(b) (a) Newtonian. Shear thinning α α TIME

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1 ABOUT COMPUTATIONS OF HELE-SHAW FLOW OF NON-NEWTONIAN FLUIDS L. KONDIC*, P. FAST**, and M. J. SHELLEY** *Departments of Mathematics and Physics, Duke University, Durham, NC 27708, **Courant Institute, New York University, New York, NY 002 ABSTRACT The æow of a æuid conæned between two solid plates èhele-shaw cellè is of considerable interest in a variety of applications. Further interest in two phase æow in this geometry stems from the close analogy between the dynamics of æuid-æuid interface and the propagation of the solidiæcation front. While the æow of Newtonian æuids is rather well understood, it is much more complicated to compute æows of non-newtonian æuids. We ænd that the dense-branching morphology of Newtonian liquids may be replaced bydendriticængers with stable tips and sidebranches, and discuss resulting length scales. INTRODUCTION Flows of complex, non-newtonian æuids are of considerable technological importance. In particular conæned thin-gap æows of non-newtonian æuids are relevant to industrial processes such as injection molding or display device design ëë. A two-phase æow in this setting is a scientiæcally important one, given the close analogy between the Saæman-Taylor instability of driven Newtonian æuid with quasi-static solidiæcation èand the Mullins-Sekerka instability ë2ëè, electrochemical deposition ë3ë, and many other physical problems. In this work, we concentrate on the interfacial dynamics of a gas bubble expanding into æuid in a radial Hele-Shaw cell èsee Fig. è. When the æuid is Newtonian, a dense branching pattern morphology is observed ë4ë, as the outcome of the nonlinear development of the Saæman-Taylor instability. It is characterized by tip-splitting of the interface and the formation of branched structures. On the other hand, experiments performed with complex liquids such as liquid crystals ë5ë, polymer solutions and melts ë4ë, clays ë6ë, and foams ë7ë, have shown that the structures reminiscent of solidiæcation ones - dendritic ængers, side branching, can be induced in these æuids. One property shared by these diæerent liquids is that they are shear thinning, and we will concentrate on this property. Elastic response of the æuid can also be an important eæect, though we will not consider it here. We start from the generalized Darcy's law governing the bulk æuid æow ë8ë. Combined with appropriate boundary conditions, this yields a nonlinear, elliptic boundary value problem èbvpè for the pressure in the driven æuid. Fully nonlinear, time dependent simulations of a bubble growing into a shear thinning æuid show that shear thinning inæuences considerably the evolution of the interface, and in agreement with experiments, can lead to the formation of ængers which do not split, and can resemble the dendritic structures observed in solidiæcation. In this paper we give an outline of our methods, combined with new results related to experimentally measurable quantities, such as observed length scales and the ænger velocity. The reader is referred to ë8, 9ë for the formulation of a Darcy's law, and more detailed explanation of the role which shear thinning plays in pattern formation, and to ë0ë for the relation of the model employed here to more general viscoelastic æuid models.

2 Fluid Gas L b Figure : Hele-Shaw cell THEORY Darcy's law for a non-newtonian æuid We use the æuid model in which non-newtonian character of æuid enters through rateof-strain dependence of the æuid viscosity ç Dv Dt =,rp + ræç çèjsj 2 ès ç ; ræv =0; èè where S is the rate-of-strain tensor, and jsj 2 = très 2 è = P ij Sij. 2 The viscosity is taken to be given by çèjsj 2 è = ç 0 è + æç 2 jsj 2 è=è + ç 2 jsj 2 è. Here ç is the relaxation time of the æuid, ç 0 its zero shear rate viscosity, and æ measures shear dependence: æ = gives Newtonian response, æé gives shear thickening, and æé shear thinning. In practice, most non-newtonian æuids are shear thinning, and we concentrate on this case. The æow in a Hele-Shaw geometry is signiæcantly simpliæed by the small aspect ratio æ = b=l ç. The Reynolds number is small, so that inertial terms could be neglected, and the velocity gradients in the short, z, direction are much larger than the lateral ones. Keeping only the terms of Oèæè, and averaging over the gap width, one obtains a generalized Darcy's law applicable to ashear thinning æuid èthe choice of appropriate sales is discussed in ë8, 9ëè u =, rp ; and ræu =0; è2è 2 ç æ èwe 2 jrpj 2 è where r is the lateral gradient operator, and u is gap averaged æuid velocity. Weissenberg number, We = ç=ç flow,measures the ratio of the time scale of the æuid, ç, andthe relevant time scale imposed by æow, ç flow. The viscosity ç æ èwe 2 jrpj 2 è is constructed from ç and shares its monotonicity properties, i.e., if the æuid is shear thinning then ç æ èwe 2 jrpj 2 è decreases monotonically with increasing argument ë8ë. Expanding bubble problem Now we apply the resulting equations to the problem of an air bubble expanding into a non-newtonian æuid. Additional nondimensional parameter, capillary number ècaè, which measures the ratio of capillary to viscous forces, enters through imposed Laplace-Young boundary conditions. To facilitate comparison with experiments èsee ë4ë and references

3 thereinè, we deæne Ca and We Ca = 2ç _ 0 R 0 R 2 0 ; and We = ç R _ 0 æb 2 b ; è3è where it is assumed that all the quantities can be varied independently. The lateral length scale is chosen as the initial radius of the bubble, R 0, and the velocity scale is the initial bubble velocity, _R 0. At the boundaries, the pressure jump ëpë is given by ëpë =Ca, ç for the pressure jump ëpë, where ç is the lateral curvature. A nonlinear BVP for the pressure follows ë9ë ræ è rp ç æ èwe 2 jrpj 2 è! =0; pj,i =, ç i Ca ; pj, e = ç e Ca ; where, i;e stand for the internal and external æuid boundaries. The motion of the interfaces follows from the requirement that they move with local æuid velocities. Numerical methods The full evolution problem, Eq. è4è, is much harder to solve than the corresponding problem for a Newtonian æuid, where the pressure is harmonic. In the non-newtonian case, p satisæes the nonlinear BVP, Eq. è4è, and must be solved for in the whole domain. Since the problem is driven by the curvature of the boundaries, high spatial resolution is required. To solve for the pressure, we use a Lagrangian grid which conforms to the interfaces and moves with the æuid. Ones the pressure is obtained, velocity of the æuid èand of the boundariesè is obtained using Darcy's law, Eq. è2è. As initial data, we take the interior interface, i as a circle perturbed with a single azimuthal mode m = 4, and the outer boundary, e as a circle. For eæciency, we impose a four-fold symmetry on the initial bubble shape, and the solution. More details about computational methods are given in ë9, 0ë. RESULTS The eæect of shear thinning on the dynamics of the interface Figure 2 shows thesimulation of an expandingbubble for Newtonian and non-newtonian æuid, plotted at equal time intervals. In Fig. 2a we observe typical pattern formation for a Newtonian æuid, where the unstable mode is growing into a petal, which widens, and then splits into two as its radius of curvature increases, in agreement with linear theory ëë. The bubble evolution in a strongly shear thinning æuid is strikingly diæerent, as is illustrated in Fig. 2b. The eæect of shear thinningisto suppress the tip splittingofthe outwardly growing petal. As the petal expands outwards, it appears to near a splitting, but then ërefocuses", leaving behind ëside-branches", and continues to grow outwards. In ë9ë it is shown that the lowest viscosity appears at the ends of the petals, and then increases sharply as one moves away from the tips, and is highest within the fjords, where it is nearly unity èthe ëzero shear" viscosity is normalized to oneè. It is this phenomena that results in the narrowed petals: shear thinning eæect increases the local æuid velocity at the tips. This eæect is limited by capillarity, which seeks to lower the length to area ratio, and which is also likely related to the production of ëside branches" left behind the advancing tip. There are three ènondimensionalè parameters which determine the dynamics of the interface: Ca, which measures the relative strength of surface tension and viscous forces, and è4è

4 Newtonian (a) Shear thinning (b) Figure 2: The snap-shots of the evolving bubble interface for èaè Newtonian æuid and èbè strongly shear thinning æuid èca = 480 for both simulations, æ = 0:5, W e = 0:5 for shear thinning oneè. TIP VELOCITY α α α =.0, Ca = 240 = 0.5, We = 0.5, Ca = 240 = 0.5, We = 0.5, Ca = TIME Figure 3: The velocity of tip propagation. Solid lines show tips which split; broken line shows non tip-splittingænger. The arrows show the point where curvature of the tip changes sign. We and æ, which determine shear thinning properties of the æuid. Flows characterized by some combination of these parameters lead to the formation of ængers whose tips do not split èas shown in Fig. 2bè; diæerent parameters might lead to production of tip splitting petals, resembling the patterns characteristic for a Newtonian æuid. Detailed discussion of the inæuence which shear thinning character of the æuid has on the pattern formation, as well as detailed parametric dependence of the observed patterns, are given in ë9, 0ë. Another eæect of shear thinning is to modify the velocity of advancing ængersèpetals. Figure 3 shows the tip velocity for a Newtonian æuid, and for two choices of parameters characterizing shear thinning æuids; the latter two diæer only in Ca. The choice Ca = 240 leads to formation of ængers which do not split, and is characterized by the tip velocity which isapproximately constant. The other choice, Ca = 600, leads to atip splitting petal, whose velocity is continuously decreasing. This eæect has been noted in the theoretical work ë5ë, where the curvature of the ænger tip was held constant artiæcially.

5 Length Scale simulation fit linear stability b/b 0 Length Scale simulation fit linear stability δp/ δp c Figure 4: The dependence of the length-scale l on plate separation b èaè, and driving pressure æp èbè. Here b 0 èaè and æp c èbè are the plate separation and the driving pressure which give Ca = 240 and We = 0:5. Linear stability results èdashedè ë9, 0ë, simulation results èdotsè and æts l = kb èaè, and l ç k 0 èæp=æp c è,=2 èbè èsolidè are shown. The constants k; k 0 are determined from the data points b = b 0 èaè, and æp = æp c èbè èæ =0:5è. Emerging length-scales A typical length-scale èlè of the patterns which develop in a radial Hele-Shaw æow for Newtonian æuids is determined by a single parameter, the capillary number Ca. For large Ca, linear stability suggests that a length-scales associated with the initial growth of the patterns is given by ç m ç 2çR q 3=Ca ëë, where ç m is the wavelength of the most unstable mode, and R is the time-dependent radius of the expanding bubble. We look into our simulation results for a similar length scaling inshear thinning æuids. In experiments ë2ë, emerging length-scales have been measured as the gap width b is varied. These results suggest that the length-scale scales roughly linearly with b. For Newtonian æuids this observation conærms the result of linear stability, since Ca ç =b 2 if the characteristic velocity is æxed independently of b. However, the æow also depends up the Weissenberg number, We, which is itself a function of b. So, one should modify both Ca and We in order to obtain realistic comparison with experimental results. These resulting length-scales are given in Fig. 3a. While it is not obvious that scaling l ç b is satisæed, there is a good qualitative agreement of the simulations and the experimental results. The driving pressure is another control parameter whose inæuence on emerging length scales can be explored. In experiments ë6, 3ë increasing the driving pressure typically decreases the observed length scales. Figure 3b compares the length scales obtained from our simulations to the results of linear stability ë9, 0ë, and toaætting function of the form l ç = p æp, where æp is the driving pressure. The motivation for this particular æt arises from analogy with Newtonian æuids where l ç = p Ca,and Ca ç æp. Here we observe that linear stabilitytheory and simulational results agree rather well atsmaller driving pressures. For larger values of æp, the length-scales resulting from linear stability analysis saturate to a constant, while the results of the simulations æt l ç = p æp very closely. We hope to verify this prediction experimentally ë4ë.

6 CONCLUSION In this paper we have shown that, under certain assumptions, æow in a Hele-Shaw cell of a complex viscoelastic æuid simpliæes to that of a generalized Newtonian æuid. Full numerical simulations of the two phase èliquidègasè æow show that shear thinning behavior of the driven æuid modiæes signiæcantly the morphology of the patterns, relative to those for Newtonian liquids, by suppressing tip-splitting. Furthermore, the varying of length scales emerging from our simulations, as parameters are changed, is in good qualitative agreement with those observed in experiments. In particular, we observe in our simulations that the typical length scale of the patterns scales with driving pressure as l ç æp,=2. This prediction, which is of considerable importance in technological processes such as injection molding oroil recovery is still to beveriæed experimentally. We are continuing our work in few directions: improvement in numerical methods, formulation of more realistic boundary conditions, and extension of our model to the æuids characterized with diæerent rheological properties. By combining experimental, theoretical and computational eæorts, we hope to further contribute to the understanding of complex æuids and pattern formation. ACKNOWLEDGMENTS This workwas supported in part by DOE Grant No. DE-FG02-88ER25053, NSF Grants No. DMR and DMS , and by ONR Grant No. N References ëë C. A. Hieber, in A. I. Isayev, ed., Injection and Compression Molding Fundamentals. Marcel Dekker, New York, 987; C. Z. Van Doorn, J. Appl. Phys. 46, 3738 è975è. ë2ë J. S. Langer, in H. E. Stanley, ed. Statistical Physics. North-Holland, 986. ë3ë J. Bockris and A. Reddy, Modern Electrochemistry, Plenum Press, 970. ë4ë K.V. McCloud and J. V. Maher, Phys. Rep. 260, 39 è995è. ë5ë A. Buka, P. Palæy-Muhoray, and Z. Racz, Phys. Rev. A 36, 3984 è987è. ë6ë H. Van Damme and E. Lemaire, in J.C. Charmet, S. Roux, and E. Guyon, editors, Disorder and Fracture, page 83. Plenum Press, New York, 990. ë7ë S. S. Park and D. J. Durian, Phys. Rev. Lett. 72, 3347 è994è. ë8ë L. Kondic, P. Palæy-Muhoray, and M. J. Shelley, Phys. Rev. E 54, 4536 è996è. ë9ë L. Kondic, M. J. Shelley, and P. Palæy-Muhoray, Phys. Rev. Lett. 80, 433 è998è. ë0ë P. Fast, L. Kondic, M. J. Shelley, and P. Palæy-Muhoray, in preparation, 998. ëë L. Paterson, J. Fluid Mech. 3, 53 è98è. ë2ë G. Daccord and J. Nittmann, Phys. Rev. Lett. 56, 336 è986è. ë3ë M. Kawaguchi, K. Makino, and T. Kato, Physica D 05, 2 è997è. ë4ë P. Palæy-Muhoray, R. Ennis, M. J. Shelley, and L. Kondic, in preparation, 998. ë5ë E. Meiburg and G. M. Homsy, Phys. Fluids 3, 429 è988è.