Potential flow past a porous body with a core of different permeability

It is well known that a uniform flow past a non-permeable rigid body does not exert a total force upon the surface of the body, however this is not the case when the body is permeable. Power et. al. (1984, 1986) first solved the problem of uniform potential flow past a two-dimensional permeable circular cylinder, with constant permeability, and found that the exterior flow exerts a drag force upon the surface of the cylinder independent of its size and secondly the problem when the uniform potential flow past a porous sphere, with constant permeability, in this case the exterior flow exerts a drag force on the sphere which is linearly dependent on the radius of the sphere. Here we will present the solution of two problems, a uniform potential flow past a porous circular cylinder and past a porous sphere, for each case the porous body is composed of two materials with different permeabilities. In both cases the total force exerted by the exterior flow upon the body is dependent on the thickness of the porous materials, and in the limit when the two permeabilities are equal, the previous results, circular cylinder and sphere, with constant permeability, are recovered. Atlhough, the mathematics involved in the solution of the present problem is simple, due to the nice boundary geometry of the bodies, the final expression for the total force found in each case is quite interesting on the way it depends on the permeability relation, in particular, in the limiting cases of a porous body with solid or hollow core.     

Slip at the surface of a sphere translating perpendicular to a plane wall in micropolar fluid

The Stokes axisymmetrical flow caused by a sphere translating in a micropolar fluid perpendicular to a plane wall at an arbitrary position from the wall is presented using a combined analytical-numerical method. A linear slip, Basset type, boundary condition on the surface of the sphere has been used. To solve the Stokes equations for the fluid velocity field and the microrotation vector, a general solution is constructed from fundamental solutions in both cylindrical, and spherical coordinate systems. Boundary conditions are satisfied first at the plane wall by the Fourier transforms and then on the sphere surface by the collocation method. The drag acting on the sphere is evaluated with good convergence. Numerical results for the hydrodynamic drag force and wall effect with respect to the micropolarity, slip parameters and the separation distance parameter between the sphere and the wall are presented both in tabular and graphical forms. Comparisons are made between the classical fluid and micropolar fluid.      

Finite element calculations of flow past a spherical bubble rising on the axis of a cylindrical tube

The velocity and pressure fields of a Newtonian fluid with homogeneous and constant physical properties flowing around a sphere on the axis of a cylindrical tube with no slip, free slip and partial slip at the sphere surface and no slip at the cylinder wall have been calculated by solving the Navier-Stokes equations and the continuity equation using the finite element technique with the penalty function method. Terminal rise velocities of spherical air bubbles in water have been calculated as function of the bubble radius and some conclusions have been drawn about the nature of the interface. Finally, the influence of the presence of a cylindrical wall on the drag force has been determined and a new empirical equation is derived for the wall correction factor for a sphere rising with free slip at its surface at low Reynolds number.     

Hypersonic flow past a sphere

The effects of dissociation or ionization of air on the analytical solution for hypersonic flow past a sphere are considered here, under certain assumptions. It has been assumed that the shock wave is in the shape of a sphere, that the density ratio across the shock is constant, that the flow behind the shock is at constant density and that dissociation or ionization only occurs behind the shock wave. Thus the effects of the compressibility of the air, variation of density ratio along the shock, and the department of the shock shape from being circular are not taken into account. Here the velocity, pressure, temperature, pressure coefficient and vorticity, etc., at any point between the shock and the surface of the sphere in the presence of dissociation or ionization are obtained. In addition, shock detachment distance, drag coefficient, stagnation point velocity gradient and sonic points on the shock and the surface have also been obtained. The results have been compared with the corresponding results obtained in the case when dissociation or ionization does not occur behind the shock.     

Flow due to the longitudinal and torsional oscillation of a cylinder

Summary An exact solution is obtained for the motion of a fluid contained in an infinité circular cylinder which is undergoing both torsional and longitudinal oscillations. An analytical expression is obtained for the viscous drag on the cylinder and the velocity is depicted graphically. Where possible, comparisons are made with corresponding results for the external problem.     

Transient elastohydrodynamic drag on a particle moving near a deformable wall

We examine the prescribed time-dependent motion of a rigid particle(a sphere or a cylinder) moving in a viscous fluid close toa deformable wall. The fluid motion is described by a nonlinearevolution equation, derived using lubrication theory, whichis solved using numerical and asymptotic methods; a local linearpressure–displacement model describes the wall. When theparticle moves from rest towards the wall, fluid trapping beneaththe particle leads to an overshoot in the normal force on theparticle; a similarity solution is used to describe trappingat early times and a multiregion asymptotic structure describesfluid draining at late times. When the particle is pulled fromrest away from the wall, a peeling process (described by a quasisteadytravelling wave) determines the rate at which fluid can enterthe growing gap between the particle and the wall, leading toa transient adhesive normal force. When a cylinder moves fromrest transversely over the wall, transient peeling motion isagain observed (especially when the wall is initially indented),giving rise to an overshoot in the transverse drag. Simulationsfor a translating sphere show highly nonlinear wall deformationscharacterized by a localized crescent-shaped ridge. Despitegenerating sharp transient deformations, we found no numericalevidence of finite-time choking events.     

A study of an arbitrary Stokes flow past a fluid coated sphere in a fluid of a different viscosity

A general method to discuss the problem of an arbitrary Stokes flow (both axisymmetric and non-axisymmetric flows) of a viscous, incompressible fluid past a sphere with a thin coating of a fluid of a different viscosity is considered. We derive the expressions for the drag and torque experienced by the fluid coated sphere and also discuss the conditions for the reduction of the drag on the fluid coated sphere. In fact, we show that the drag reduces compared to the drag on a rigid sphere of the same radius when the unperturbed velocity is either harmonic or purely biharmonic, i.e., of the form ${r^2\vec{\textbf{v}}}$ , where ${\vec{\textbf{v}}}$ is a harmonic function. Previously Johnson (J Fluid Mech 110:217–238, 1981), who considered a uniform flow showed that the drag on the fluid coated sphere reduces compared to the drag on the uncoated sphere when the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4. We show that this result is true when the undisturbed velocity is harmonic or purely biharmonic, uniform flow being a special case of the former. However, we illustrate by an example that the drag may increase in a general Stokes flow even if this ratio is greater than 4. Moreover, when the unperturbed velocity is harmonic or purely biharmonic, and the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4 for a fixed value of the viscosity of the ambient fluid, we determine the thickness of the coating for which the drag is minimum.     

Dissociation effect on the hypersonic flow past a circular cylinder

The effects of dissociation of air on hypersonic flow past a circular cylinder at zero angle of incidence are considered under the assumptions that the shock wave is in the shape of a circular cylinder, the density ratio across the shock is constant, the flow behind the shock is at constant density and dissociation occurs only behind the shock wave. In the present paper, the velocity, pressure and drag coefficients, vorticity, shock detachment distance, stagnation point velocity gradient and sonic points on the shock and the surface have been obtained in the presence of dissociation. The results have been compared with the corresponding results obtained in the case when dissociation dose not occur and the corresponding results in the case of the sphere in the presence of dissociation.     

On the hydrodynamic interaction of two circular cylinders oscillating in a viscous fluid

The present paper deals with the plane flow fields induced by two parallel circular cylinders with radiia andb oscillating in a direction which is i) parallel or ii) perpendicular to the plane containing their axes. The effect of the cylinders' hydrodynamic interaction on steady streaming has been studied analytically at high frequency by the method of matched asymptotic expansions.It is found that ifa=b the steady streaming is directed symmetrically to the cylinders while whenab (in the case i)) the secondary steady flow is directed towards the larger cylinder and one of the outer steady vortices disappears.It is shown in case i) that the drag force acting on each cylinder is smaller than the same force experienced on a single cylinder with the same radius which is placed in an unbounded oscillating flow. When the cylinder radii are equal, the drag is greater on the forward cylinder than on the rear one.In contrast, in case ii), wherea=b, it is shown that the drag on each of the two cylinders is greater than the drag acting on a single cylinder with the same radius placed in an unbounded oscillating stream and also each of the cylinders experiences a repulsive force in a direction perpendicular to the oscillating flow.     

Flow of a micropolar fluid through a circular cylinder subject to longitudinal and torsional oscillations

The internal flow of a micropolar fluid inside a circular cylinder which is subject to longitudinal and torsional oscillations is investigated. Analytical expressions of the fluid velocity and micro-rotation are obtained. Explicit expressions of the shear stresses and drag force acting at the wall of the cylinder are derived as well. A numerical analysis followed to examine the effect of the micropolar fluid on the two components of the velocity field through graphical curves. In addition, the magnitude of the tangential drag is computed and compared with the case of a classical fluid.     

The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

Analytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier-Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen-like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall-bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bending-torsion vibration of a partially submerged cylinder with an arbitrary cross-section

An analytical method is presented to investigate the bending-torsion vibration characteristics of a cylinder with an arbitrary cross-section and partially submerged in water. The compressibility and the free surface waves of the water are considered simultaneously in the analysis. The exact solution of structure–water interaction is obtained mathematically. Firstly, the analytical expression of the velocity potential of the water is derived by using the method of separation of variables. The unknown coefficients in the velocity potential are determined by the longitudinal and circumferential Fourier expansions along the outer surface of the cylinder and are expressed in the form of integral equations including the unknown dynamic bending deflection and torsional angle of the cylinder. Secondly, the force and torque acting on the cylinder per unit length, provided by the water, are obtained by integrating the water dynamic pressure along the circumference of the cylinder. The general solution of bending-torsion vibration of the cylinder under the water dynamic pressure is derived analytically. The integral equations included in the velocity potential of the water can be solved exactly. Finally, the eigenfrequency equation of cylinder–water interaction is obtained by means of the boundary conditions of the cylinder. Some numerical examples for elliptical columns partially submerged in water are provided to show the application of the present method.     

Slow viscous flow through a membrane built up from porous cylindrical particles with an impermeable core

This paper concerns the slow viscous flow through an aggregate of concentric clusters of porous cylindrical particles with Happel boundary condition. An aggregate of clusters of porous cylindrical particles is considered as a hydro-dynamically equivalent to solid cylindrical core with concentric porous cylindrical shell. The Brinkman equation inside the porous cylindrical shell and the Stokes equation outside the porous cylindrical shell in their stream function formulations are used. The drag force acting on each porous cylindrical particle in a cell is evaluated. In certain limiting cases, drag force converges to pre-existing analytical results, such as, the drag on a porous circular cylinder and the drag on a solid cylinder in a Happel unit cell. Representative results are then discussed and presented in graphical forms. The hydrodynamic permeability of the membrane built up from porous particles is evaluated. The variation of hydrodynamic permeability with different parameters is graphically presented. Some new results are reported for flow pattern in the porous region. Being in resemblance with the model of colloid particles with a coating of porous layers due to adsorption phenomenon, results obtained through this model can be useful to study the membrane filtration process.     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

The loss of stability of a two-step rod when it hits a rigid barrier

The critical precollision velocity of a two-step rod of finite length when it collides with a rigid obstacle, leading to a loss of its stability, is calculated by an analytical solution of the wave equation using d’Alembert's method. The critical force and velocity are calculated using Euler's formula for a static load.     

Thermal radiation effects on magnetohydrodynamic free convection heat and mass transfer from a sphere in a variable porosity regime

A mathematical model is presented for multiphysical transport of an optically-dense, electrically-conducting fluid along a permeable isothermal sphere embedded in a variable-porosity medium. A constant, static, magnetic field is applied transverse to the cylinder surface. The non-Darcy effects are simulated via second order Forchheimer drag force term in the momentum boundary layer equation. The surface of the sphere is maintained at a constant temperature and concentration and is permeable, i.e. transpiration into and from the boundary layer regime is possible. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite difference scheme. Increasing porosity (ε) is found to elevate velocities, i.e. accelerate the flow but decrease temperatures, i.e. cool the boundary layer regime. Increasing Forchheimer inertial drag parameter (Λ) retards the flow considerably but enhances temperatures. Increasing Darcy number accelerates the flow due to a corresponding rise in permeability of the regime and concomitant decrease in Darcian impedance. Thermal radiation is seen to reduce both velocity and temperature in the boundary layer. Local Nusselt number is also found to be enhanced with increasing both porosity and radiation parameters.     

Optimal Recovery Methods for Solutions of the Dirichlet Problem that are Exact on Subspaces of Spherical Harmonics

We consider the optimal recovery problem for the solution of the Dirichlet problem for the Laplace equation in the unit d-dimensional ball on a sphere of radius ρ from a finite collection of inaccurately specified Fourier coefficients of the solution on a sphere of radius r, 0 < r < ρ < 1. The methods are required to be exact on certain subspaces of spherical harmonics.

     

蠕动流中圆球阻力系数的入口效应

本文研究了圆球在半无穷长圆管入口处的蠕动流。得到了速度分布,压力分布和流函数的无穷级数形式的分析解.采用配置法将无穷级数截断并确定出级数中各项系数.求出了均匀入口流绕静止圆球以及圆球以瞬时速度在管内静止流体中运动这两种情形下圆球的阻力系数以及圆球表面上的应力分布.结果表明,当圆球在入口处运动时会遭受到较无穷圆管内为大的阻力.本文还对配置法的收敛性进行了数值实验.试验证明,该法具有好的收敛性.     

The virtual mass of a sphere in a suspension of spherical particles

The problem of the virtual mass of a sphere, moving in an ideal incompressible fluid when there are other identical spherical particles of arbitrary mass present is considered. A solution is constructed for the velocity potential of the fluid in the form of the superposition of perturbation fields, introduced into the flow by each of the particles. The perturbation fields are obtained in the form of functional series, the coefficients of which are mutually consistent by a defined system of equations. An explicit expression is obtained for the hydrodynamic force acting on the sphere in the form of a function of the coordinates of all the particles. A simple analytical dependence of the mean value of the force and the virtual mass of the sphere on the particle-to-fluid density ratio in a first approximation of the volume fraction of the dispersed phase is obtained for a statistically uniform distribution of the dispersed particles in the suspension, using the procedure of averaging over their different possible configurations in space.     

A solution for the problem of stokes flow past a porous sphere

The problem of a general non-axisymmetric Stokes flow of a viscous fluid past a porous sphere is considered. The expressions for the velocity and pressure, both inside and outside the sphere are given, when the flow outside satisfies the Stokes equations and the flow inside the sphere is governed by Darcy's law. The expressions for drag and torque are given. It is found that the drag is greater or smaller than the drag in the rigid case, depending on whether the undisturbed velocity is a pure biharmonic or a harmonic respectively. The torque is same as in the rigid case.