PHYSICS OF FLUIDS VOLUME 14, NUMBER 3 MARCH 2002 Hydrodynamic stability of a suspension in cylindrical Couette flow Mohamed E. Ali Mechanical Engineering Department, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia Deepanjan Mitra, John A. Schwille, and Richard M. Lueptow Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208 ͑Received 8 October 2001; accepted 11 December 2001͒ A linear stability analysis was carried out for a dilute suspension of rigid spherical particles in cylindrical Couette flow. The perturbation equations for both the continuous fluid phase and the discontinuous particle phase were decomposed into normal modes resulting in an eigenvalue problem that was solved numerically. At a given radius ratio, the theoretical critical Taylor number at which Taylor vortices first appear decreases as the particle concentration increases. Increasing the ratio of particle density to fluid density above one decreases the stability. However, using an effective Taylor number based on the suspension density and viscosity largely accounts for this effect. The axial wave number is the same for a suspension as it is for a pure fluid. Experiments using neutrally buoyant particles in a Taylor–Couette apparatus show that the flow is more stable as the particle concentration increases. The reason that the theory does not fully capture the physics of the flow should be addressed in future research. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1449468͔ I. INTRODUCTION rotation of the inner cylinder are also thought to inhibit par- ticles from plugging the pores of the filter.26 The linear stability of circular Couette flow in the annu- Nearly all research on the stability of cylindrical Couette lus between a rotating inner cylinder and a concentric, fixed flow has been done for simple Newtonian fluids with a few outer cylinder has been studied from both theoretical and notable exceptions. Recently there has been interest in the experimental standpoints. The instability appears as pairs of stability of a viscoelastic fluid in Taylor–Couette flow.27 For counter-rotating, toroidal vortices stacked in the annulus. example, using a simple viscoelastic constitutive equation, Taylor1 conducted a simple flow visualization experiment to Khayat found that the flow is destabilized as the fluid elas- confirm his analytic prediction for the onset of the instability. ticity is increased.28 However, experimental results indicate Chandrasekhar,2 DiPrima and Swinney,3 Kataoka,4 and both stabilizing and destabilizing effects of viscoelasticity, Koschmeider5 provide extensive summaries of the abundant depending upon the polymer solution used.29 Fewer studies research on this topic since Taylor’s pioneering work. have addressed a suspension as a complex fluid in cylindrical The stability of Taylor vortex flow is altered when com- Couette flow. Nsom carried out a stability analysis for a sus- plexity is added to the system. For instance, an axial flow in pension of rigid fibers using rheological coefficients in the the annulus stabilizes the circular Couette flow so that the stress tensor to account for the non-Newtonian effects.30 Re- transition to supercritical Taylor vortex flow occurs at a sults indicate that increasing the concentration of fibers in- higher Taylor number.6–8 Likewise, a radial flow in the an- creases the stability of the system. Yuan and Ronis consid- nulus between differentially rotating porous cylinders also ered the stability of colloidal crystals in cylindrical Couette affects the stability of the Taylor vortex flow.9–11 flow using a Stokes drag interaction between the particles Our interest in the effect of a complex fluid, specifically and the fluid.31 Although the Taylor instability is suppressed a dilute suspension, on the stability of Taylor–Couette flow as the colloidal crystal lattices become more rigid, other lat- is motivated by the processing of a suspension in a Taylor– tice instabilities appear. Although not directly related to the Couette reactor cell12–15 or in a rotating filter device during stability of a suspension, there recently has been substantial dynamic filtration.16–24 In Taylor–Couette reactors, chemi- interest in tracking particle motion in both nonwavy and cally reacting species are dispersed or exposed uniformly to wavy Taylor–Couette flow.32–37 This work has focused on chemical catalysts by the vortical motion. In rotating filter particle paths and enhanced diffusion without regard to the devices, an axial flow introduces a suspension into the annu- effect of the particles in the suspension on the stability of the lus between a rotating porous inner cylinder and a stationary flow. Finally, Dominguez-Lerma et al. detected a nonperiodi- nonporous outer cylinder. Filtrate passes radially through the cally time-dependent nonuniformity in the size of the vorti- porous wall of the rotating inner cylinder, while the concen- ces when a low concentration of flakes was used to visualize trate is retained in the annulus. The Taylor vortices appearing Taylor–Couette flow in vertical apparatus, but they did not in the device are believed to wash the filter surface of the note how the flakes affected the critical Taylor number.38 inner cylinder clean of particles thus preventing the plugging The processing of suspensions in Taylor–Couette reactor of pores of the filter medium.25 Centrifugal forces acting on cells and in rotating filtration devices has motivated us the particles in suspension and the shear resulting from the to address the effect of particles on the stability of Taylor– 1070-6631/2002/14(3)/1236/8/$19.001236 © 2002 American Institute of Physics Downloaded 14 Mar 2002 to 129.105.69.150. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp Phys. Fluids, Vol. 14, No. 3, March 2002 Stability of a suspension in cylindrical Couette flow 1237 ͒ ␣͑ץ ץ ␣ץ Couette flow. In this study we apply linear hydrodynamic p 1 pV*pz stability analysis to determine the critical Taylor number for ϩ ͑␣ r*V* ͒ϩ ϭ0, ͑2a͒ *zץ r* p prץ *t* rץ the transition from stable cylindrical Couette flow to vortical 2 *p* nFץ V* 1ץ V* V*␪ץ *Vץ flow when a dilute concentration of a discontinuous phase of pr ϩ * prϪ p ϩ * pr ϭϪ ϩ r , ␳ ␣ ץ ␳ ץ V pz ץ V pr ץ rigid spherical particles is present in the fluid. In addition, we t* r* r* z* p r* p p provide results of simple experiments on the stability of a ͑2b͒ suspension of neutrally buoyant particles in cylindrical Cou- ץ ץ ץ ette flow. While the stability of the flow of a suspension in a V*p␪ V*p␪ V*prV*p␪ V*p␪ nF␪* ϩV* ϩ ϩV* ϭ , ͑2c͒ z* ␣ ␳ץ r* r* pzץ t* prץ Taylor–Couette device is a much simpler problem than that in a Taylor–Couette reactor cell or in a rotating filtration p p ץ ץ ץ ץ device that have axial flow and other complications, our in- V*pz V*pz V*pz 1 p* nFz* ϩV* ϩV* ϭϪ Ϫgϩ , ͑2d͒ z* ␣ ␳ץ z* ␳ץ r* pzץ t* prץ tent is to provide insight into the stability of the flow in these devices. p p p ␳ where (V*pr ,V*p␪ ,V*pz) are the particle velocities, p is the ␣ particle density, and p is the volume fraction of particles. ␣ ␣ ϩ␣ ϭ f is the volume fraction of the fluid such that f p 1. II. ANALYTICAL FORMULATION ␣ ӷ␣ ␣ We consider very dilute suspensions ( f p), so f can be The conservation equations for the system are based on assumed constant and approximately equal to unity. It there- the traditional two-fluid formulation, appropriate for a dilute fore does not appear in the fluid phase equations. Note that concentration of monodisperse rigid particles.39 In this for- the volume fraction of particles is related to the number den- 40 mulation, concentration-weighted forms of the continuity sity by and Navier–Stokes equations in cylindrical coordinates n␲␾*3 ␪ ␣ ϭ (r, ,z) are used for the continuous fluid phase and the dis- p , perse particle phase. The flow is assumed to be steady and 6 incompressible. The cylinders are assumed to be infinitely where ␾* is the diameter of the particle. The last term on the long with the inner cylinder rotating and the outer cylinder right-hand side of the momentum equations for both phases fixed. For axisymmetric flow, the dimensional form of the is the Stokes drag and added mass, expressed as continuity and Navier–Stokes equations for the fluid phase ץ are ␳ ␲͑␾*͒3 ϭ ␲␮␾ ͑ Ϫ ͒ϩ f ͑ Ϫ ͒ Fi* 3 * V*fi V*pi V*fi V*pi *tץ 12 ץ ץ 1 V*fz ͑r*V* ͒ϩ ϭ0, ͑1a͒ z* for iϭr,␪,z, ͑3͒ץ r* frץ *r ␮ 41 ץ 2 ץ ץ V*fr V*fr V*f ␪ V*fr where is the dynamic viscosity of the carrier fluid. The ϩV* Ϫ ϩV* z* drag force acting on the particles due to the fluid is equal inץ r* r* fzץ t* frץ magnitude but has opposite sense to the force acting on the p* nF* fluid due to the particles, thus coupling the fluid phase andץ 2V* 1ץ ץ 1 ץ ϭ␯ͫ ͩ ͑ * ͒ͪ ϩ frͬϪ Ϫ r .␳ , particle phase equations ץ ␳ 2 ץ r*V fr ץ ץ r* r* r* z* f r* f The equations are nondimensionalized by using the fol- ͑ ͒ 1b lowing scheme: ץ ץ ץ V*f ␪ V*f ␪ V*frV*f ␪ V*f ␪ ␾ ϩV* ϩ ϩV* r* z* * , z* rϭ , zϭ , ␾ϭץ r* r* fzץ t* frץ d d d 2ץ ץ ץ 1 V*f ␪ nF␪* ϭ␯ͫ ͩ ͑ * ͒ͪ ϩ ͬϪ ͑ ͒ V V ␪ V 1 ͒ ͑ ͒ ␳ , 1c jr ϭ j ϭ jz ϭ ͑ ϭ 2 ץ r*V f ␪ ץ ץ r* r* r* z* f ⍀ where j f ,p , 4 V*jr V*j␪ V*jz 1r1 2ץ ץ ץ ץ ץ ץ V*fz V*fz V*fz 1 V*fz V*fz ϩV* ϩV* ϭ␯ͫ ͩ r* ͪ ϩ ͬ p* t* z*2 ϭ ϭץ *rץ *rץ *z* rץ r* fzץ t* frץ p ␳ ⍀2 2 , t ␶ .