Drag on a single fluid sphere translating in power-law liquids at moderate Reynolds numbers
Introduction
Owing to their overwhelming theoretical and pragmatic significance, in recent years considerable effort has been directed at investigating the drag behaviour of single bubbles and rigid spheres in a great variety of non-Newtonian fluids including purely viscous, viscoplastic and viscoelastic fluids. An inspection of the recent reviews of the pertinent literature (Chhabra, 2006) shows that the sedimentation behaviour of rigid particles in purely viscous fluids (mostly power-law) has been studied most thoroughly, followed by that in viscoplastic and in viscoelastic liquids. Based on a combination of the numerical and experimental studies, reliable results are now available on drag coefficient—Reynolds number behaviour for a sphere falling in unconfined fluids up to about Reynolds number of 1000, which is nowhere near the wide range of conditions encompassed by the corresponding literature for Newtonian fluids (Clift et al., 1978, Michaelides, 2006). At the other extreme is the case of gaseous bubbles rising freely in non-Newtonian fluids. Here too, based on a combination of analytical and/or numerical results coupled with experimental data, reliable information on their drag behaviour in power-law fluids is now available up to about the Reynolds number of 500 (Rodrigue, 2001a, Rodrigue, 2001b, Rodrigue, 2004; Chhabra, 2006, Dhole et al., 2007). However, it is appropriate to add here that the behaviour of rising gas bubbles in non-Newtonian liquids differs significantly from that of solid spheres at one hand and that of bubbles in Newtonian fluids at the other hand, as noted in detail elsewhere (Clift et al., 1978, Chhabra, 2006). These differences stem from the ability of the bubble to deform thereby exhibiting a great variety of shapes determined by the net resultant forces acting on the bubbles. Furthermore, the surface active agents also influence the bubble dynamics in a significant manner (Thorsen et al., 1968).
In contrast, much less is known about the intermediate case of a liquid droplet undergoing steady translation in a quiescent power-law continuous medium. Typical examples entailing the translation of a droplet in non-Newtonian continuous phase include the use of polymer solutions in enhanced oil recovery operations wherein the oil droplets become suspended in polymer solutions and/or during their pipeline transport in the form of oil-in-polymer solution type emulsions, or in scores of personal care products (lotions, shampoos, creams), food-stuffs, pharmaceutical products wherein polymeric thickening agents are used to enhance their stability for extended shelf-life. Irrespective of whether the separation in such emulsions is desirable or not, the need to estimate the free settling velocity of a drop in power law continuous medium arises frequently in process design calculations. From a theoretical standpoint, this case differs from that of a rigid sphere and of a gas bubble in so far that one needs to solve the governing equations for the continuous phase only in the latter cases. On the other hand, for a fluid sphere, one needs to solve the governing equations for both inner and outer phases which are coupled via the continuity of the velocity and tangential stress at the interface. Due to the highly non-linear form of the viscous terms, theoretical solution is not possible even for the slow translation (at vanishingly small values of the Reynolds number) of a Newtonian fluid sphere in the simple power-law continuous phase, when the non-linear intertial terms in momentum equations are altogether neglected. Therefore, the progress in this area has been rather slow from theoretical and numerical stand points. This work aims to elucidate the role of power-law rheology on the drag behaviour of a Newtonian fluid sphere undergoing steady translation in a quiescent power-law continuous phase over a wide range of the Reynolds number and of the ratio of the viscosity of the two phases. It is, however, appropriate to begin with a short review of the previously available scant literature on this subject.
Section snippets
Previous work
Ever since the celebrated analysis of Hadamard (1911) and Rybzynski (1911) for the creeping motion of a fluid sphere in an unconfined incompressible Newtonian fluid, considerable research effort has been extended in studying the drag behaviour of Newtonian-spherical and non-spherical-drops in another immiscible Newtonian medium focusing on a variety of aspects including the drag coefficient–Reynolds number relationship, wall effects, prediction of shape, wake dynamics, etc. Consequently over
Problem statement and description
Depending upon the relative magnitudes of the inertial, viscous and surface tension forces, the fluid spheres (drops and bubbles) are known to undergo shape changes during their free fall (or rise) in inelastic power-law fluids, though gas bubbles show greater deviations than the drops under otherwise identical conditions. In Newtonian liquids with moderate to high values of surface tension, significant deviations from the spherical shape of drops do not occur up to about the Reynolds numbers
Numerical details
The governing equations (1), (2a), (2b), subject to the boundary conditions outlined in (8a), (8b), (8c), (8d), (8e), (8f), (8g), (8h) have been solved by a finite difference method based SMAC-implicit algorithm implemented on a staggered grid arrangement, a simplified version of MAC method due to Harlow and Welch (1965) which has been adopted for power-law fluids. The application of an implicit algorithm on a staggered grid arrangement avoids the numerical instability commonly encountered at
Results and discussion
The scaling of the governing equations and of the boundary conditions suggests this flow to be governed by five dimensionless groups, namely, Reynolds number , characteristic viscosity ratio , power-law index , density ratio and drag coefficient . In this study, numerical results are obtained to elucidate the roles of all these parameters except that of the density ratio which was set equal to unity. The influence of the density ratio has been explored extensively by Juncu
Conclusions
The steady settling behaviour of a Newtonian viscous fluid sphere in a quiescent power-law fluid has been investigated numerically using a finite difference method based SMAC implicit solver on a staggered grid arrangement. The formation of a wake or a return-flow region in the rear of the drop depends primarily on the values of k, and . Irrespective of the values of and , recirculation or flow separation does not occur for . For and , as the value of k
Notation
total drag coefficient, dimensionless friction drag coefficient, dimensionless pressure drag coefficient, dimensionless mesh size in r-direction, dimensionless mesh size in streamwise (-) direction, ° time-step size, dimensionless k characteristic viscosity ratio, , dimensionless m power-law consistency index, n power-law behaviour index, dimensionless p pressure, dimensionless QUICK quadratic upstream interpolation for convective kinematics r radial distance, dimensionless R drop radius,
References (51)
- et al.
Settling and transport of spherical particles in power-law fluids at finite Reynolds number
Journal of Non-Newtonian Fluid Mechanics
(1994) - et al.
Drag and mass transfer in multiple drop slow motion in power-law fluids
Chemical Engineering Science
(1986) A numerical study of steady viscous flow past a fluid sphere
International Journal of Heat and Mass Transfer
(1999)- et al.
Sedimentation in emulsions of mono-size droplets at moderate Reynolds numbers
Chemical Engineering Research & Design
(2006) A stable and accurate convective modeling procedure based on quadratic upstream interpolation
Computer Methods in Applied Mechanics and Engineering
(1979)- et al.
Creeping flow of a power-law fluid past a fluid sphere
International Journal of Multiphase Flow
(1976) - et al.
A numerical study of the motion of a spherical drop rising in shear-thinning fluid systems
Journal of Non-Newtonian Fluid Mechanics
(2003) - et al.
Dynamic processes in a deformed drop rising through shear-thinning fluids
Journal of Non-Newtonian Fluid Mechanics
(2005) - et al.
The effect of viscoelasticity on the translation of a surfactant covered Newtonian drop
Journal of Non-Newtonian Fluid Mechanics
(1992) - et al.
Flow around a spherical drop at intermediate Reynolds numbers
Applied Mathematics and Mechanics
(1976)
Determination de la trainee engendree par une sphere fluide en translation
Chemical Engineering Journal
On the terminal velocity of circulating and oscillating liquid drops
Chemical Engineering Science
Hydrodynamics of creeping motion of an ensemble of power-law liquid drops in an immiscible power-law medium
International Journal on Engineering Science
Shape of Newtonian liquid drop moving through an immiscible quiescent non-Newtonian liquid
Chemical Engineering and Processing
Motion of liquid drops in rheologically complex fluids
Canadian Journal of Chemical Engineering
Transport phenomena
Viscous flows past spherical gas bubbles
Journal of Fluid Mechanics
Bubbles, Drops, and Particles in Non-Newtonian Fluids
Wall effects on terminal velocity of small drops in Newtonian and non-Newtonian fluids
Canadian Journal of Chemical Engineering
Bubbles, Drops and Particles
Flow of power-law fluids past a sphere at intermediate Reynolds numbers
Industrial & Engineering Chemistry Research
Flow and shape of drops in non-Newtonian fluids
Transactions of the Society of Rheology
Drag coefficients of viscous spheres at intermediate and high Reynolds numbers
Trans. ASME, Journal of Fluid Engineering
Cited by (37)
On enhancing interfacial mass transport through microextraction in dispersed droplet systems
2023, International Journal of Heat and Mass TransferNumerical investigation of particle cloud sedimentation in power-law shear-thinning fluids for moderate Reynolds number
2022, Chemical Engineering ScienceCitation Excerpt :Lockyer et al. (1980) and Graham and Jones (1994) formulated simple correlations between the drag coefficient and Reynolds number for the settling of a single particle in power-law fluids. Kishore et al. (2007) later expanded the range of Reynolds number covered by these correlations. Comprehensive reviews on single-particle motions in various complex fluids were conducted by Chhabra (2007) and Chhabra and Richardson (2008).
Critical Reynolds numbers of shear-thinning fluids flow past unbounded spheres
2018, Powder TechnologyCitation Excerpt :Finally, in summary, for the case of Newtonian fluid flow past spheres, adequate information is now available pertaining to the critical Reynolds numbers for the onset of steady axisymmetric and non-axisymmetric separated flows. However, analogous information is not available either on the basis of experimental studies or numerical simulations even for simple power-law type non-Newtonian fluids let alone other complex non-Newtonian fluids though steady axisymmetric momentum and heat/mass transfer results are available up to Re = 200 [5,9,10]. Therefore, this work is aimed to numerically investigate these critical Reynolds numbers of the shear-thinning type power-law non-Newtonian fluids flow past spheres using 3D CFD simulations.
Dynamics of an air bubble rising in a non-Newtonian liquid in the axisymmetric regime
2017, Journal of Non-Newtonian Fluid MechanicsForced convective heat transfer from spheres to Newtonian fluids in steady axisymmetric flow regime with velocity slip at fluid-solid interface
2016, International Journal of Thermal SciencesCitation Excerpt :The fully converged temperature field is then used to evaluate the near surface kinematics such as the isotherm contours, local and average Nusselt numbers as described in previous section for various combinations of pertinent dimensionless parameters. Further, more details of this solver can be found elsewhere [23,31,69–78]. The present heat transfer problem is solved using a segregated approach.
Numerical study on interaction between two bubbles rising side by side in CMC solution
2013, Chinese Journal of Chemical Engineering