Drag on two coaxial rigid spheres moving along the axis of a cylinder filled with Carreau fluid

Abstract

The boundary effect on the drag on two identical, rigid spheres moving along the axis of a long cylinder filled with a Carreau fluid for Reynolds number ranges from 0.1 to 40 is investigated. The influences of the key parameters of the problem under consideration, including the separation distance between two spheres, the relaxation time constant and the power-law index of a Carreau fluid, the Reynolds number, and the ratios (radius of sphere/radius of cylinder), on the drag acting on two spheres are investigated. We show that the boundary effect for the present case is more significant than that for the corresponding Newtonian fluid. The presence of the cylinder has the effect of enhancing the convective motion in the rear part of a sphere, thereby forming wakes and a reverse flow field, and this phenomenon is enhanced by the shear-thinning nature of a fluid. If the boundary effect is insignificant, the shear-thinning nature of a fluid has the effect of reducing the deviation of the ln(drag coefficient)-ln(Reynolds number) curve from a Stokes'-law-like relation. On the other hand, if it is significant, this deviation has a local minimum as the shear-thinning nature of a fluid varies.

Introduction

Evaluating the drag on a particle as it translates through a fluid is a classic problem. Sedimentation, which is often used to estimate the physical properties of a particle, for instance, involves this problem. The fluidized bed often adopted in chemical plants is another typical example where the drag on a particle needs be evaluated. In practice, two important factors usually should be considered in the calculation of the drag on a particle, namely, the presence of a boundary such as container wall, and the influence of neighboring particles, especially when the concentration of particles is appreciable. In general, the former yields an extra retarding force on a moving particle, and the closer the particle to a boundary the slower its movement [1], [2]. The latter depends largely on the concentration of particles, the relative orientation of neighboring particles, and the operating conditions such as Reynolds number.

The presence of a boundary and the neighboring particles on the drag on a particle has been investigated by many investigators. Stimson and Jeffery [3] presented a complete solution for the slow motion of two spheres parallel to their line of centers in an unbounded viscous fluid. Happel and Pfeffer [4] studied experimentally the falling of two spheres in a viscous liquid. Zhu et al. [5] developed a micro-force measuring system to directly measure the drag on two interacting particles at a medium large Reynolds number. Applying the same technique, Liang et al. [6] and Chen and Wu [7], [8] measured the drag on two interacting rigid spheres in a Newtonian fluid. Greenstein [9] analyzed the drag on two spherical particles translating in a cylindrical tube filled with an incompressible viscous fluid. Maheshwari et al. [10] investigated the effect of blockage ratio (tube diameter/sphere diameter) on the steady flow and heat transfer characteristics of an incompressible Newtonian fluid over a sphere and an in-line array of three spheres placed at the axis of a tube.

Many fluids encountered in industrial processes are of non-Newtonian nature where the rheological behavior of a fluid can differ appreciably from that of a Newtonian fluid [11], [12], [13], [14]. In these cases, it is expected that the drag on a particle in the former can also be quite different from that in the latter. For instance, the drag on a particle in a shear-thinning fluid is expected to be smaller than that in the corresponding Newtonian fluid. Several attempts have been made recently regarding the behavior of a particle in a non-Newtonian fluid [15], [16], [17], [18], [19], [20]. Missirlis et al. [21], for instance, studied the wall effect on the motion of a sphere in a shear-thinning power-law fluid at a low Reynolds number. Ceylan et al. [22] evaluated the drag on a rigid particle settling in a power-law fluid. Machač et al. [23], [24] conducted the sedimentation of both spherical and non-spherical particles in a Carreau fluid in the creeping flow regime. Jaiswal et al. [25] modeled an unbounded slow flow of a power-law non-Newtonian fluid through an assemblage of spheres. Zhu et al. [26] investigated the drag on two interacting rigid spheres in a power-law pipe flow with a large (pipe diameter/sphere diameter) ratio. Hsu et al. [27] modeled theoretically the sedimentation of a spheroid in a cylindrical tube filled with a Carreau fluid for a low to medium value of Reynolds number. Daugan et al. [28], [29] conducted an experimental study on the settling of two or three identical particles along their line of centres in a shear-thinning fluid at low Reynolds numbers. Maheshwari et al. [30] analyzed the hydrodynamic interaction between two rigid spheres tangentially translating in a power-law fluid.

The objective of this study is to analyze simultaneously the effects of particle–particle interaction, the presence of a boundary, and the non-Newtonian nature of a fluid on the drag on a particle. We consider the case of two identical, coaxial, rigid spheres moving along the axis of a long cylinder filled with a Carreau fluid. The influences of the separation distance between two spheres, the distance between spheres and cylinder wall, the Reynolds number, and the properties of a Carreau fluid, on the drag acting on the spheres are investigated.