### Abstract

- Top of page
- Abstract
- Introduction
- Model Description
- Results and Discussion
- Conclusions
- Notation
- Acknowledgments
- Literature Cited

A numerical method is utilized to examine the steady and transient mass/heat transfer processes that involve a neutrally buoyant liquid sphere suspended in simple shear flow at low Reynolds numbers is described. By making use of the known Stokes velocity field, the convection-diffusion equations are solved in the three-dimensional spherical coordinates system. For the mass transfer either outside or inside a liquid sphere, Sherwood number Sh approaches an asymptotic value for a given viscosity ratio at sufficiently high Peclet number Pe. In terms of the numerical results obtained in this work, two new correlations are derived to predict Sh at finite Pe for various viscosity ratios. © 2013 American Institute of Chemical Engineers *AIChE J*, 60: 343–352, 2014

### Introduction

- Top of page
- Abstract
- Introduction
- Model Description
- Results and Discussion
- Conclusions
- Notation
- Acknowledgments
- Literature Cited

Multiphase systems exist extensively in natural and industrial processes. Dispersed phases, such as bubbles, liquid droplets, and solid particles, would translate and rotate in flow fields; inclusion of a dispersed phase makes modeling of transport and chemical reactions more complicated. Among others, the study on the behavior of a single bubble, drop, or particle has a fundamental significance to the industrial scale-up. There have been many investigations on the translation and mass/heat transfer of a single particle in uniform flows.1978, 1996 In contrast, the studies are insufficient on the transport process around a bubble, drop, or particle in shear flow. As shear flow is expected in practically all chemical reactors and particularly in stirred tanks, understanding the transport around a dispersed phase in shear flow is of great importance. The rotation of a spherical liquid drop in simple shear flow, and the transport processes associated with this motion are the focus of this work.

Frankel and Acrivos1968 and Acrivos1971 studied the rates of heat and mass transfer from small cylinders and solid spheres in simple shear flow in the lower limit of Reynolds number by the singular perturbation method. When the direction of the flux is from the particle to infinite bulk phase, they derived the asymptotic formulas which related the Nusselt number *Nu* to the Peclet number *Pe* in the limit of , where the Nusselt number is , the Peclet number ( for heat transfer), *h* the heat transfer coefficient, *K*_{t} the thermal conductivity, the characteristic magnitude of the velocity gradient, *D* the mass diffusivity, *α* the thermal diffusivity, and *a* the sphere radius. For the heat transfer from a cylinder, they found Nusselt number approached a asymptotic value at , whereas it is 4.5 for a solid sphere.

Robertson and Acrivos1970, 1970 considered experimentally the momentum and heat transfer from a cylinder immersed in a uniform shear field. Because the closed streamlines surrounded the freely rotating cylinder, the asymptotic Nusselt number was found to be 2.65, in reasonable agreement with the theoretical value of 2.865 obtained by Frankel and Acrivos.1968

Subramanian and Koch2006, 2006 and Subramanian et al.2011 found that the microscale inertia would break the closed streamlines near a solid sphere immersed in a simple shear flow, leading to the deviation from the Stokes asymptote. Through a boundary layer analysis, they arrived at a correlation in the limit of *Re* << 1 and *PeRe* >> 1.

Yang et al.2011 applied the boundary layer analysis method coupled with numerical simulation to solve the transport process from a solid sphere suspended in simple shear flow at large *Pe* and *Re* ≤ 10. Their numerical results at high *Pe* and *Re* << 1 were consistent with the theoretical results derived by Subramanian and Koch.2006, 2006 When , their Nusselt number at high *Pe* also approached the asymptotic value of 4.5, in agreement with Acrivos' analysis.1971

For a spherical drop in simple shear creeping flow, Leal1973 used a perturbation method to calculate the temperature field in both phases in the limit of . Here, the far field is a linear temperature profile with a gradient normal to the velocity at far upstream and downstream boundaries. The final result is an expression for evaluating the effective thermal conductivity of a dilute suspension. Li et al. found the spiraling streamline structure at the 2-D plane near a liquid sphere immersed in a simple shear flow at low and moderate *Re* by numerical method.

The streamlines outside a drop are open in uniform flows.1978 Such a pattern of the streamlines will lead to Sherwood number *Sh* increasing with increasing Peclet number, where the Sherwood number is and *k* is the mass transfer coefficient. Aforementioned findings reveal that there is a significant difference between the shear flow case and the uniform flow one. If theoretical analysis is used to investigate the transport process of a single sphere immersed in simple shear creeping flow, the range of *Pe* is restricted to either very low or infinite values. Most previous research works on drops in shear flow were focused on the particle motion1958, 1966 and the transport process of a dilute suspension.1973 Limited information has been reported concerning the characteristics of mass/heat transfer from or to a drop in simple shear flow, particularly at large *Pe*.

In this work, the convection-diffusion equation is solved by a finite difference method2005 in order to examine the solute concentration near the surface of a fluid sphere immersed in simple shear creeping flow over a range of Peclet numbers; such results of mass transfer have not been previously reported. The validity of the numerical method was checked by computing the rate of heat/mass transfer of a translating drop.

### Conclusions

- Top of page
- Abstract
- Introduction
- Model Description
- Results and Discussion
- Conclusions
- Notation
- Acknowledgments
- Literature Cited

In terms of the known Stokes velocity field at small Reynolds numbers, steady-state mass/heat transfer outside a liquid sphere and transient transport inside a liquid sphere in simple shear creeping flow are investigated by numerical simulations.

For the external transport problem, our simulations show that *Sh* would reach an asymptotic value for sufficiently large *Pe*_{1} at any viscosity ratio, which is related with the closed streamlines around the sphere. The value of viscosity ratio also influences the rate of mass transfer. When *λ* = 100, the transport behavior is close to that of solid spheres. This characteristic asymptotic value differs substantially from the case in uniform or extensional flows.

For the transient internal problem inside a liquid sphere, the simulation results indicate that *Sh* also approaches an asymptotic value. However, the convective effect is weak on the mass transfer inside the sphere at high viscosity ratios, which is related to the special topology of streamlines.

This study deals with shear flow around an isolated drop. The present results are useful for the dilute suspensions with insignificant settling that are mostly relevant for mist and aerosol situations. In other applications, shear plus slip may be a more realistic situation. The physical problem of the synergistic effect of shear and slip should consider the relative directions of the shear and the slip flows and the ratio of the shear rate to the slip velocity, and it is a problem that requires much more effort to be fully characterized and comprehended.