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Wall effects on a spherical particle
Abstract
The motion of a solid spherical particle at the instant it passes the centre of a spheroidal container has been investigated using the no-slip condition at the surfaces. The shape of this spheroidal container is assumed to deviate slightly from that of a sphere at the surface. An expression for the stream function describing the flow field has been obtained to the first order in the parameter characterizing the deviation and this in turn has been utilized to evaluate the drag on the particle. Wall effects are then examined and special cases deduced.
... It was also observed that Newtonian liquid spheroid experiences more wall effects than the Reiner-Rivlin liquid spheroid. Ramkissoon and Rahaman (2003) studied the wall effects of a spheroidal container on the motion of a solid spherical particle. They (Ramkissoon and Rahaman 2003) concluded that the wall effects increases if the container becomes more and more spheroidal. ...
... Ramkissoon and Rahaman (2003) studied the wall effects of a spheroidal container on the motion of a solid spherical particle. They (Ramkissoon and Rahaman 2003) concluded that the wall effects increases if the container becomes more and more spheroidal. Keh and Chang (1998) investigated the effect of boundary surfaces on the creeping motion of a spherical aerosol particle situated at the center of a concentric spherical cavity. ...
The present article deals with the analytical study of translational and rotational motion of a porous spherical particle with quadratically increasing permeability inside a concentric spherical cavity filled with incompressible Newtonian fluid, under the creeping flow conditions. The flow fields in clear fluid and porous regions are governed by Stokes equation and generalized Darcy’s law (Brinkman equation) together with mass conservation, respectively. Closed form solutions for translational and rotational mobilities of the particle are obtained with the help of drag and torque acting on the particle surface. The particle mobility inside a cavity attains a maximum value of 1. However, the presence of cavity wall slows down the particle motion as a result the particle mobility becomes smaller than unity. The effect of cavity wall on the mobility is significant when the gap between the particle surface and cavity wall is less. Various limiting cases are obtained which agree with earlier existing results. The results are explained with the aid of graphs for better clarity.
... Vainshtein et al. [38] studied the creeping motion over and inside a permeable spheroid. Ramkissoon and Rahaman [39] tackled the flow of fluid over an impermeable sphere immersed in a spheroidal vessel by assuming the no-slip condition. Besides, the drag executed on the sphere covered by a spheroidal cell was evaluated. ...
The perspective of the current analysis is to represent the incompressible viscous flow past a low permeable spheroid contained in a fictitious spheroidal cell. Stokes approximation and Darcy’s equation are adopted to govern the flow in the fluid and permeable zone, respectively. Happel’s and Kuwabara’s cell models are employed as the boundary conditions at the cell surface. At the fluid porous interface, we suppose the conditions of conservation of mass, balancing of pressure component at the permeable area with the normal stresses in the liquid area, and the slip condition, known as Beavers-Joseph-Saffman-Jones condition to be well suitable. A closed-form analytical expression for hydrodynamic drag on the bounded spheroidal particle is determined and therefore, mobility of the particle is also calculated, for both the case of a prolate as well as an oblate spheroid. Several graphs and tables are plotted to observe the dependence of normalized mobility on pertinent parameters including permeability, deformation, the volume fraction of the particle, slip parameter, and the aspect ratio. Significant results that influence the impact of the above parameters in the problem have been pointed out. Our work is validated by referring to previous results available in literature as reduction cases.
... Vainshtein et al [12] handled the problem of creeping flow past and within a permeable spheroidal particle. Ramkissoon and Rahaman [13] analysed the motion of a solid spherical particle in a spheroidal vessel and obtained the drag force acting on the inner sphere. Srinivasacharya [14] focussed on the motion of a porous sphere in a spherical vessel by using Brinkman's equation for the porous region. ...
The present paper studies the impact of applied uniform transverse magnetic field on the flow of incompressible conducting fluid around a weakly permeable spherical particle bounded by a spherical container. Analytical solution of the problem is obtained using Happel and Kuwabara cell models. The concerned flow is parted in two regions, bounded fluid region and internal porous region, to be governed by Stokes and Darcy’s law respectively. At the interface between the fluid and the permeable region, the boundary conditions used are continuity of normal component of velocity, Saffman’s boundary condition and continuity of pressure. For the cell surface, Happel and Kuwabara models together with continuity in radial component of the velocity has been used. Expressions for drag force, hydrodynamic permeability and Kozeny constant acting on the spherical particle under magnetic effect are presented. Representation of hydrodynamic permeability for varying permeability parameters, particle volume fraction, slip parameter and Hartmann numbers are represented graphically. Also, the magnitude of Kozeny constant for weakly permeable and semipermeable sphere under a magnetic effect has been presented. In limiting cases many important results are obtained.
... As an alternative approach, many researchers considered fully computational scheme in their investigations [9][10][11]. Kosinski et al. [12,13] provides a wide range of numerical results on this subject. ...
In this paper, a Lagrangian-Lagrangian numerical simulation method for transient hydrodynamics of solid particles in an enclosure is presented. In this numerical scheme, we solve the fluid phase using a mesoscale method of lattice Boltzmann scheme. The particle motion is governed by Newton's law thus following the Lagrangian approach. The dynamics of solid particles in a lid-driven cavity was investigated at a wide range of Reynolds numbers. The results of this study suggest that the particle trajectories are critically dependent on the magnitude of Reynolds Numbers and the vortex behavior in the cavity. Comparisons with other previous studies demonstrate the application diversity of the present scheme.
... The paper [16] investigated the motion of inner Reiner-Rivlin (non-Newtonian) fluid sphere in a spherical container and they deduced that cross viscosity μ c is to reduce the wall effects. Two years later, [17] also investigated the problem of a solid spherical particle in a spheroidal container. They obtained the expression for the drag on the inner sphere and examined the wall effects and concluded that as the deformation of the spheroidal container increases the wall effects also increase. ...
An analysis is carried out to study the flow characteristics of creeping motion of an inner non-Newtonian Reiner-Rivlin liquid spheroid r = 1+ ∑_{k=2}^∞α_kG_k(cos θ), here α_k is very small shape factor and G_k is Gegenbauer function of first kind of order k, at the instant it passes the centre of a rigid spherical container filled with a Newtonian fluid. The shape of the liquid spheroid is assumed to depart a bit at its surface from the shape a sphere. The analytical expression for stream function solution for the flow in spherical container is obtained by using Stokes equation. While for the flow inside the Reiner-Rivlin liquid spheroid, the expression for stream function is obtained by expressing it in a power series of S, characterizing the cross-viscosity of Reiner-Rivlin fluid. Both the flow fields are then determined explicitly by matching the boundary conditions at the interface of Newtonian fluid and non-Newtonian fluid and also the condition of impenetrability and no-slip on the outer surface to the first order in the small parameter ε, characterizing the deformation of the liquid sphere. As an application, we consider an oblate liquid spheroid r = 1+2εG_2(cos θ) and the drag and wall effects on the body are evaluated. Their variations with regard to separation parameter, viscosity ratio λ, cross-viscosity, i.e., S and deformation parameter are studied and demonstrated graphically. Several well-noted cases of interest are derived from the present analysis. Attempts are made to compare between Newtonian and Reiner-Rivlin fluids which yield that the cross-viscosity μ_c is to decrease the wall effects K and to increase the drag D_N when deformation is comparatively small. It is observed that drag not only varies with λ, but as η increases, the rate of change in behavior of drag force increases also.
... For this purpose, a new equation for the Fidleris and Whitmore (1961) graphical method, described in the next section, was used. This analysis assumed the side of square and diameter of cylindrical settling columns to be equivalent, notwithstanding the report by Ramkissoon and Rahaman (2003) that wall effects are more pronounced in cylindrical than in square cross-sectional columns. After correction for wall effects we found no evidence of a difference between the data from different settling columns. ...
Single relations that can be used to calculate both the terminal settling velocities of spheres and the equivalent diameter of particles from their settling velocities are developed. The literature going back to Newton is reviewed and the relations developed tabulated. It is shown how the standard drag curve has developed into the dimensionless velocity versus dimensionless diameter curve. No relations that cover the full range that can conveniently be used for both velocity and diameter calculation were found, however a relation by Concha and Almendra covers most of the range.The standard drag curve data are constructed by utilizing 535 data points available in the literature in a Reynolds number range of 2.4 × 10−5 to 2 × 105. The settling velocities are corrected for experiments in finite width columns that do not satisfy the infinite medium dimensions. The data are converted to the dimensionless diameter and dimensionless velocity terms, which is more convenient for calculation purposes.The data are analyzed using piecewise cubic functions. Data from sources with excessive scatter and a few outliers are removed leaving 443 data points. The resulting piecewise cubic can be used to obtain velocity from diameter or diameter from velocity. To give an algebraic expression a hyperbola is fitted to the data giving an expression that can be solved to give explicit relations for both dimensionless velocity and dimensionless diameter. This provides an accuracy that compares well with expressions given in the literature.
The steady axisymmetric flow of an incompressible couple stress fluid past a solid sphere located at the center of a hypothetical spherical cavity is analytically investigated using the cell model technique. Here we assume that the inner sphere is solid and the outer one is fictitious. On the surface of the inner sphere, boundary conditions are the vanishing of normal velocity, slip boundary condition, and nil couple stress condition have been used. On the fictitious surface of the outer spherical cell, boundary conditions are Happel’s model, Kuwabara’s model, Kvashnin’s model, and Cunningham’s model applied. The couple stress fluid flow is governed in the cavity. The drag force and the wall correction factor acting on a slip sphere are evaluated. In special cases, we have discussed the drag force acting on the slip sphere in an unbounded case and the wall correction factor experienced by viscous fluid for all four models in a bounded case. Variations of the wall correction factor versus separation parameter for different values of the couple stress and slip parameters with a fixed value of the couple stress viscosity ratio parameter are presented graphically. Here we have also discussed the results of consistent couple stress theory. The tabulated results show that the wall correction factor increases monotonically as the couple stress viscosity ratio, slip, and separation parameters increases. It has the largest values when an interaction between the spherical cavity and inner sphere is very close and the smallest values when they are far from each other.
The problem of steady rotation of a porous approximate sphere located at the center of an approximate spherical container has been investigated. 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 torque experienced by the porous approximate spherical particle in the presence of cavity is obtained and wall correction factor is calculated. The special cases of rotation of a porous sphere and a solid sphere in a spherical container are obtained from the present analysis.
The problem of the creeping flow through a spherical droplet with a non-homogenous porous layer in a spherical container has been studied analytically. Darcy’s model for the flow inside the porous annular region and the Stokes equation for the flow inside the spherical cavity and container are used to analyze the flow. The drag force is exerted on the porous spherical particles enclosing a cavity, and the hydrodynamic permeability of the spherical droplet with a non-homogeneous porous layer is calculated. Emphasis is placed on the spatially varying permeability of a porous medium, which is not covered in all the previous works related to spherical containers. The variation of hydrodynamic permeability and the wall effect with respect to various flow parameters are presented and discussed graphically. The streamlines are presented to discuss the kinematics of the flow. Some previous results for hydrodynamic permeability and drag forces have been verified as special limiting cases.
This paper presents an analytical study of creeping motion of a permeable sphere in a spherical container filled with a micro-polar fluid. The drag experienced by the permeable sphere when it passes through the center of the spherical container is studied. Stream function solutions for the flow fields are obtained in terms of modified Bessel functions and Gegenbauer functions. The pressure fields, the micro-rotation components, the drag experienced by a permeable sphere, the wall correction factor, and the flow rate through the permeable surface are obtained for the frictionless impermeable spherical container and the zero shear stress at the impermeable spherical container. Variations of the drag force and the wall correction factor with respect to different fluid parameters are studied. It is observed that the drag force, the wall correction factor, and the flow rate are greater for the frictionless impermeable spherical container than the zero shear stress at the impermeable spherical container. Several cases of interest are deduced from the present analysis.
The creeping motion of a non-Newtonian (Reiner-Rivlin) liquid sphere at the instant it passes the center of an approximate spherical container is discussed. The flow in the spheroidal container is governed by the Stokes equation, while for the flow inside the Reiner-Rivlin liquid sphere, the expression for the stream function is obtained by expressing it in the power series of a parameter S, characterizing the cross-viscosity. Both the flow fields are then determined explicitly by matching the boundary conditions at the interface of Newtonian fluid and non-Newtonian fluid, and also the condition of imperviousness and no-slip on the outer surface. As an application, we have considered an oblate spheroidal container. The drag and wall effects on the liquid spherical body are evaluated. Their variations with regard to the separation parameter ℓ, viscosity ratio λ, cross-viscosity S, and deformation parameter ε are studied and demonstrated graphically. Several renowned cases are derived from the present analysis. It is observed that the drag not only varies with ε, but as ℓ increases, the rate of change in behavior of drag force also increases. The influences of these parameters on the wall effects has also been studied and presented in a table.
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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.
The goal of this study is to determine the effect of transient vortex structure on solid particles and explain this behaviour in terms of particles' trajectories. In the present study, an alternative numerical scheme was proposed to predict the fluid flow and coupled with a Lagrangian scheme on the prediction of solid phase. The dynamics of solid particle in a lid-driven cavity was investigated at a wide range of Reynolds numbers. The results of this study suggest that the particle trajectories are critically dependence on the magnitude of Reynolds numbers and the vortex behaviour in the cavity. Good comparisons with the experimental and previous studies demonstrate the multidisciplinary applications of the present scheme.
In this paper, a Lagrangian–Lagrangian numerical simulation method for transient hydrodynamics of solid particles in an enclosure is presented. In this numerical scheme, we solve the fluid phase using a mesoscale method of lattice Boltzmann scheme. The particle motion is governed by Newton's law thus following the Lagrangian approach. The dynamics of solid particles in a lid-driven cavity was investigated at a wide range of Reynolds numbers. The results of this study suggest that the particle trajectories are critically dependent on the magnitude of Reynolds Numbers and the vortex behavior in the cavity. Comparisons with other previous studies demonstrate the application diversity of the present scheme.
The problem of steady rotation of a composite sphere located at the centre of a spherical container has been investigated. A composite particle referred to in this paper is a spherical solid core covered with a permeable spherical shell. The Brinkman’s model for the flow inside the composite sphere and the Stokes equation for the flow in the spherical container were used to study the motion. The torque experienced by the porous spherical particle in the presence of cavity is obtained. The wall correction factor is calculated. In the limiting cases, the analytical solution describing the torque for a porous sphere and for a solid sphere in an unbounded medium are obtained from the present analysis.
The creeping motion of a porous approximate sphere with an impermeable core at the instant it passes the center of a spherical container is discussed. The flow in the spherical container is governed by the Stokes equation. The flow inside the porous approximate sphere is governed by Brinkman’s equation. The boundary conditions used at the interface are stress jump condition for tangential stresses, continuity of the normal stresses and velocity components. The drag experienced by the porous approximate spherical particle is obtained. The wall correction factor is calculated. The variations of drag coefficient and wall correction factor are studied with respect to permeability, separation parameters, deformation parameters and stress jump coefficient.
Research on numerical schemes on fluid–solid interaction has been quite intensive in the past decade. The difficulties associated with accurate predictions of the interaction at specific spatial and temporal levels. Traditional computational fluid dynamics schemes are struggling to predict at high level of accuracy for this type of problem. Hence, in the present study, an alternative numerical scheme was proposed to predict the fluid flow and coupled with a Lagrangian scheme on the prediction of solid phase. The dynamics of solid particle in a lid-driven cavity was investigated at a wide range of Reynolds numbers. The results show that the particle trajectories are critically dependence on the magnitude of Reynolds numbers and the vortex behavior in the cavity. Good comparisons with the experimental and previous studies demonstrate the multidisciplinary applications of the present scheme.
The creeping motion of a porous sphere at the instant it passes the center of a spherical container has been investigated. The Brinkman's model for the flow inside the porous sphere and the Stokes equation for the flow in the spherical container were used to study the motion. The stream function (and thus the velocity) and pressure (both for the flow inside the porous sphere and inside the spherical container) are calculated. The drag force experienced by the porous spherical particle and wall correction factor is determined. To cite this article: D. Srinivasacharya, C. R. Mecanique 333 (2005).
This paper investigates first the Stokes’ axisymmetrical translational motion of a spheroid particle, whose shape differs slightly from that of a sphere, in an unbounded micropolar fluid. A linear slip, Basset-type, boundary condition has been used. The drag acting on the spheroid is evaluated and discussed for the various parameters of the problem. Also, the terminal velocity is evaluated and tabulated for the slip, deformity, and micropolarity parameters. Secondly, the motion of a spheroidal particle at the instant it passes the centre of a spherical envelope filled with a micropolar fluid is investigated using the slip condition at the surface of the particle. The analytical expressions for the stream function and microrotation component are obtained to first order in the small parameter characterizing the deformation. As an application, we consider an oblate spheroidal particle and the drag acting on the body is evaluated. Its variation with respect to the diameter ratio, deformity, micropolarity, and slip parameters is tabulated and displayed graphically. Well-known cases are deduced, the wall effect is then examined and comparisons are attempted between the classical fluid and micropolar fluid.PACS Nos.: 47.45.Gx, 47.15.Gf, 47.50.–d
In this paper we study the steady flow, at low Reynolds number R , past a pervious sphere with a source at its centre. Assuming the source strength s also to be small, we use the method of matched asymptotic expansions to generate the inner and outer solutions pertaining to the problem. Graphs depicting the streamlines have been drawn to exhibit the effects of s and R . Also, the expression of the calculated drag experienced by the sphere shows that the effect of a source () s o is to decrease the drag.
- H Ramkissoon
H. Ramkissoon, J. Math. Sci. 8 (1997) 59.
1. Wall Correction Factor for a sphere moving in a spheroidal container
- Fig
Fig. 1. Wall Correction Factor for a sphere moving in a spheroidal container.
- E Cunningham
E. Cunningham, Proc. Roy. Soc. (London) A 83 (1910) 357.
- W E Williams
W.E. Williams, Phil. Mag. 29 (1915) 526.