi-linearization and multilevel discretization. The effects of some physical parameters on the flow and heat transfer are discussed and presented through tables and graphs. The present investigations may be beneficial in the flow and thermal control of polymeric processing.     

Fundamental solutions for axi-symmetric translational motion of a microstretch fluid

The fundamental solution for the axi-symmetrictranslational motion of a microstretch fluid due to a concentrated point body force is obtained.A general formula for thedrag force exerted by the fluid on an axi-symmetric rigid particle translating in it is then deduced.As an application to theobtained drag formula,this paper has discussed the problemof creeping translational motion of a rigid sphere in a microstretch fluid.The slip boundary condition on the surfaceof the spherical particle is applied.The drag force and theother physical quantities are obtained and represented graphically for various values of the micropolarity and slip parameters.     

Flow of a viscous fluid past a heterogeneous porous sphere at low Reynolds numbers

A flow past a heterogeneous porous sphere is investigated by using the perturbation theory. The flow through the sphere is divided into two zones, which are fully saturated with the viscous fluid, and the flow in these zones is governed by the Brinkman equation. The space outside the sphere, where a clear fluid flows, is also divided into two zones: the Navier–Stokes zone and the Oseen flow zone. The solutions on the interface inside the sphere are matched with the condition proposed by Merrikh and Mohammad. The stream function in the Navier–Stokes zone is matched with that on the sphere surface by the condition proposed by Ochoa-Tapia and Whitaker. It is found that the drag on the spherical shell decreases as the permeability toward the sphere boundary increases.     

Forces on a porous sphere spinning in a fluid flow

Exact expressions are found for the drag (modified Stokes force) and the lift (modified Magnus force) on a porous sphere spinning slowly in a viscous fluid flowing slowly and uniformly past it.     

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 motion of a slip spherical particle in a circular cylindrical pore

A combined analytical–numerical study for the creeping flow caused by a spherical fluid or solid particle with a slip-flow surface translating in a viscous fluid along the centerline of a circular cylindrical pore is presented. To solve the axisymmetric Stokes equations for the fluid velocity field, a general solution is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The boundary conditions are enforced first at the pore wall by the Fourier transforms and then on the particle surface by a collocation technique. Numerical results for the hydrodynamic drag force acting on the particle are obtained with good convergence for various values of the relative viscosity or slip coefficient of the particle, the slip parameter of the pore wall, and the ratio of radii of the particle and pore. For the motion of a fluid sphere along the axis of a cylindrical pore, our drag results are in good agreement with the available solutions in the literature. As expected, the boundary-corrected drag force for all cases is a monotonic increasing function of the ratio of particle-to-pore radii, and approaches infinity in the limit. Except for the case that the cylindrical pore is hardly slip and the value of the ratio of particle-to-pore radii is close to unity, the drag force exerted on the particle increases monotonically with an increase in its relative viscosity or with a decrease in its slip coefficient for a constant ratio of radii. In a comparison for the pore shape effect on the axial translation of a slip sphere, it is found that the particle in a circular cylindrical pore in general acquires a lower hydrodynamic drag than in a spherical cavity, but this trend can be reversed for the case of highly slippery particles and pore walls.     

SERIES PERTURBATIONS APPROXIMATE SOLUTIONS TO N-S EQUATIONS AND MODIFICATION TO ASYMPTOTIC EXPANSION MATCHED METHOD

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅａｓｙｍｐｔｏｔｉｃｅｘｐａｎｓｉｏｎｍａｔｃｈｅｄｍｅｔｈｏｄｗａｓｄｅｖｅｌｏｐｅｄｏｎｔｈｅｂａｓｉｓｏｆＰｌａｎｄｔｌ’ｓＢｏｕｎｄａｒｙＬａｙｅｒｓＴｈｅｏｒｙ[1- 3].Ｉｎ 1 960’ｓｌａｔｅｒ,ｔｈｅｖａｌｉｄｍａｔｃｈｅｄｐｒｉｎｃｉｐｌｅｓｗｅｒｅａｄｖａｎｃｅｄｂｙＶａｎＤｙｋｅａｎｄＫａｐｌｕｎ[4 ,5 ],ｒｅｓｐｅｃｔｉｖｅｌｙ .Ｂｕｔｔｈｅａｐｐｌｉｃａｂｌｅｒａｎｇｅｏｆｔｈｅｍｅｔｈｏｄｗａｓｒｅｓｔｒｉｃｔｅｄｂｙｔｈｅｄｉｆｆｉｃｕｌｔｙｔｈ…     

Slow Motion of a Porous Eccentric Spherical Particle-in-Cell Models

A combined analytical?Cnumerical method is presented for the quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous eccentric spherical particle-in-cell models. The flow inside the porous particle is governed by the Brinkman model and the flow in the fictitious envelope region is governed by Stokes equations. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based on both the porous particle and fictitious spherical envelope. Boundary conditions on the particle??s surface and fictitious spherical envelope that correspond to the Happel, Kuwabara, Kvashnin, and Cunningham/Mehta-Morse models are satisfied by a collocation technique. The drag of these eccentric porous particles relative to the drag experienced by a centered porous particle are investigated as functions 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 and a coefficient that is proportional to the inverse of the permeability. In the limits of the motions of the porous particle in the concentric position with 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.     

Flow past an axisymmetric body embedded in a saturated porous mediumÉcoulement derrière un corps à symétrie axiale plongé dans un milieu poreux saturé

The problem of viscous fluid past an axisymmetric body embedded in a fluid saturated porous medium is studied using the Brinkman's extension. A general formula for the drag on the body is derived in the form of a limit of an expression involving the stream function characterizing the flow. The flow past an axisymmetric approximate sphere is also considered. The stream function in this case is obtained in terms of Bessel functions and Gegenbauer's functions. The drag acting on the body is evaluated by using the formula derived. Its variation is studied with respect to geometric and permeability parameters. The special cases of flow past a sphere and a spheroid are obtained from the present analysis. To cite this article: D. Srinivasa Charya, J.V. Ramana Murthy, C. R. Mecanique 330 (2002) 417–423.     

Slow motion of a permeable sphere in a viscous fluid

Flow of a viscous fluid past a permeable sphere is investigated in the Stokes approximation. An example of such a flow is flow past a perforated or meshed spherical surface. The elements of the sphere contain rigid impermeable sections and openings through which the fluid can flow. The interaction of the sphere with the flow is described by two drag coefficients, which established the connection between the flow velocity of the fluid at the sphere and the stress tensor on it. The dependence of the flow pattern and also the drag and flow rate of the fluid on these coefficients is investigated. In special cases, the obtained solution describes flow past solid and liquid spheres.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 165–167, September–October, 1982.     

Stokes flow past a porous sphere with an impermeable core

Stokes flow of a viscous, incompressible fluid past a porous sphere with an impermeable core using Darcy law for the flow in the porous region is discussed. The formulae for drag and torque are found by deriving the corresponding Faxen's laws. It is found that torque is always less than that on a solid sphere and it does not depend on the radius of the impermeable core. Some illustrative examples are discussed.     

圆柱管道内运动的小球受到的黏滞阻力的数值分析

为了更好地运用落球法测量研究流体的黏滞系数,研究小球在黏性流体中下落的受力情况,本文对小球在充满黏性流体的圆柱管道的下落过程进行分析。利用COMSOL4.4仿真模拟,建立了合理的仿真模型,并分析了小球受到的黏滞阻力与小球的大小、下落位置的关系。结果表明:选择速度项二阶近似、压强项一阶近似的离散化方法,可以得到和理论值非常相符的仿真结果;当下落过程中小球球心始终在圆柱轴线上时,小球受到的黏滞阻力相对于Stokes力的修正系数,是小球半径与圆柱管道半径的比例函数,本文得到了更大范围的符合理论解的修正系数;当下落过程中小球的球心偏离圆柱轴线时,对于同样大小的小球,黏滞阻力、压强力、黏性力均随着球心到轴线的距离先减小后增大,且具有不同的极小值点。     

A transient natural convection of micropolar fluids over a vertical cylinder

A transient free convective boundary layer flow of micropolar fluids past a semi-infinite cylinder is analysed in the present study. The transformed dimensionless governing equations for the flow, microrotation and heat transfer are solved by using the implicit scheme. For the validation of the current numerical method heat transfer results for a Newtonian fluid case where the vortex viscosity is zero are compared with those available in the existing literature, and an excellent agreement is obtained. The obtained results concerning velocity, microrotation and temperature across the boundary layer are illustrated graphically for different values of various parameters and the dependence of the flow and temperature fields on these parameters is discussed. An increase in the vortex viscosity tends to increase the magnitude of microrotation and thus decreases the peak velocity of fluid flow. An increase in the vortex viscosity in micropolar fluids is shown to decrease the heat transfer rate.     

Drag and Separation Flow Past Solid Sphere with Porous Shell at Moderate Reynolds Number

Axisymmetric viscous, two-dimensional steady and incompressible fluid flow past a solid sphere with porous shell at moderate Reynolds numbers is investigated numerically. There are two dimensionless parameters that govern the flow in this study: the Reynolds number based on the free stream fluid velocity and the diameter of the solid core, and the ratio of the porous shell thickness to the square root of its permeability. The flow in the free fluid region outside the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by a Darcy model. Using a commercially available computational fluid dynamics (CFD) package, drag coefficient and separation angle have been computed for flow past a solid sphere with a porous shell for Reynolds numbers of 50, 100, and 200, and for porous parameter in the range of 0.025–2.5. In all simulation cases, the ratio of b/a was fixed at 1.5; i.e., the ratio of outer shell radius to the inner core radius. A parametric equation relating the drag coefficient and separation point with the Reynolds number and porosity parameter were obtained by multiple linear regression. In the limit of very high permeability, the computed drag coefficient as well as the separation angle approaches that for a solid sphere of radius a, as expected. In the limit of very low permeability, the computed total drag coefficient approaches that for a solid sphere of radius b, as expected. The simulation results are presented in terms of viscous drag coefficient, separation angles and total drag coefficient. It was found that the total drag coefficient around the solid sphere as well as the separation angle are strongly governed by the porous shell permeability as well as the Reynolds number. The separation point shifts toward the rear stagnation point as the shell permeability is increased. Separation angle and drag coefficient for the special case of a solid sphere of radius r = a was found to be in good agreement with previous experimental results and with the standard drag curve.     

Slow motion of a slip spheroid along its axis of revolution

A combined analytical and numerical study of the Stokes flow caused by a rigid spheroidal particle translating along its axis of revolution in a viscous fluid is presented. The fluid is allowed to slip at the surface of the particle. The general solution for the stream function in prolate and oblate spheroidal coordinates can be expressed in an infinite-series form of semi-separation of variables. The slip boundary condition incorporating the shear stress at the particle surface is applied to this general solution to determine its unknown coefficients of the leading orders. The solution of these coefficients can be either numerical results obtained from a boundary-collocation method or explicit formulas derived analytically. The drag force exerted on the spheroidal particle by the fluid is evaluated with good convergence behavior for various values of the slip parameter and aspect ratio of the particle. The agreement between our hydrodynamic drag results and the relevant numerical solutions obtained previously using a singularity method is excellent. Although the drag force acting on the translating spheroid normalized by that on a corresponding sphere with equal equatorial radius increases monotonically with an increase in the axial-to-radial aspect ratio for a no-slip spheroid, it decreases monotonically as this aspect ratio increases for a perfect-slip spheroid. The normalized drag force exerted on a spheroid with a given surface slip coefficient in between the no-slip and perfect-slip limits is not a monotonic function of its aspect ratio. For a spheroid with a fixed aspect ratio, its drag force is a monotonically decreasing function of the slip coefficient of the particle.     

Viscous fluid flow induced by rotational-oscillatory motion of a porous sphere

Viscous fluid flow induced by rotational-oscillatorymotion of a porous sphere submerged in the fluid is determined. The Darcy formula for the viscous medium drag is supplementedwith a term that allows for the medium motion. The medium motion is also included in the boundary conditions. Exact analytical solutions are obtained for the time-dependent Brinkman equation in the region inside the sphere and for the Navier–Stokes equations outside the body. The existence of internal transverse waves in the fluid is shown; in these waves the velocity is perpendicular to the wave propagation direction. The waves are standing inside the sphere and traveling outside of it. The particular cases of low and high oscillation frequencies are considered.     

Slow motions of a circular cylinder experiencing slip near a plane wall

A theoretical study is presented for the two-dimensional creeping flow caused by a long circular cylindrical particle translating and rotating in a viscous fluid near a large plane wall parallel to its axis. The fluid is allowed to slip at the surface of the particle. The Stokes equations for the fluid velocity field are solved in the quasi-steady limit using cylindrical bipolar coordinates. Semi-analytical solutions for the drag force and torque acting on the particle by the fluid are obtained for various values of the slip coefficient associated with the particle surface and of the relative separation distance between the particle and the wall. The results indicate that the translation and rotation of the confined cylinder are not coupled with each other. For the motion of a no-slip cylinder near a plane wall, our hydrodynamic drag force and torque results reduce to the closed-form solutions available in the literature. The boundary-corrected drag force and torque acting on the particle decrease with an increase in the slip coefficient for an otherwise specified condition. The plane wall exerts the greatest drag on the particle when its migration occurs normal to it, and the least in the case of motion parallel to it. The enhancement in the hydrodynamic drag force and torque on a translating and rotating particle caused by a nearby plane wall is much more significant for a cylinder than for a sphere.     

Stokes flow past a porous spheroid embedded in another porous medium

A study is made with an analysis of an incompressible viscous fluid flow past a slightly deformed porous sphere embedded in another porous medium. The Brinkman equations for the flow inside and outside the deformed porous sphere in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in small parameter characterizing the deformation. The flow through the porous oblate spheroid embedded in another porous medium is considered as the particular example of the deformed porous sphere embedded in another porous medium. The drag experienced by porous oblate spheroid in another porous medium is also evaluated. The dependence of drag coefficient and dimensionless shearing stress on the permeability parameter, viscosity ratio and deformation parameter for the porous oblate spheroid is presented graphically and discussed. Previous well-known results are then also deduced from the present analysis.