Hydrodynamic Stability of a Suspension in Taylor Couette Flow

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 particles 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–

͒ ␣͑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 ␶ . f 1r1 *p* nFץ 1 Ϫ Ϫ Ϫ z ͑ ͒ g ␳ . 1d Here, the characteristic length scale is the gap width between ץ ␳ f z* f ϭ Ϫ the two cylinders d r2 r1 , where r1 and r2 are the inner In these equations, the asterisk indicates dimensional vari- and outer radii of the cylinders, respectively. The velocities ␪ ables. (V*fr ,V*f ␪ ,V*fz) are the fluid velocities in the r*, *, are nondimensionalized using the characteristic velocity of ⍀ ⍀ and z* directions, respectively, t* represents time, p* is the rotation 1r1 where 1 is the rotational speed of the inner ␳ ␶ϭ 2 ␯ pressure, f is the fluid density, n is the number density of cylinder. d / is the characteristic viscous time scale of the particles, and F* is defined shortly. The equations for the the problem. Upon nondimensionalizing the continuity and particulate phase take on a similar form except that the vis- Navier–Stokes equations the parameters that appear are the ⑀ϭ␳ ␳ cous terms are absent. The dimensional form of the continu- density ratio, p / f , and the Taylor number, Ta ϭ⍀ ␯ ␯ ity and Navier–Stokes equations for the particulate phase are 1r1d/ , where is the kinematic viscosity of the carrier

Downloaded 14 Mar 2002 to 129.105.69.150. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp 1238 Phys. Fluids, Vol. 14, No. 3, March 2002 Ali et al.

fluid. This form of the Taylor number, often called the rotat- 18A A␴ ¯V ing Reynolds number, is used because it is simple and con- ͩ DD Ϫk2Ϫ␴Ϫͩ ϩ ͪͪu͑r͒ϩ2Ta v͑r͒ * ␾2 2 r sistent with the form used in other studies.42 The nondimensional equations are similar to those used 18A A␴ ϭTa͑D␻͑r͒͒Ϫͩ ϩ ͪ u ͑r͒, ͑7b͒ by Dimas and Kiger to analyze the linear stability of a ␾2 2 p particle-laden mixing layer.40 Implicit in this derivation are the following assumptions: ͑1͒ The scale of the particle mo- 18A A␴ ͩ DD Ϫk2Ϫ␴Ϫͩ ϩ ͪͪv͑r͒ϪTa͑D ¯V͒u͑r͒ tion relative to the fluid motion is very small. ͑2͒ The particle * ␾2 2 * Reynolds number is always less than unity, so that the drag ␴ force on the particle is described by Stokes law. ͑3͒ The 18A A ϭϪͩ ϩ ͪ v ͑r͒, ͑7c͒ particles are spherical and rigid. ͑4͒ The suspension is suffi- ␾2 2 p ciently dilute to prevent hydrodynamic interactions between 1 18A A␴ particles. ͑5͒ The Faxen correction to the Stokes drag is neg- ͩ Ϫ 2Ϫ␴ϩ Ϫͩ ϩ ͪͪ ͑ ͒ DD k 2 2 w r ligible. ͑6͒ The Bassett history force is negligible in compari- * r ␾ 2 son to the Stokes drag. 18A A␴ A linear stability analysis is performed by separating the ϭ ␻͑ ͒Ϫͩ ϩ ͪ ͑ ͒ ͑ ͒ ik Ta r ␾2 w p r , 7d variables into mean and perturbation components such that 2 where the differential operators D and D are defined as ϭ ͒ ϭ¯ ͒ϩ ͒ * V fr uЈ͑r,z,t , V f ␪ V͑r vЈ͑r,z,t , d d 1 ͑ ͒ϭ ͑ ͒ ͑ ͒ϭ ͑ ͒ϩ ͑ ͒ ϭ ͒ D , D . V fz wЈ͑r,z,t , dr * dr r For the case of zero concentration (Aϭ0), these equations V ϭuЈ͑r,z,t͒, V ␪ϭ¯V͑r͒ϩvЈ͑r,z,t͒, ͑5͒ pr p p p reduce to those for Taylor–Couette flow of a simple fluid.2 ϭ Ј͑ ͒ A similar treatment of the particulate phase equations V pz w p r,z,t , results in expressions for u p(r), v p(r), and w p(r) in terms ϭ ϩ Ј͑ ͒ ␣ ϭ ϩ Ј͑ ͒ of the fluid velocity perturbations, p P p r,z,t , p A a p r,z,t . ¯ 2TaVP0 Here, the primed ͑Ј͒ variables are the perturbation compo- u ͑r͒ϭͭ ͮ v͑r͒ p r͑͑␴ϩP ͒2ϩ2Ta2͑͑¯V/r͒D ¯V͒͒ nents, A is the concentration of particles in the undisturbed 0 * ¯ ϭ ␴ϩ ͒ state, and the stable velocity profile is given by V(r) C1r P0͑ P0 ϩ ϩͭ ͮ u͑r͒ C2 /r, where the constants C1 and C2 are functions of the ͑͑␴ϩ ͒2ϩ 2͑͑¯ ͒ ¯ ͒͒ ␩ϭ P0 2Ta V/r D V radius ratio, r1 /r2 . * ␴ϩ ͒ Next, the perturbations are expressed as normal modes Ta͑ P0 Ϫͭ ͮ D␻͑r͒, ͑8a͒ of the form ⑀͑͑␴ϩP ͒2ϩ2Ta2͑͑¯V/r͒D ¯V͒͒ 0 * uЈ vЈ wЈ ͑ikzϩ␴t͒ Ta P D ¯V ϭ ϭ ϭe , ͑ ͒ϭϪͭ 0 * ͮ ͑ ͒ u͑r͒ v͑r͒ w͑r͒ v p r u r ͑͑␴ϩP ͒2ϩ2Ta2͑͑¯V/r͒D ¯V͒͒ 0 * Ј Ј Ј ͑␴ϩ ͒ u p v p w p ϩ␴ P0 P0 ϭ ϭ ϭe͑ikz t͒, ͑6͒ ϩͭ ͮ v͑r͒ ͒ ͒ ͒ 2 2 u p͑r v p͑r w p͑r ͑͑␴ϩP ͒ ϩ2Ta ͑͑¯V/r͒D ¯V͒͒ 0 * Ј Ta2 D ¯V pЈ a p ϭe͑ikzϩ␴t͒, ϭe͑ikzϩ␴t͒, ϩͭ * ͮ D␻͑r͒, ͑8b͒ ␻͑ ͒ ͑ ͒ ⑀͑͑␴ϩP ͒2ϩ2Ta2͑͑¯V/r͒D ¯V͒͒ r a p r 0 * ␻ P ik Ta where u(r), v(r), w(r), u p(r), v p(r), w p(r), (r), and ͑ ͒ϭ 0 ͑ ͒Ϫ ␻͑ ͒ ͑ ͒ w p r ␴ϩ ͒ w r ⑀ ␴ϩ ͒ r , 8c a p(r) are the amplitudes of the corresponding disturbances, k ͑ P0 ͑ P0 ␴ϭ␴ is the axial wave number of the disturbance, and r ϩ ␴ where i i is an amplification factor. The wave number and the amplification factor are nondimesionalized as kϭk*d and 18 ␴ ␴ϭ␴ ␶ P ϭ ϩ . * . 0 ␾2⑀ 2⑀ Equations ͑5͒ and ͑6͒ are then substituted into the nondimensionalized governing equations and the stable flow The particle velocity components ͑8͒ are then substituted terms subtracted out. Linearization of the equations by dis- into ͑7͒ and the equations simplified to a set of 12 nonlinear carding higher order terms results in the final form of the first-order ordinary differential equations. This system of disturbance equations, which for the fluid phase can be writ- equations was solved using the boundary-value problem soft- 43 ten as ware package SUPORT in combination with the nonlinear equation solver SNSQE.44,45 Computations were performed in D u͑r͒ϭϪikw͑r͒, ͑7a͒ double precision. Extensive code testing of the SUPORT pack- *

FIG. 1. Sample of the stability curves for several radius ratios ͑Aϭ0.05, ⑀ϭ1, ␾ϭ0.004͒. FIG. 2. Critical Taylor number for several radius ratios as a function of particle concentration A for neutrally buoyant particles ͑⑀ϭ1, ␾ϭ0.004͒. ͑*͒ ␩ϭ0.45, ͑᭹͒ ␩ϭ0.65, ͑᭡͒ ␩ϭ0.75, ͑᭿͒ ␩ϭ0.85, ͑ࡗ͒ ␩ϭ0.95. age with the SNSQE solver has been previously reported.46–49 The eigenvalue problem may be written in the implicit func- tional form the validity of our procedure. The critical Taylor number and wave number for Taylor–Couette flow of a simple fluid F͑Ta, A,␾,k,⑀,␩,␴͒ϭ0. ͑9͒ are also recovered for ⑀ approaching zero, corresponding to The parameters A, ␾, k, ⑀, and ␩ are usually fixed and solu- inertialess particles, further confirming the validity of our tion of the ordinary differential equations is obtained by it- numerics. ␴ eration on the eigenvalue pair (Ta, i) using the procedure The range of parameters that was considered is consis- outlined in Ali and Weidman.48,49 Since we seek to find the tent with physically realizable systems. Particle density ra- ␴ neutral stability conditions, r is set to zero. At fixed radius tios were used that are consistent with gas bubbles in a liquid ratio ␩, a search is conducted over all wave numbers k to find ͑⑀ϭ0.001͒, neutrally buoyant particles ͑⑀ϭ1͒, heavy particles the minimum Taylor number, denoted as Tac . A sample of in a liquid ͑⑀ϭ10͒, and particles about the density of water in neutral stability curves for ␩ϭ0.45, 0.65, 0.75, and 0.85 are a gas ͑⑀ϭ833͒. In the case of gas bubbles, we do not account given in Fig. 1 for Aϭ0.05, ⑀ϭ1, and ␾ϭ0.004. For specific for the deformability of the bubbles or the deviation from ⑀ ␾ ␩ values of A, , , and , critical conditions Tac , kc , and Stokes drag due to a fluid disperse phase. Particle radius to ␴ ( i)c are those for which the Taylor number is a minimum. gap widths ranged from ␾ϭ0.0002 ͑the gap corresponds to ϭ For the example in Fig. 1, Tac 63.04, 69.75, 79.82, 100.79 5000 particle diameters͒ to ␾ϭ0.02 ͑the gap corresponds to in an ascending order of ␩. In the vicinity of the minimum, 50 particle diameters͒. Assuming that the velocity difference the increments in ⌬k were taken to be 0.001. The wave between the particle and fluid is at least one order of magni- number of the neutral curves in Fig. 1 are 3.172, 3.143, tude smaller than the surface velocity of the inner cylinder 3.135, 3.130 in ascending order of ␩. ͑probably an overestimate͒, the particle Reynolds number is To assure the validity of our procedure we compared our always less than 0.4 justifying the use of the Stokes drag in results for a simple fluid by assuming zero concentration the formulation. Furthermore, a simple order of magnitude in the analysis (Aϭ0) to previously published results.50 analysis of the ratio of the Bassett history force to the Stokes The similarity of the values for the critical Taylor numbers drag goes like ␾, indicating that it is reasonable to neglect Tac and critical wave numbers kc , shown in Table I, confirm the particle history. Particle concentrations up to Aϭ0.05 were considered, noting that our assumptions requiring a dilute concentration make our results most dependable at lower TABLE I. Comparison of critical Taylor number and wave number obtained concentrations than this. Radius ratios from ␩ϭ0.45 to for pure Taylor–Couette flow from Rectenwald et al. ͑Ref. 50͒ with the ␩ϭ present study. 0.99 were considered.

␩ Ta ͑Ref. 50͒ ͑ ͒ ͑ ͒ k ͑present study͒ c Tac present study kc Ref. 50 c III. RESULTS AND DISCUSSION 0.400 68.2963 68.2965 3.1835 3.183 0.500 68.1860 68.1863 3.1625 3.162 The critical Taylor number for transition from stable cy- 0.600 71.7154 71.7157 3.1483 3.148 lindrical Couette flow to supercritical Taylor vortex flow for 0.700 79.4903 79.4907 3.1389 3.139 a suspension of neutrally buoyant particles is shown in Fig. 2 0.750 85.7760 85.7765 3.1354 3.135 as a function of the particle concentration for radius ratios 0.800 94.7331 94.7336 3.1326 3.133 ranging from ␩ϭ0.45 to ␩ϭ0.95. As the particle concentra- 0.900 131.6139 131.6145 3.1288 3.129 0.975 260.9483 260.9499 3.1270 3.127 tion increases, the flow becomes less stable. For all conditions studied the critical Taylor number decreases by about

Downloaded 14 Mar 2002 to 129.105.69.150. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp 1240 Phys. Fluids, Vol. 14, No. 3, March 2002 Ali et al.

TABLE II. Critical wave numbers at different concentrations in the range 0рAр0.05 for several radius ratios.

͑␩ϭ ͒ ͑␩ϭ ͒ ͑␩ϭ ͒ ͑␩ϭ ͒ ͑␩ϭ ͒ kc 0.95 kc 0.85 kc 0.75 kc 0.65 kc 0.45 3.127 3.130 3.135 3.143 3.172

7% as the particle concentration increases from zero to the maximum concentration. This result for spherical particles is different from theoretical predictions for rigid fibers.30 In that case, the fibers stabilize the flow. The differing results could be a consequence of the different shape of the particles or the different analytical techniques ͑two-fluid model versus non- Newtonian rheological coefficients͒. The critical wave number is not altered by the concentration of particles. Table II provides the wave number for FIG. 3. The effect of density ratio ⑀ on critical Taylor number for several neutrally buoyant particles for concentrations ranging from ͑ ͒ ⑀ϭ ͑ ͒ ⑀ϭ ͑ ͒ ⑀ϭ ͑ ͒ ⑀ϭ ͑᭡͒ ϭ ϭ radius ratios — • — 0.001, — 1, ––– 10, --- 833; A 0toA 0.05. This result is independent of density ratio ␩ϭ0.75, ͑᭿͒ ␩ϭ0.85, ͑ࡗ͒ ␩ϭ0.95. and particle size ratio. The effect of the particle density on the critical Taylor number is shown in Fig. 3 for three radius ratios. It is appar- lor number based on the bulk effective density and viscosity ent that the more dense particles have a much greater effect of the suspension. The effective density is readily calculated on the stability of the flow than less dense particles. As the based on the particle concentration and the density ratio. The particle concentration increases from 0 to 5%, the decrease in effective density can be as much as 42 times the fluid density the critical Taylor number is negligible for ⑀ϭ0.001, but as for dense particles at the highest concentration considered. large as 35% for ⑀ϭ10. The destabilizing effect is even more The effective viscosity is more difficult to specify. Several striking for particles in a gas ͑⑀ϭ833͒, where the critical correlations, both theoretical and empirical, exist to calculate Taylor number decreases to only a small fraction of its value the effective viscosity of a suspension.39 For simplicity, we ␮ ϭ␮ ϩ as the particle concentration increases. use the Einstein formulation such that eff (1 2.5A). The greater effect of heavy particles on the stability may Thus, the effective viscosity can be as much as 12.5% greater be attributed to their inertia, which results in an increased than the fluid viscosity for the concentrations that we consid- degree of coupling between the fluid and particle phases. ered. Figure 4 indicates the dependence of the effective criti- This effect is most evident if considered in terms of the cal Taylor number, based on the effective bulk density and Stokes number, defined as viscosity, on the concentration of particles for density ratios ⑀ϭ ⑀ϭ ␳ ␾ 2 ⍀ from 0.001 to 833 and the three radius ratios that were p * r1 1 Stϭ . shown in Fig. 3. Using the effective critical Taylor number 18␮d The Stokes number represents the ratio of the viscous relax- ␳ ␾ 2 ␮ ation time for the particle ( p * /18 ) to the time scale of ⍀ the flow (d/r1 1). It indicates the ability of the particles to respond to the motion of the carrier fluid.40,51 For small values of the Stokes number, particles accurately follow the carrier fluid, whereas for large values the particles do not closely follow the fluid motion because of their inertia. The Stokes number for ⑀ϭ833 ranges from 10Ϫ2 to 10Ϫ1 depending on concentration and radius ratio, whereas for the lower density ratios, the Stokes number is one to six orders of magnitude less. Thus, for the smaller density ratios, the inertia of the particles is so small that the particles have little effect on the stability. For the largest density ratio, the particles interact more strongly with the flow and reduce the stability. This naturally poses the possibility of accounting for the effect of the density ratio in the form of the critical Taylor number. The Taylor numbers in Figs. 2 and 3 are based on the fluid properties ͑density and viscosity͒. However, the FIG. 4. The effect of particle concentration on the effective critical Taylor number. The effective Taylor number is based on the density and viscosity presence of particles alters the effective density and viscosity ͑ ͒ ⑀ϭ ͑ ͒ ͑ ͒ ⑀ϭ of the suspension -•- 0.001 nearly hidden by the solid line , — 1, of the suspension. This suggests the use of an effective Tay- ͑–––͒ ⑀ϭ10, ͑---͒ ⑀ϭ833.

FIG. 5. Critical Taylor number as a function of density ratio for several FIG. 6. Experimental results obtained using neutrally buoyant nylon par- radius ratios ͑Aϭ0.005, ␾ϭ0.004͒; ͑᭹͒ ␩ϭ0.65, ͑᭡͒ ␩ϭ0.75, ͑᭿͒ ␩ϭ0.85, ticles ͑␩ϭ0.824, ␾ϭ0.0038͒. ͑ࡗ͒ ␩ϭ0.95.

ϭ nearly collapses the data for this wide range of density ratios quantity (Aflakes 0.000 05) of silicon dioxide coated mica at all three radius ratios. Apparently, the effective suspension reflective flakes ͑Flamenco Superpearl, Mearl Corp.͒ were density, which plays a much greater role than the effective added to the suspension to make the vortices visible. The viscosity in the effective Taylor number, accounts for the concentration of the flakes was a factor of 20–100 times less degree to which particles follow the fluid flow and thereby than the nylon particle concentration. The temperature in the affect the stability. annulus was monitored during the experiments to permit The effect of the density ratio ⑀ on the critical Taylor temperature correction of the density and viscosity. number is shown in Fig. 5 for a single concentration of A The Taylor number was increased from rest in a stepwise ϭ0.005 and several radius ratios. The density ratio has little fashion using small increments ͑⌬TaϷ1 near the theoretical effect on the critical Taylor number until it is quite large. transition point͒. The system was allowed to equilibrate for ͑Values for 10Ͻ⑀Ͻ833 were not calculated, since they do at least 10 min after each change in Ta. Transition was said to not represent physically realizable systems.͒ occur at the minimum Taylor number where vortices were Finally, we calculated the critical Taylor number for par- observed to fill the entire annulus. ͑Here we refer to the fluid ticle sizes of 0.0002р␾р0.02 over the range of concentra- Taylor number, not the effective Taylor number, though there tions and density ratios that were considered. The critical is little difference between them for ⑀ϭ1.͒ Precursor vortices Taylor number and critical wave number were independent appeared first near the end caps and then in various parts of of ␾. the annulus slightly below the critical Taylor number, consistent with previous results for simple fluids.52 The Taylor number was increased until the vortices filled the annulus IV. EXPERIMENTAL RESULTS FOR A NEUTRALLY and then decreased until they were visible only in parts of the BUOYANT SUSPENSION annulus. In this way, we could bracket the transition from We performed a limited number of experiments for sus- stable to Taylor–Couette flow to be 98.6ϽTaϽ100.3 for A ϭ ϭ pensions of neutrally buoyant particles to determine how 0. The theoretical value is Tac 100.73 for the radius ratio well the linear stability analysis captures the physics of the of the setup.50 Although the wavelength of the vortices was problem. The experiments were conducted using a standard not measured precisely, it was clearly quite near the expected Couette cell with a fixed outer cylinder, stepper motor driven value for square vortical cells. When observing transition for ϭ inner cylinder, and fixed end caps such that 2r1 4.97 nonzero particle concentrations, a similar method to bracket Ϯ ϭ Ϯ 0.013 cm, 2r2 6.03 0.013 cm. The radius ratio was the transition was used to determine the critical Taylor num- ␩ϭ0.824 and the aspect ratio was ⌫ϭH/dϭ96, where H ber. The critical Taylor number for transition from Taylor ϭ50 cm, is the height of the column. The concentration of vortex flow to wavy vortex flow was determined in a similar the aqueous glycerol solution was matched to the approxi- way. Transition to wavy vortex flow was said to occur when mate density of the nylon particles ͑␳Х1.1 g/cm3͒.Itwas wavy vortices filled the entire annulus. For zero particle con- difficult to exactly match the density of all particles, due to centration the wavy transition could be bracketed as 1.24 Ϯ ␮ ͑ Ͻ Ͻ variability in the particles. The 20 5 m particles Goodfel- Ta/Tac 1.25. Previous experiments show significant vari- low LS194265, Cambridge, UK͒ were used in volume con- ability in the transition depending on the aspect ratio and р р 53 centrations of 0 A 0.005. Surfactant in the form of deter- experimental conditions. The range is 1.15 to 1.28 Tac for gent was added in very small amounts ͑Х1.5 ml/1000 ml of radius ratios near that in our experiments. suspension͒ to assure suspension of the particles. A small The effect of concentration on transition is shown in Fig.

Downloaded 14 Mar 2002 to 129.105.69.150. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp 1242 Phys. Fluids, Vol. 14, No. 3, March 2002 Ali et al.

6 for Aр0.005, concentrations that are much lower than the was altered from that used in Eq. ͑3͒. ͑5͒ The linear stability concentrations in previous figures. Clearly, the presence of analysis necessarily neglects nonlinear terms. However, it neutrally buoyant particles stabilizes the flow for both the seems unlikely that a subcritical instability related to these transition from stable flow to Taylor vortex flow and from terms would occur altering the theoretical results. ͑6͒ The Taylor vortex flow to wavy vortex flow. Obviously, the ex- Bassett history force acting on the particles is assumed to be perimental results differ from the linear stability analysis. negligible based on an order of magnitude analysis. How- There may be a maximum in the data for Aϭ0.004 for both ever, a recent study has suggested that the motion of a par- transitions with a slight downward trend for greater concen- ticle itself may be unstable when the Bassett history force is trations that might be consistent with the theoretical predic- included.54 The way in which this would impact the two- tion. Unfortunately, the transition became more difficult to fluid model used here is unclear. ͑7͒ The linear stability detect visually as the concentration increased, so that it was analysis assumes that the stable concentration of particles is impossible to detect for AϾ0.005. Thus, it is not clear if the uniform ͓A in Eq. ͑5͒ is independent of r, z, and t͔. Given the maximum vaguely evident in Fig. 6 really occurs or is a uniform shear in stable Couette flow, it seems unlikely that result of the difficulty in identifying the transition to vortical shear-induced diffusion would result in a concentration gra- flow at high particle concentrations. We further note that the dient, but clearly the wall-exclusion effect results in a non- experimental results displayed a consistent stabilizing trend uniform concentration profile near the walls of the annulus with increasing concentration even for very low concentra- that was not included in the model. In addition, once vortices tions (Aϭ0.001). occur the concentration of particles is unlikely to be uniform. We are confident that our experiments were repeatable It is likely that one of the two latter points, the omission of and robust. While it was challenging to determine the exact history term or the assumption of a uniform concentration of Taylor number for transition by visualizing the flow, the sta- particles in the theory, are the most likely causes of the bilizing effect was quite clear. Our visual observation of the theory not capturing the physics to properly match the ex- first transition coinciding with the predicted theoretical value periments. for a fluid with no nylon particles gives us further confidence in our method. Furthermore, our visual observation of the V. SUMMARY transition to wavy vortices with no nylon particles is consistent with previous measurements. We note that it was quite The results of the theoretical analysis of the stability of a difficult to match the fluid density to the particle density ͑a suspension in cylindrical Couette flow indicate that the flow small fraction of the particles always sank while a similar is destabilized by the presence of a dispersed species. The small fraction floated͒. However, the bulk of the particles degree of destabilization depends on the density ratio be- were neutrally buoyant, so we doubt that density differences tween the dispersed phase and the continuous phase. More affected the results. dense particles result in more of a destabilizing effect. Most There are some differences between the experiments and likely more dense particles ͑such as solid particles in a gas͒ the stability analysis that could be speculated as reasons for are less likely to follow the fluid motion, based on the par- the differing analytical and experimental results. ͑1͒ Al- ticle Stokes number. This results in a stronger interaction though our experimental setup had a reasonably large aspect between the disperse and continuous phases through the ratio, end effects, which are not included in the theory, could Stokes drag term ͑since the velocity difference is greater͒.It play a role. However, this seems unlikely given that end is quite logical to argue that this disrupts the stability more effects do not alter the critical Taylor number for particle- easily than neutrally buoyant or very light particles. The ef- free flows except when the aspect ratio is quite small. ͑2͒ The fect of the density ratio can be readily taken into account, at model assumes monodisperse particles, whereas a reasonably least to a first approximation, by converting the calculated narrow distribution of polydisperse particles were used in the critical Taylor number, which is based on the fluid properties, experiments. However, the negligible dependence of the to an effective Taylor number based on the bulk suspension model results on particle size suggests that this is not a prob- density and viscosity. This permits the estimation of a critical lem. ͑3͒ The model assumes spherical particles, whereas the Taylor number that is nearly independent of the density ratio. experiments included flakes ͑in small concentration͒ for vi- Nevertheless, our preliminary experiments with neutrally sualization and used particles that were not exactly spherical. buoyant particles indicate a stabilizing effect of particles. Since similar flakes have been used by many researchers to This could be argued as a logical result given that the par- visualize transition to vortical flow without adverse affects, ticles increase the effective viscosity slightly and thereby re- we can only assume that the small concentration of flakes is quire a larger Taylor number based on fluid properties to inconsequential. ͑4͒ It is possible that the detergent added to have the effective Taylor number reach the critical value. aid in the dispersion of particles affected the stability of the However, this effect should be quite small for the experimen- flow. However, the amount of detergent was so small that the tal particle concentrations. viscosity and density of the fluid were unaffected. Further- The original motivation for this work was the flow of more, the experiment at zero particle concentration using the suspensions in Taylor–Couette reactor cells or rotating fil- water-glycerol-detergent solution matched the theoretical ters. In these devices, the base flow is substantially more prediction quite well indicating no fluid non-Newtonian ef- complex due to the necessary axial or radial flows to carry fects. The only possibility is that the detergent altered the the process fluid into and out of the cell. The results pre- interaction between phases so that the interfacial force term sented here provide some insight with regard to these appli-

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 1243 cations. In particular, our results suggest that an estimate of 23H. B. Winzeler and G. Belfort, ‘‘Enhanced performance for pressure- the conditions for the flow to become unstable may be based driven membrane processes: The argument for fluid instabilities,’’ J. ͑ ͒ on the effective Taylor number. However, the differences be- Membr. Sci. 80,35 1993 . 24R. M. Lueptow and A. Hajiloo, ‘‘Flow in a rotating membrane plasma tween the experimental and theoretical results prevent any separator,’’ ASAIO J. 41, 182 ͑1995͒. conclusions with regard to the stabilizing or destabilizing 25J. R. Hildebrandt and J. B. 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