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.     

Stokes flow with slip and Kuwabara boundary conditions

The forces experienced by randomly and homogeneously distributed parallel circular cylinder or spheres in uniform viscous flow are investigated with slip boundary condition under Stokes approximation using particle-in-cell model technique and the result compared with the no-slip case. The corresponding problem of streaming flow past spheroidal particles departing but little in shape from a sphere is also investigated. The explicit expression for the stream function is obtained to the first order in the small parameter characterizing the deformation. As a particular case of this we considered an oblate spheroid and evaluate the drag on it.     

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.     

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.     

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.     

Drag force and virtual mass of a cylindrical porous shell in potential flow

We study the two-dimensional potential flow due to a circular cylinder in motion relative to an unbounded fluid. The cylinder consists of a thin, circular porous shell with fluid inside. The full nonlinear hydrodynamic problem is solved by Fourier expansion of Green's theorem. The truncated series is determined numerically by sampling points around the circle. A dimensionless shell parameter is introduced. For homogeneous porous shells, a maximal drag force occurs at the value 0.433 for the shell parameter, but the virtual mass is a monotonous function of the shell parameter. For an inhomogeneous shell, we have found a maximal value for the virtual mass which is 5% above the value for a rigid cylinder. Some of the results may be relevant to offshore engineering, especially in connection with porous coating of platform legs to reduce the total force.     

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.     

Stokes drag on axially symmetric bodies: a new approach

In this paper a new approach to evaluate the drag force in a simple way on a restricted axially symmetric body placed in a uniform stream (i) parallel to its axis, (ii) transverse to its axis, is advanced when the flow is governed by the Stokes equations. The method exploits the well-known integral for evaluating the drag on a sphere. The method not only provides the value of the drag on prolate and oblate spheroids and a deformed sphere in axial flow which already exists in literature but also new results for a cycloidal body, an egg shaped body and a deformed sphere in transverse flow. The salient results are exhibited graphically. The limitations imposed on the analysis because of the lack of fore and aft symmetry in the case of an eggshaped body is also indicated. It is also seen that the analysis can be extended to calculate the couple on a body rotating about its axis of symmetry.     

Hydrodynamics of a particle impact on a wall

The problem of a particle impacting on a wall, a common phenomenon in particle-laden flows in the minerals and process industries, is investigated computationally using a spectral-element method with the grid adjusting to the movement of the particle towards the wall. Remeshing is required at regular intervals to avoid problems associated with mesh distortion and the constantly reducing maximum time-step associated with integration of the non-linear convective terms of the Navier–Stokes equations. Accurate interpolation between meshes is achieved using the same high-order interpolation employed by the spectral-element flow solver. This approach allows the full flow evolution to be followed from the initial approach, through impact and afterwards as the flow relaxes. The method is applied to the generic two-dimensional and three-dimensional bluff body geometries, the circular cylinder and the sphere. The principal case reported here is that of a particle colliding normally with a wall and sticking. For the circular cylinder, non-normal collisions are also considered. The impacts are studied for moderate Reynolds numbers up to approximately 1200. A cylindrical body impacting on a wall produces two vortices from its wake that convect away from the cylinder along the wall before stalling while lifting induced wall vorticity into the main flow. The situation for a sphere impact is similar, except in this case a vortex ring is formed from the wake vorticity. Again, secondary vorticity from the wall and particle plays a role. At higher Reynolds number, the secondary vorticity tends to form a semi-annular structure encircling the primary vortex core. At even higher Reynolds numbers, the secondary annular structure fragments into semi-discrete structures, which again encircle and orbit the primary core. Vorticity fields and passive tracer particles are used to characterize the interaction of the vortical structures. The evolution of the pressure and viscous drag coefficients during a collision are provided for a typical sphere impact. For a Reynolds number greater than approximately 1000 for a sphere and 400 for a cylinder, the primary vortex core produced by the impacting body undergoes a short-wavelength instability in the azimuthal/spanwise direction. Experimental visualisation using dye carried out in water is presented to validate the predictions.     

Numerical study on viscoelastic fluid flow past a rigid body

Viscoelastic non-Newtonian fluids can be achieved by adding a small amount of polymer additives to a Newtonian fluid. In this paper, numerical simulations are used to investigate the influence of such polymer additives on the behavior of flow past a circular cylinder. A numerical method is proposed that discretizes the non-linear viscoelastic system on a uniform Cartesian grid, with a penalization method to model the presence of the cylinder. The drag of the cylinder and the flow behavior under the effect of different Reynolds numbers ($\mathrm{Re}$), Weissenberg numbers (Wi) and polymer viscosity ratios (ε) are studied. Numerical results show that different flow characteristics are exhibited in different parameter zones. The polymer viscosity ratio plays an important role at low Weissenberg and Reynolds numbers, but as the Reynolds and Weissenberg numbers increase, the influence of ε weakens. The drag force of the cylinder is mostly affected by the Reynolds and Weissenberg numbers. At low Reynolds numbers, the drag of the cylinder and the flow fields are only affected by a large value of Wi when the elastic forces are strong. Non-trivial drag reduction occurs only when there is vortex shedding in the wake flow, whereas drag enhancement happens when the vortex shedding is inhibited.     

Creeping flow of a sphere nearby a cylinder

We present a three-dimensional solution of a sphere nearby an infinite cylinder at low Reynolds number. We utilize the Lamb’s general solution based on spherical harmonics and develop a framework based on cylindrical harmonics to solve the flow field around the sphere and outside the cylinder, respectively. The solution is solved semi-analytically by considering geometrical parameters, including sphere radius, sphere velocity, separation distance and cylinder radius. The drag force coefficients of the sphere which are dependent on the distance between the cylinder surface and the sphere, as well as the velocity contours in the vicinity of the sphere, are analyzed. We also provide an analytical formula to calculate the drag force. The analytical formula has good quantitative agreement with the semi-analytical solution when the radius of the cylinder is smaller than the sphere. Such analysis can give insights into the details of the complex interaction between the sphere and cylinder.     

Stokes flow past a porous sphere using Brinkman's model

A general non-axisymmetric Stokes flow past a porous sphere in a viscous, incompressible fluid is considered. The flow inside the sphere is governed by Brinkman's equations. A representation for velocity and pressure for the Brinkman's equations is suggested and a method of finding the flow quantities is given. Faxén's laws for drag and torque for the flow past a porous sphere are also given.     

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.     

通过可渗透近球体的轴对称流动

提出了可渗透近球体轴对称流动的分析方法.用修正边界条件的办法反映可渗透性.用正规摄动法求解了Stokes方程,达到ε的2阶修正.ε是描述不变形球体半径偏差的小参数.计算了阻力和流量,并从几何方面和表面渗透性方面考查了计算结果.还尝试将此理论应用于过滤供水问题.小型的生态学上重要的水生生物体的过滤器,被模型化为轴对称可渗透物体,用扁球体或近球体建立了该问题的初级模型.     

Flow past a porous sphere at small Reynolds number

low of an incompressible viscous fluid past a porous sphere has been discussed. The flow has been divided in three regions. The Region-I is the region inside the porous sphere in which the flow is governed by Brinkman equation with the effective viscosity different from that of the clear fluid. In Regions II and III clear fluid flows and Stokes and Oseen solutions are respectively valid. In all the three regions Stokes stream function is expressed in powers of Reynolds number. Stream function of Region II is matched with that of Region I at the surface of the sphere by the conditions suggested by Ochao-Tapia and Whitaker and it is matched with that of Oseen’s solutions far away from the sphere. It is found that the drag on the sphere reduces significantly when it is porous and it decreases with the increase of permeability of the medium.Received: February 7, 2002; revised: April 8, 2003 / June 9, 2004     

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.     

Creeping flow about a slightly deformed sphere

The problem of slow streaming flow of a viscous incompressible fluid past a spheroid which departs but little in shape from a sphere with mixed slip-stick boundary conditions, is investigated. The explicit expression for the stream function is obtained to the first order in the small parameter characterising the deformation. The case of an oblate spheroid is considered as a particular example and the force on this non-spherical body is evaluated. It is found that the parameter ₁, which arises in connection with the boundary condition, has significant effect upon the hydrodynamic force. In fact, it is shown that, the force is a quadratic function of this parameter up to the first order of deformation. Also, it is observed that the drag in the present case is less than that of the Stokes resistance for a slightly oblate spheroid. Some other special cases are also deduced from the present result. A brief discussion of the results to other body shapes is presented.     

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.     

Magnetic field effects on Newtonian and non-Newtonian ferrofluid flow past a circular cylinder

The changes in the flow properties under the action of electromagnetic body forces are investigated numerically for ferrofluid flow past a circular cylinder. Ferrofluid is modeled as both a Newtonian and a non-Newtonian Power-Law fluid. Magnetic forces are applied by placing magnets at different locations on the surface of the cylinder. The magnetostatic effects on the structure of the wake region, on drag reduction and on vortex formation length and frequency are shown and compared in terms of Reynolds number, interaction parameter, Power-Law index and magnet location. It is shown that the increase in the interaction parameter reduces drag for both Newtonian and non-Newtonian model. This decrease is observed to be higher for shear thinning and lower for shear thickening fluid compared to Newtonian case. It is also shown that vortex street formation in the wake region behind the cylinder may be delayed under high magnetic effects. The Strouhal number is higher for shear thinning case at both low and high Reynolds numbers, and lower for shear thickening case at high Reynolds numbers, compared to Newtonian fluid. The vortex formation frequency also decreases under the action of the magnetic field in all cases, however the vortex formation length increases. Placing the magnet towards the front region of the cylinder increases considerably the drag coefficient for both Newtonian and non-Newtonian model. This increase in drag coefficient is higher in the shear thinning fluid and lower in the shear thickening fluid compared to the Newtonian case.