Coating Flows of Viscous Liquids on arotatingverticaldisc

Transcription

1 Coating Flows of Viscous Liquids on arotatingverticaldisc Submitted in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY in CHEMICAL ENGINEERING by Nilesh H. Parmar Roll No. 345 Supervisor: Prof. Mahesh S. Tirumkudulu Department of Chemical Engineering Indian Institute of Technology, Bombay 7

2 INDIAN INSTITUTE OF TECHNOLOGY BOMBAY Certificate of course work This is to certify that Mr. Nilesh H. Parmar (Roll No. 345 was admitted to the candidacy of the Ph.D. Degree on /7/3, after successfully completing all the courses required for the Ph.D. Degree Programme. The details of the course work done are given below. Sr. No. Course Code Course Name Credits Mode CL 6 Advance Transport Phenomena 6. Credit CL 65 Advance Reaction Engineering 6. Credit 3 CLS 8 Seminar 4. Credit 4 CL 6 Multiphase Flow Systems 6. Credit Total credits. Date: Place: IIT Bombay Dy. Registrar (Academic

3 Abstract Thin films of liquids occur widely in nature including the flow of blood in the body, lava flows and linings of mammalian lungs and tear films in theeyes. Thin films are also widely encountered in industrial processes related to paints, pharmaceuticals, electronics, packaging and polymer. They also appear in various modern technological applications such as microchip production where spin coating is used to coat silicone substrates with photoresist. Here, the photoresist is allowed to spread on a rotating horizontal surface (of silicone under the action of centrifugal force to get uniform thin film laminates. Thin film flowsalsoappearin extrusion coating process in which a polymeric material is extruded on to another polymeric material to form composite laminates. Thin liquid films show variety of interesting dynamics based on the Reynolds number of the flow. In the absence of electrical and magnetic forces, the dynamics of inviscid thin films are determined by the balance between inertia, gravity and surface tension forces. They appear in very fast or rapidly-changing flows of low viscosity liquids past solid substrate. On the other hand, the dynamics of viscous thin films, under the zero Reynolds number approximation, are determinedby balancing viscous, gravity and surface tension forces. Examples include the flow of viscous liquid down an inclined plane or the flow exterior to arotatingcylinder. The presence of a deformable boundary, air-liquid interface, in thin film flows leads to wave motion in the films; the waves can travel and steepen under certain conditions for high flow rates. Wave motion sometimes results inquasiperiodicor chaotic structures. The fundamental difficulty in solving thin film flows arise due to the presence of moving contact line, air-solid-liquid interface, in the flows. The i

4 classical no slip boundary condition fails to predict the advancing contact line. There are different approaches to circumvent the so-called contact line paradox in thin film flows. The common approach is using a precursor film model, which basically assumes that the solid surface is prewetted with the liquid. Spreading of thin liquid film on a rotating horizontal surface havebeenex- tensively studied in the literature. Here in this work, we studied coating flows of viscous liquids on a rotating vertical disc. The motivation for the work comes from experiments performed on particular non-newtonian liquids in which an uniformly coated thin film of liquid on a rotating vertical disc collected into a ring like structure (donut shaped. Experiments were performed with aknownvolumeof liquid and at varying rotation rates such that the inertial effects were negligible. Liquid injected on the rotating disc initially coats the surface uniformly, which then redistributes itself such that at steady state a significant amount collected into a circular ring, off center with the axis of rotation. Beyond a critical rotation speed and for a given liquid volume, the ring formation did not occur. Thering formation was not observed in the case of Newtonian liquids. In order to better understand the ring formation phenomenon, detailed experiments were performed with viscous Newtonian liquids at varying disc rotation speeds and liquid volumes, and the thickness profile at steady state was measured as a function of the spatial coordinates. AlubricationanalysisforNewtonianliquidresultedinatime evolution equation for the film thickness that accounted for gravity, surface tension and viscous forces. In the absence of surface tension, the lubrication equation at steady state was solved analytically to obtain constant height contours that were circular and symmetric about the horizontal axis. Following an analysis by Batchelor (Batchelor, 956 for inviscid flow with closed streamlines, we show that the effects of surface tension need to be included to obtained a unique thickness profile. A perturbation solution accounting for surface tension was derived whose predictions compared well with the experiments for small liquid volumes or high rotation rates. Further, the timeevolutionequaii

5 tion with surface tension terms was solved numerically using time-marchingfinite difference scheme. The predicted thickness profiles are in excellent quantitative agreement with those obtained experimentally for moderate volumes. Experiments with Newtonian liquids also showed that though the maximum liquid supported by the rotating disc varied with rotation rate and liquid viscosity, the numerical value of the dimensionless number signifying the ratio of gravitytoviscous force was same in all the cases. Shear rate calculations for the Newtonian liquid along with the rheological measurements for the non-newtonian liquids suggest that the shear thinning nature of the non-newtonian liquids may be the cause for the observed ring formation. Hence, the lubrication analysis was extended for shear thinning liquids, which resulted in a time evolution equation for the free surface. The equations were derived for two different shear thinning rheology, namely, power law and Ellis model liquids, and solved numerically using time marching finite difference scheme. Although, the predicted film thickness profiles do not show the ring formation phenomenon observed experimentally, depletion of the liquid from the high shear rate regions to the surrounding regions was observed. At steady state, the time evolution equation for the power lawliquidsintheab- sence of surface tension terms was also solved analytically to obtain the constant height contours. Key words : thin film flow, coating flow, lubrication analysis. iii

6 Contents Abstract i Nomenclature xi Introduction. Motivation and Scope of the work Experiments Lubrication analysis for Newtonian liquids Lubrication analysis for shear thinning liquids Organization of the thesis Coating flow of viscous Newtonian liquids on a rotating vertical disc. Introduction Experiments Model: Lubrication approximation Locating constant height contours using the method of characteristics Model accounting for surface tension Results and Discussion Without surface tension Conclusions Ring shaped structure in free surface flow of non-newtonian liquids 35 iv

7 3. Experiments Experiments with shampoo Experiments with an aqueous colloidal dispersion of latex particles Experiments with an aqueous solution of CTAB and sodium salicylate Shear thinning hypothesis for the ring formation Conclusions Lubrication approximation for power law and Ellis model liquids Lubrication approximation for power law liquids Lubrication approximation for an Ellis model liquid Results and Discussion Power law and Ellis model fits to the shampoo viscosity data at 3 o C Constant height contours for power law liquids from the method of characteristics Numerical solutions Conclusions Conclusions and Future work Conclusions Future scope of work Appendix I 77 Appendix II 83 References 84 List of publications 9 Acknowledgements 9 v

8 List of Figures. The schematic of the ring disc reactor (reproduced from Zanfir et al. ( The schematic of the PET reactor (a front and (b side view (reproduced from Afanasiev and Wagner (accessed August 7, The schematic of the experimental setup, (a Front view of the rotating disc (b Side view of the set up A photograph of the experimental setup Interpolated thickness profile on a 7 cm diameter circular discrotating clockwise at rpm and coated with 5 ml of silicone oil of viscosity, 35 Pa s. The x and y coordinates are non-dimensionalized by the disc radius while the thickness is rendered dimensionless by the average film thickness Representation of constant height contours for Newtonian liquids..5 Constant thickness contours for (a α =.58, (b α = Comparison of (a thickness profile (Φ(q predicted by (5 with that obtained numerically ((8, without the surface tension effects and (b constant thickness contours, obtained using the quadratic profile for Φ(q, for α = vi

9 .7 Comparison of numerical results with (solid line and without (dashed line surface tension effects with experiments ( alongdifferent constant x and y lines. The dotted line represents the asymptotic solution. Here, the results are shown for α =.9,h =.3mm and Ca =6.56,whichcorrespondstorotationrateof4rpm,liquid volume of 5 ml and µ =35Pa sona7cmdiameterdisc Comparison of numerical results with (solid line and without (dashed line surface tension effects with experiments ( alongdifferent constant x and y lines. The dotted line represents the asymptotic solution. Here, the results are shown for α =.58, h =.3mm and Ca =8.9whichcorrespondstorotationrateofrpm,liquid volume of 5 ml and µ =35Pa sona7cmdiameterdisc Comparison of numerical results with (solid line and without (dashed line surface tension effects with experiments ( alongdifferent constant x and y lines. The dotted line represents the asymptotic solution. Here, the results are shown for α =.6, h =.3mm and Ca =4.45whichcorrespondstorotationrateofrpm,liquid volume of 5 ml and µ =35Pa sona7cmdiameterdisc Values of α for maximum possible liquid volume supported by a rotating disc of diameter 7 cm as a function of rotation rate for two different viscosity liquids width = 4 cm Small angle oscillatory shear measurements for (a shampoo and (b colloidal dispersion Stress ramp experiment for shampoo Dynamics of ring formation for 6 ml of shampoo coating a 9 cm diameter disc rotating at rpm after (a min, (b 3 min, (c 4 min and (d 7 min at 3 o C. The arrow shows the direction of rotation. 4 vii

10 3.5 Ring formation on a 9 cm diameter disc for 6 ml of shampoo at rotation rates of (a rpm, (b rpm, and (c 3 rpm at 3 o C. The arrow shows the direction of rotation Ring formation on a 9 cm diameter disc rotating at rpm for shampoo volumes (a 4 ml, (b 5 ml and (c 6 ml at 3 o C. The arrow shows the direction of rotation Measured nondimensional film thickness profile along y =fora shampoo experiment (at 3 o C on a 9 cm diameter disc rotating clockwise at rpm for 5 ml of shampoo volume (h =.78 cm Ring formation on a 9 cm diameter disc rotating at rpm after min for 6 ml of aqueous colloidal dispersion at 8 o C Steady shear viscosity for CTAB - sodium salicylate solution measured at 3 o Conaconeandplategeometry(fromWon-Jongand Yang ( Small angle oscillatory shear measurements for CTAB - sodium salicylate solution (from Won-Jong and Yang ( Measured nondimensional film thickness profile along y =foran experiment with CTAB - sodium salicylate solution (at 3 o C on a7cmdiameterdiscrotatingclockwiseat.5rpmfor.5mlof liquid volume (h =.65 cm (a Power law fit for the shampoo rheology. Regimes of ring formation for varying α corresponding to experiments with varying disc diameters, rotation rates and volumes of the shampoo at (b 5 o C and (c 3 o C Power law fits for the shampoo rheology Ellis model fits for the shampoo rheology viii

11 4.3 Constant thickness contours obtained using Φ(q values,alongθ = π, fromexperimentallymeasuredfilmthicknessprofile(figure3.7. The symbols (H =.53, (H =.9 and (H =.88 correspond to the measurements. Here, m =.76 Pa.s n and n =.96 which corresponds to α = Comparison of numerical results for the power law liquids ((- - --m =.76Pa s n, n =.96 and α =.77, (- - - m =.76Pa s n, n =.33andα =.38 and ( m =.76Pa s n, n =.andα =.56 with that of a Newtonian liquid (solid line, m =.76Pa sandα =.7 along different constant x and y lines. Here, the results are shown for h =.943mmand Ca =3.57whichcorrespondstoarotationrateofrpm,liquid volume of 6 ml on a 9 cm diameter disc Comparison of numerical results for the power law liquids (( m =.76 Pa s n, n =.96 and α =.5, (- - - m =.76Pa s n, n =.33andα =.44and( m =.76Pa s n, n =.andα =.63withthatofaNewtonianliquid(solidline, m =.76Pa sandα =.4alongdifferentconstantx and y lines. Here, the results are shown for h =.943mmandCa =.78 which corresponds to a rotation rate of rpm, liquid volume of 6 ml on a 9 cm diameter disc Comparison of numerical results for the Ellis model liquids (( µ =4.65Pa sandλ =.78 and ( µ =4.65Pa sand λ =3.withthatofaNewtonianliquid(solidline,µ =4.65Pa s along different constant x and y lines. Here, the results are shown for α =.569,h =.943mm,τ / =.4,andCa =.which corresponds to rotation rate of a rpm, liquid volume of 6 ml on a 9cmdiameterdisc ix

12 4.7 Comparison of numerical results for the Ellis model liquids (( µ =4.65Pa sandλ =.78 and ( µ =4.65Pa sand λ =3.withthatofaNewtonianliquid(solidline,µ =4.65Pa s along different constant x and y lines. Here, the results are shown for α =.4, h =.943mm,τ / =.447,andCa =.which corresponds to rotation rate of rpm, liquid volume of 6 ml on a9 cm diameter disc A typical cell of the mesh One dimensional representation of boundary nodes in x direction Comparison of the measured thickness profile along θ =,π for shampoo at 3 o C(seefig.(3.7withthepredictionof(5.9. Here, m =.76 Pa.s n and n =.96 which corresponds to α = x

13 Nomenclature γ ij τ components of the strain rate tensor, s second invariant of the stress tensor, Pa τ / value of the second invariant of the stress tensor when µ = µ /, Pa τ ij components of a stress tensor, Pa τ ij components of the deviatoric stress tensor, Pa Ē p r second invariant of the strain rate tensor, s pressure, Pa radial coorditate, m ū θ tengential velocity, m/s ū r radial velocity, m/s ū z velocity in the z direction, m/s x, ȳ, z λ Cartesian coordinates, m constant in the Ellis model equation µ zero shear viscosity, Pa s τ second invariant of the stress tensor, dimensionless τ ij components of a stress tensor, dimensionless Ca Capillary number xi

14 g gavitational acceleration, m/s H h film thickness, dimensionless film thickness, m h average film thickness, m m n P p q power law coefficient, Pa s n power law index characteristic pressure scale, Pa pressure, dimensionless initial condition for r for a given characteristic curve, dimensionless q max location of the maximum film thickness, dimensionless R r Re s t radius of the disc, m radial coordinate, dimensionless Reynold s number variable along a characteristic curve, dimensionless time, s U characteristic velocity in the radial and tengential direction, m x, y, z Cartesian coorditates, dimensionless Greek symbols α ratio of gravitational force to viscous force µ viscosity, Pa s Ω Φ rotation rate, rpm constant height along a characteristic curve, dimensionless xii

15 Φ(q max maximum film thickness, dimensionless φ x liquid flux in x direction, dimensionless φ y liquid flux in y direction, dimensionless ρ density, m 3 /s θ tengential coorditate xiii

16 Chapter Introduction Thin film flows are encountered in a variety of settings such as spreading of paints, coatings and adhesives on substrates, gravity currents under water, lava flows, linings of mammalian lungs. They appear in paper making machines asflowonthe rollers, in paint industries on a paint-coated rollers as an alternative to brushes for spreading the paint, in many engineering machinery as thin layer of coating on rotating shafts and in photographic industry as flow on rotating horizontal substrate. They also appear in various modern technological applicationssuchas microchip production and microfluidic devices. The work by Landau and Levich (94 is one of the earliest pioneering work on free boundary problemforfalling and rising thin film flows on vertical and inclined planes. Their work lead to the prediction of thickness and shape of the thin film on the substrate being pulled out from a liquid reservoir. Emslie et al. (958 studied spreading of a thin axisymmetric film of Newtonian liquids on a horizontal rotating disc, a situation which is encountered during the spin coating process. Exact solution of the evolution of the film due to centrifugal and viscous forces was obtained. The method of characteristics was used to obtain the characteristic curves and thesurfacecontoursat successive times for fixed initial liquid distribution. The knowledge of the initial surface profile is required to obtain profiles at subsequent times. Surface contours for different initial distributions were constructed and the solutionshowedthat the initially non-uniform free surface profiles tend to become increasingly uniform

17 during spinning. Experimental studies by Fraysse and Homsy (994 and Melo et al. (989 show that the profile of the spreading films in spin coating become almost flat except near the contact line where the surface tension effects aresignificant.the work of Fraysse and Homsy (994 aims to make systematic and accurate observations of the axisymmetric spreading rate over a fairly wide rangeofconditions and compares the results with available approximate theories. Wetting properties of the liquid, surface tension and contact angle, have been studied for varying liquid volumes and rotation rates. The theoretical analysis oftheprobleminthe asymptotic limit of weak surface tension effects showed that, intheouterregion away from the contact line, the evolution of the film thickness profileisdescribed by the solution obtained on balancing the centrifugal and viscous forces (Emslie et al., 958. However, in the inner region close to the contact line, the film thickness profile is obtained by balancing the surface tension and the centrifugal forces. This description was first proposed by Huppert (98 forflowofviscous current down a slope and the first valid leading order composite solutions were obtained by Moriarty et al. (99. The latter analyzed two dimensional gravity driven draining and the axisymmetric spin coating problem and obtained excellent agreement between their asymptotic solutions and numerical solutionsofthegoverning lubrication equation for fairly weak surface tension. Wilson et al. ( reconsidered the spin coating problem for the cases in which surface tension and moving contact line effects are significant. Numerical solution in the case of weak but finite surface tension was obtained and compared with the experimental results by Fraysse and Homsy (994 for the evolution of drop radius prior to the onset of the fingering instability. They also present a detailed analytical description of the no surface tension and weak surface tension asymptotic solutions. While qualitative agreement with the experimental results for the evolution of drop radius prior to the onset of instability was obtained by asymptotic solution, quantitative agreement was obtained only on inclusion of weak but finite surface tension. Myers and Charpin ( studied the effect of Coriolis forces on thesameproblem

18 and showed that Coriolis force has no effect on the height of the fluidfilm. The review of Oron et al. (997 describes thin film flows that include effects of viscous, surface tension and body forces, thermocapillarity, evaporation, and van der Waals attraction. The review addresses interesting dynamic phenomena such as wave propagation, wave steepening, development of chaotic responses, spreading of fronts and development of fingers displayed by thin films subjected to mechanical, thermal or structural factors. It presents a unified mathematical theory which takes advantage of the disparity of the length scales in the problem and asymptotically reduces the full set of governing equations and boundary conditions to a simplified, highly nonlinear evolution equation ortoasetofequations. This reduces the mathematical complexity of the system without compromising on the physics of the problem. The derivation of time evolution equation or equations for the film thickness are presented for various flow problems involving free surface. The cases reducing to a single highly nonlinear evolution equation involve films with constant inter-facial stress and constant surface tension,filmswithcon- stant surface tension and gravity only, films with van der Walls forces and constant surface tension only, films on a thick substrate, films on a horizontal cylinder and films on a rotating disc. Problems which reduce to a set of nonlinear evolution equations are the dynamics of free liquid films, bounded films with an interfacial viscosity and dynamics of surfactants in bounded and free films. The review calls for careful experiments to verify the presented theory. The geometry closest to the one considered in this thesis is the flow of viscous liquids over a horizontal rotating cylinder. Puzzled by the question of maximum amount of honey that can be supported by a rotating knife, Moffatt (977 studied the dynamics of viscous thin films on the outer surface of a horizontal rotating cylinder using lubrication approximation. The condition for the existence of a steady state solution of the Navier-Stokes equation was obtained when part of the fluid domain is bounded by a rigid surface and part by a free surface. Experiments are described to demonstrate the existence of the steady state for a suitable range of parameters and to show the evolution of an instability. The maximumamount 3

19 of Newtonian liquid that can be supported on the outer surface ofahorizontal rotating cylinder for a given rotation rate was predicted, which comes from the balance between viscous and gravitational forces. Almost all of the previously mentioned studies address the flow of Newtonian liquid only, while there are few papers where non-newtonian behavior is considered. Fraysse and Homsy (994 and Spaid and Homsy (996 investigated the stability of viscoelastic coatings in the presence of a moving contact line. Similarly, the flow of power law and Bingham model liquids on a shallow layer of fluid mud have been investigated by Liu and Mei (994 and Ng and Mei(994. Perazzo and Gratton (3 studied thin film flow of a power law liquid on an inclined plane under the action of gravity and viscous forces. Spreading of Ellis model liquids and liquids with significant normal stress differences with constant viscosity have been analyzed by Neogi and Ybarra (. In this work, we report experimental and theoretical investigation of flow of Newtonian and non-newtonian liquids on a rotating vertical disc. A variation of this geometry is a rotating disc reactor where in a vertical rotating disc is immersed partially in a pool of liquid and is used for many reactions to enhance gas liquid contacting (Zanfir et al., 7. Figure. shows the schematic of a rotating disc reactor reproduced from Zanfir et al. (7. The reactor consisted of a stainless steel disc mounted on a horizontal shaft, accommodated in a cylindrical shell. The disc was partially immersed in the liquid phase. The rotating disc carry a thin film of liquid on its upper part, which could be brought in contact with a gas phase used for stripping. The main advantage of the rotating disc reactor is that the interfacial gas-liquid area is constant and known. Also, as the contact between the phases is generated by maintaining a thin filmofliquidon the disc surface it minimizes the mass transfer resistances related to the liquid phase. The hydrodynamics of the phases can be decoupled by modification of the size and dimensions of the gas shell such that it can accommodate wide range of gas flow rates, unlike conventional agitated reactors that are limited by impeller flooding. The rotating disc reactor (RDR gives high performance in the removal 4

20 of dissolved pollutants at the expense of less energy in comparison with other conventional methods (Friedman et al., 979. For this application the RDR is known as rotating biological contactor. The reactor is used to denitrify high concentration nitrogen compounds in the waste water. The disc surface alternatively comes into contact with the air and the waste water and works as anaeration device for waste water treatment. The RDR is also used for photocatalytic degradation of chlorinated phenols and chlorinated aromatics (Hamill et al., and in citric acid production by aerobic fermentation using a bacterial culture (Sakurai et al., 997. Another application of RDR includes production of thermoplastic polymer such as polycarbonates which requires continuous removal of byproducts by gas stripping from the reaction mixture (Woo et al.,. Glatzer et al. (998 and Glatzer and Doraiswami ( designed RDR for liquid-liquid reactions. The recent work of Hardcre et al. ( presents the heterogeneously catalyzed oxidation of cinnamyl alcohol in toluene and ionic liquidsonrdr. The vertical disc geometry is also a part of the polycondensation reactors used for the production of PET (polyethylenterephthalat. The reactor consist of a horizontal cylinder which is partially filled with polymer melt and contains disks rotating about the horizontal axis of the cylinder, thus picking up and spreading the melt in form of a thin film over a large area of the disks (Figure., reproduced from Afanasiev and Wagner (accessed August 7, 7 The film profilesand thickness of the melt will vary from disk to disk, since the viscosity increases with the degree of polymerization. In spite of the several industrial applications of the thin film flow on a rotating vertical disc, it has not received much attention in the literature while the related case of spin coating, wherein a thin film of liquid spreads over arotatinghorizontal substrate by the action of centrifugal force has been well studied (Emslie et al., 958; Fraysse and Homsy, 994; Myers and Charpin, ; Wilson et al.,. Here, gravity acts perpendicular to the plane of the substrate leading to an axisymmetric film thickness profile that spreads radially outwards. In contrast, gravity acts parallel to the disc surface in our case leading to a non-axisymmetric 5

21 Figure.: The schematic of the ring disc reactor (reproduced from Zanfir et al. (7 (a (b Figure.: The schematic of the PET reactor (a front and (b side view (reproduced from Afanasiev and Wagner (accessed August 7, 7. film thickness profile. 6

22 . Motivation and Scope of the work Apart from its industrial importance, the study is motivated byexperimentsper- formed on particular non-newtonian liquids where an initial uniformcoatingon the rotating vertical disc redistributed such that most of the liquid accumulated into a ring like structure (donut shaped in a section of the disc. The ring formed on the left (right half plane for clockwise (counter-clockwise rotation of the disc. In order to better understand this phenomenon, we performed similar experiments with Newtonian liquids and found that although the aforementioned inhomogeneity was absent, the film thickness exhibited spatial variation. The main objective of this work is to understand the ring formation phenomenon observed in the case of particular non-newtonian liquids. This includes understanding the role of driving forces on the disc surface for Newtonian liquids and predicting the spatial variation in the film thickness profile observed experimentally. It is important to note that the flow of Newtonian viscous liquids on vertical rotating discs has not been studied before, let alone the interesting ring formation phenomenon observed in case of non-newtonian liquids. A summary of the work carriedoutis given below. Detailed experiments with Newtonian and non-newtonian liquids on the rotating vertical disc. Lubrication analysis for Newtonian liquids to predict the film thickness profile on the surface on the disc. Analytical and numerical solution of the time evolution equation for the free surface for Newtonian liquids. Comparison of the predicted film thickness profiles with experiments for Newtonian liquids. Aperturbationsolutionofthetimeevolutionequationforthe free surface for Newtonian liquids. 7

23 Prediction of the maximum amount of Newtonian liquid that can be supported on the disc surface for a given rotation rate and confirmation with experiments. Lubrication analysis for shear thinning liquids (power law and Ellis model liquids. Analytical solution of the time evolution equation for the free surface for power law liquids to predict contours of constant heights. Numerical solution of the time evolution equation for the free surface for power law and Ellis model liquids.. Experiments As mentioned previously, the flow of viscous liquids over rotating vertical discs has not been considered before. So, experiments formed a significant part of the investigation. Experiments were performed for varying liquid volumes, rotation rates and liquid viscosities with the film thickness profile measured with an xyz traverse system. The latter was specifically designed for the flowgeometrysoasto enable accurate measurement of the thickness profile. The Reynolds number for the flow in all the cases was sufficiently small to neglect the inertial effects. Consequently, a balance of viscous, gravitational and surface tension forces supports the liquid on the vertical surface and determines the film thickness profile. Since thin film flows involving moving contact lines are difficult to analyze, it was ensured that the contact line was pinned at the periphery of the disc. Experiments were performed with Newtonian liquids and a range of non-newtonian liquids that exhibited shear thinning and elastic properties. Some of the non-newtonian liquids showed ring formation at steady state where most of the liquid collectedintoa ring like structure in a section of a rotating disc. 8

24 .3 Lubrication analysis for Newtonian liquids The flow was first theoretically analyzed via lubrication analysis for the Newtonian liquids which resulted in a time evolution equation for the film thickness. On neglecting the surface tension effects, the equation was solved analytically to obtain constant height contours under steady state conditions. Interestingly, height contours were circular and symmetric about the horizontal axis. Next, we show that inclusion of surface tension effects is a must to obtained auniquefilmthickness profile. An asymptotic analysis on inclusion of the surface tension resulted in an approximate solution for the thickness profile. The time evolution equation in the presence of surface tension term was solved numerically using time marching finite difference scheme to obtain the steady state solution. Since the contact line was pinned at the disc edge, the no-flux boundary condition at the edge ensured volume conservation of the liquid. The numerical results areinclosequantitative agreement with the experiments even for low rotation rates/large volumes. Experiments also showed that there exists a maximum liquid volume supported by the rotating disc and that the numerical value of the dimensionless number signifying the ratio of gravity and viscous force at the maximum volume was the same irrespective of liquid viscosity or rotation rate. Finally, shearratecalculationsfor Newtonian liquid along with the rheological measurements for the non-newtonian liquids suggest that the shear thinning behavior of the non-newtonian liquids could be the cause for the ring formation phenomenon..4 Lubrication analysis for shear thinning liquids The lubrication analysis for Newtonian liquids was extended forpowerlawand Ellis model liquids to investigate the cause of the ring formation observed in experiments. The resulting time evolution equation for the free surface was solved numerically using time marching finite difference scheme. Although, the predicted 9

25 film thickness profiles do not show the ring formation phenomenon observed experimentally, depletion of liquids from higher shear rate regions due to decrease in the viscosity was captured. The predicted film thickness profiles are compared with the Newtonian liquids and differences are elucidated..5 Organization of the thesis The theoretical and experimental investigation of the thin film flow of viscous Newtonian liquids on the rotating vertical disc is presented inchapter.next,in the chapter 3 experiments with the non-newtonian liquids are demonstrated. It describes the details of the observed ring formation phenomenon. The lubrication analysis for the power law and Ellis model liquids and comparison of the predicted film thickness profiles with Newtonian liquid are documented in chapter 4. Finally, chapter 5 presents the conclusions of the work carried out followed by the scope of future work.

26 Chapter Coating flow of viscous Newtonian liquids on a rotating vertical disc Abstract We study the flow of Newtonian viscous liquids coating a vertical rotating disc in the creeping flow regime. Experiments with viscous Newtonian liquids were performed at varying disc rotation speeds and liquid volumes, and the thickness profile at steady state was measured as a function of the spatial coordinates. A lubrication analysis for Newtonian liquid resulted in a time evolutionequationfor the film thickness that accounted for gravity, surface tension and viscous forces. In the absence of surface tension, the lubrication equation at steady state was solved analytically to obtain constant height contours that were circular and symmetric about the horizontal axis. The constant height contours predicted by the analytical solution compared well with those obtained experimentally for small liquid volumes or high rotation rates. However, for obtaining a unique solution, the surface tension effects need to be included. An asymptotic analysis accounting for surface tension resulted in an approximate solution. Further, the time evolution equation with surface tension term was solved numerically using time-marching

27 finite difference scheme. The predicted thickness profiles areinexcellentquan- titative agreement with those obtained experimentally for moderate volumes of silicone oil. Experiments also showed that though the maximum liquid supported by the rotating disc varied with rotation rate and liquid viscosity, the numerical value of the dimensionless number signifying the ratio of gravity to viscous force was same in all the cases.. Introduction In this study, we report experimental and theoretical investigation of flow of viscous Newtonian liquids coated on a vertical rotating disc. As discussed in the first chapter, in the the related case of spin coating, gravity acts perpendicular to the plane of the substrate leading to an axisymmetric film thickness profile that spreads radially outwards (Emslie et al., 958; Fraysse and Homsy, 994; Wilson et al., ; Myers and Charpin,. In contrast, gravity acts parallel to the disc surface in our case leading to a non-axisymmetric thickness profile with the contact line pinned at the perimeter of the disc. The Reynolds numberfortheflow was sufficiently small to neglect the inertial effects. A balance of viscous, gravitational and surface tension forces supports the Newtonian liquid on the vertical surface and determines the film thickness profile. For a stationary vertical surface, afilmofliquidofthicknessh drips under gravity with a velocity gh. Here, g /mu is the gravitational constant, h is the film thickness and µ is the liquid viscosity. Thus the maximum thickness that can be supported by a rotating discwithout µrω the liquid dripping off is given by.itthusfollowsthatthemaximumliquid g µωr that a rotating disc can support is proportional to 5. g Since the thin film flows involving moving contact lines are difficult to analyze, it was ensured that the contact line was pinned at the periphery of the disc in all experiments. Experiments were performed for varying liquid volumes, rotation rates and liquid viscosities with the film thickness profile measured with an xyz traverse system.

28 . Experiments Figures. and. present, respectively, a schematic diagram and a photograph of the experimental setup. A vertical flat stainless steel disc was attached to a stepper motor via a :5 reduction pulley so that the disc could berotatedover awiderangeofrotationrates. Liquidvolumesforexperiments were selected so that the liquid coated the disk completely with the contact line pinned at the circumference of the disc. The latter was ensured by fabricating discs with sharp edges. The thickness profile of the film at steady state was measured accurately using an xyz micrometer traverse of least count of. mm. This in turn was placed on an xy traverse with a coarse movement of mm in each directionand aleastcountofmm(figure.. Aneedlewasattachedtothexyz traverse to probe the entire surface of disc. Since the accuracy of the final measurement depends on the alignment of the traverse with respect to the vertical disc, a detailed calibration was performed prior to the experiments to determine the error in thickness measurement. At a fixed y, theneedlewasmadetocontactthediscsurfaceatanumberofpoints along the x direction (figure.. This was repeated at various vertical positions (for different y onthediscsurface. Themaximumvariationinthediscposition over the entire surface was found to be less than ±.5 mm. Once the average disc surface position (z co-ordinate of the disc surface was known, the liquid film thickness profile was determined by noting the z co-ordinate of the liquid film (the free surface at various locations on the disc and then subtracting the disc surface position from it. Experiments were performed with transparent silicon oils (Newtonian of viscosities 5 Pa sand35pa sandsurfacetension3mn/matvaryingrotation speeds, disc diameters and liquid volumes. Each experiment involved injecting a fixed quantity of oil on to the rotating disc so that the liquid coated the entire disc. The volumes chosen for the experiments were such that the film thickness over most of the disc surface was much greater than the error in measurement. For all experiments, the film thickness was measured at steadystateatmorethan 3

29 Figure.: The schematic of the experimental setup, (a Front view of the rotating disc (b Side view of the set up. Figure.: A photograph of the experimental setup. 6 points on the disc surface and interpolated using MATLAB c to obtain the full thickness profile. Steady state was said to be achieved when the thickness profile 4

30 measured at different times were identical. Figure.3 shows the interpolated values for an experiment performed on a 7 cm diameter disc rotating at rpm and coated with 5 ml of silicone oil. Experiments with liquid volumes ranging from 3 to 7 ml were performed on discs with diameters of 7 and 8 cm at rotation rates of to5rpm. Figure.3: Interpolated thickness profile on a 7 cm diameter circular disc rotating clockwise at rpm and coated with 5 ml of silicone oil of viscosity, 35 Pa s. The x and y coordinates are non-dimensionalized by the disc radius while the thickness is rendered dimensionless by the average film thickness. The thickness data points were integrated over the whole domain to determine the total volume of liquid. In all experiments, the numerically determined liquid volume differed slightly (3% error from the amount actually injected, thereby confirming the accuracy of the measurement system. The measured thickness 5

31 profile (figure.3 shows more liquid in the region x<wheregravityandviscous forces oppose each other compared to x>. Also, the profile appears symmetric about the horizontal axis. Asymmetry was observed for large volumes/low rotation rates where the thickness in the region x<, y<washigherthanthatforx<, y>. In the next section, we develop a lubrication approximation to predict the film thickness profile for a viscous liquid coating a vertical rotating disc..3 Model: Lubrication approximation The starting point for modeling coating film flows are the Navier-Stokes equation for an incompressible liquid (in cylindrical coordinates, ( ūr ρ t +ū ū r r r + ūθ ū r r θ +ū ū r z z ūθ = r p [ ( r + µ r r r ( rū r + r ū r θ + ū r z r ū ] θ ρgsinθ (. θ ( ūθ ρ t +ū ū θ r r + ūθ r r p θ + µ [ r ( r ū θ θ +ū ū θ z z + ūrū θ = r + r ū θ θ r ( rū θ ( ūz ρ t +ū ū z r r + ūθ ū z r p z + µ [ r θ +ū ū z z = z ( r ū z r r + ū θ z + r ū z θ + r ū ] r ρgcosθ (. θ + ū ] z z (.3 where ū r,ū θ and ū z are the liquid velocities in the r, θ and z direction, respectively, p is the pressure, ρ is the density, g is the gravitational force per unit mass and µ is the viscosity. The characteristic length scale in r and z directions are the radius of the disc (Randtheaveragefilmthickness(h, respectively, and the characteristic time scale is the inverse of the angular velocity (Ω. The characteristic velocity in the radial and angular direction is U (= ΩR whilethatintheaxialdirection ( z ish Ω. With P as the characteristic pressure scale, we non-dimensionalize the Navier-Stokes equation to obtain, ( ( ( ur ρωrh h µ R t + u r ( h [ + R r( r r (ru r u r r + u θ u r r + r u r θ θ + u z r u θ θ 6 ] u ( r z u θ h = P r RµU ( ρgh + u r z µu p r sinθ (.4

32 ( ( ρωrh h µ R ( h [ + R r( r ( ρωrh µ ( h + R ( h R ( uθ t + u r r (ru θ ( uz t + u r [ r r u θ r + u θ u θ r + r u θ θ ( u z r + u θ r r u z r θ + u z + r u r θ ] u z θ + u z u θ z + u ru θ r + u θ z u z = z ( ρgh µu ( h = P RµU ( PR µu p r θ cosθ (.5 p z + u ] z + u z r θ z. (.6 The inertial terms on the left hand side of the equations are neglected for ( h R Re wherere ρωrh µ. Since the flow is generated from a balance of viscous and gravitational forces, we set P µru ( h h and drop the viscous terms of O ( ( h R as R. These assumptions simplify the momentum equations to give, = p ( r + u r ρgh z sinθ (.7 µu = r p θ + u θ z ( ρgh µu cosθ (.8 = p z. (.9 Further, if the surface tension effects are neglected, we obtain p = (atmospheric everywhere. The corresponding boundary conditions are that of no-slip at the disc surface (z =, u r = u z =andu θ = r, (. the zero tangential stress condition at film surface (z = H(r, θ which, under the lubrication approximation, reduces to (Oron et al., 997 u r z = u θ z and the kinematic condition at the film surface (z = H(r, θ, u z = H t + u H r r + u θ =, (. r H θ. (. 7

33 Here, H(r, θ h(r, θ/h is the dimensionless film thickness. On integrating (.7 and (.8 twice with respect to z, andapplyingtheaboveboundaryconditions, we have ( z u r = H(r, θz α sinθ (.3 ( z u θ = H(r, θz α cosθ r (.4 ( ρgh where α µu represents the balance of gravity and viscous forces. Next, the continuity equation is integrated over the thickness of the film to give ( ( H t + H(r,θ ru r dz + H(r,θ u θ dz =. (.5 r r r θ z= On substituting the expressions for u r and u θ from (.3 and (.4 respectively, we obtain the evolution equation for the free surface, H t ( α r r r 3 H3 sinθ (α H3 cosθ + rh = (.6 r θ 3 z= which at steady state reduces to ( H r + αh cosθ θ r + αh sinθ H r =. (.7 Since the flux of liquid at r =iszero,wehave H(r,θ z= u r r= dz = H 3 r= sinθ =. (.8 Note that since H, (.7 implies H θ θ= =. The remaining condition arises from the requirement that the volume of liquid be conserved, π H(r, θrdθdr =. (.9 π r= θ=.3. Locating constant height contours using the method of characteristics The steady state equation (.7 was solved via the method of characteristics to determine the constant height contours. This involves reducing the PDE on the (r, θ planetoasetofodesalongaparametriccurve(calledthecharacteristic curve parametrized by a variable, say s. On specifying the initial data (at 8

34 s =, we can generate a family of characteristics that gives the solution of (.7. Equation (.7 can be rewritten as H θ + αh sinθ ( + αh cosθ r so that the characteristic equations are given by, dθ ds dr ds = H r ==dh ds, (. = and (. αh sinθ (. (. + αh cosθ r Equations (. and (. constitute the characteristic equations of (. and are solved by setting θ = π, r = q and H =Φ(q ats =, [Φ(q] = q r α(q + rcosθ. (.3 Since the maximum film thickness is expected in the region, π <θ< 3π,theinitial condition for θ was specified at θ = π. Interestingly, the above solution suggests that the contours of equal height are all circles with centers atx = α[φ(q] and corresponding radii, q α[φ(q].nowtolocateaparticularcharacteristiccurve, we fix the value of q and the corresponding height, Φ(q, and then solve (.3 for r at various values of θ. Figure.4presentsasketchofatypicalheightprofilein terms of a constant height contours. The figure shows that the constant height contours continuously shrink with increasing q (from q = and finally reduce to a point along θ = π which corresponds to the maximum height for a given value of α. The location of the maximum thickness can be determined by equating the roots of (.3 along θ = π. ThusifΦ(q max isthemaximumheightforagivenα, itwillbe located at q max = α[φ(q max ].Itisalsoclearfromthefigurethatalltheconstant height contours cross θ = π between q = q max and q =. Therefore,aknowledge of the height profile (i.e. Φ(q whichisalongθ = π betweenq = q max and q = is sufficient to obtain the full height profile. However, it is not possible to know apriori the film thickness profile along θ = π that will satisfy (.9 for a given α. This is evident from (.3 in the limit of α whichgivescircularheight contours of unknown heights centered about the origin. The situation is analogous 9

35 to that for inviscid flows in confined regions with closed streamlines where the velocity distribution remains indeterminate unless additional information such as viscosity is available Batchelor (956. It will be show in the next section that inclusion of the surface tension term leads to a unique thickness profile. Presently, we will assume a quadratic thickness profile between q max and of the form, g R s = contour for maximum height contours of constant height Figure.4: Representation of constant height contours for Newtonian liquids Φ(q =a(q + b(q + c (.4 and determined the coefficients from the following conditions, Φ( =, Φ(q max = qmax α and Φ(q max q =. While it may appear that there is no justification for choosing a quadratic profile, analysis with the surface tension term suggests (see next section that this is indeed correct in the limit of α. The above boundary conditions give the thickness profile as Φ(q = (q (q max q qmax (q max α. (.5

36 Since the film thickness profile has to satisfy the volume conservation condition (.9, we use the fact that the contours are all circles and rewrite the same condition in terms of q and Φ(q, = Φ(qmax ( q α[φ(q] dφ. (.6 The quadratic approximation (.5 can be used to write q in terms of Φ(q, q = q max +(q max Φ(q (.7 Φ(q max which, on substitution in (.6, gives a fifth order polynomial in Φ(q max, = α [Φ(q max] 5 + α 5 [Φ(q max] 3 + Φ(q max (.8 The above equation was solved numerically to determine the roots for a given value of α. Forα<, four of the roots are complex conjugates while the fifth is apositiverealnumber. Further,since<q max α[φ(q max ] <, the maximum possible value for α is This implies that for a disc of a given radius and a Newtonian fluid of known properties, the upper limit of α (= α max corresponds to the maximum amount of liquid that can be supported by a rotating disc. As will be shown later, the experimentally determined value of α for the maximum liquid supported by a rotating disc is close to this value..3. Model accounting for surface tension We retain the pressure term in the evolution equation to include the effect of surface tension term on the thickness profile. The pressure is related to the interface curvature through the normal stress boundary condition at the interface which, for thin films depends only on r and θ and is given by, ( where κ(r, θ isthedimensionlesscurvature,ca p = ɛ3 κ(r, θ (.9 Ca µu o γ is the capillary number, γ is the surface tension, and ɛ ( h R. On substituting the above expression in (.7 and (.8 and repeating the steps outlined in the previous section gives, H t ] [r (αsinθ H3 ɛ3 κ [ ] H 3 (αcosθ ɛ3 κ + rh = r r 3 Ca r r θ 3 Ca r θ (.3

37 The above conservation equation is of the form H t + Q =where Q is the height averaged fluid flux. In the absence of surface tension and for α =,the streamlines of this flux are all circles centered about the origin and will coincide with the constant height contours. However, to determine the full film thickness profile, we introduce a small surface tension so that the total flux is, Q = Q o + ɛ3 Ca H 3 κ (.3 3 where, Q o = rh(rê θ.followingbatchelor(956,weconsidersteadyflowand integrate Q =overanareaenclosedbyastreamline, so that A } { Q o + ɛ3 H3 Ca 3 κ da = (.3 C κ dl = (.33 n for every streamline which implies that the curvature is a constant and the film thickness profile will be a segment of a sphere, where n is the unit normal and l is the length of the streamline. Note that H(r comesoutoftheabovelineintegral as the height contours and the streamlines are both circles centered at the origin for α =. The result of constant curvature implies a quadratic thickness profile and hence the quadratic thickness assumption for Φ(q intheprevioussection for predicting the profiles for small α. Forfinitevaluesofα, thestreamlinesand the constant height contours will not coincide and consequently the integration is much more complex. We instead solve the problem by posing an asymptotic expansion in powers of α and ζ (ζ = ɛ3 Ca whileassumingζ<<α<<, H(r, θ; α, ζ = H (r+αh (r, θ+α H (r, θ... +ζ ( H (r, θ+αh (r, θ+α H 3 (r, θ (.34 On substituting (5.5 in (.3 and then comparing coefficients of α n,wehave at α : at α : H θ = and so H = H (r, H θ = H dh dr sin θ gives H = H dh cos θ, dr

38 at α : at α 3 : ( H θ = rh 4 dh sin θ cos θ ( H 4 dh r r dr r θ dr cos θ H (r, θ = r d ( H 4 dh dr r dr 4 cosθ + f (r H 3 θ = ( r ( 3H r r 3 H +3H H sinθ ( cosθ ( 3H r θ 3 H +3H H gives Note that the constant of integration obtained on integrating with respect to θ at O(α hasbeensettozerosincechangingthedirectionofrotationwouldchange the sign of this term. Further, we set rf (rdr =, so that the the volume conservation (.9 reduces to, H (rrdr =. (.35 Next, we consider the terms involving surface tension with the curvature following a similar expansion in α, κ = κ + ακ + α κ +... Since the curvature at leading order is independent of θ, κ = ( r H, r r r collecting coefficients of leading order in surface tension gives, at ζ H : = ( rh 3 κ. θ r r r However, the thickness profile is periodic in θ implying that, H = and rh 3 Further, the boundary condition of, κ r =constant zero height, H as r, dh symmetry at leading order, dr and κ as r, and r zero flux, rh 3 κ r as r 3

39 implies that κ is a constant. Applying (5.6 gives, H (r =( r and H (r, θ = 6r( r cos θ. (.36 Now, f (r isdeterminedbycollectingtermsato(αζ ando(αζ, H at αζ : = ( r H3 κ + H3 κ θ r r 3 r 3r θ at αζ : H = θ r r rq r + r θ q θ where q r = H H sin θ + H H κ r + H3 κ 3 r, and ( H H. q θ = H H cos θ + r κ θ + H3 3 Substituting the expression of H and H from (5.8 in the equation obtained at O(αζ givesh on integration, κ θ H =48r( r (6 3r sin θ. (.37 In order to calculate H,wefirstnotethatitshouldbeperiodicinθ which implies that the θ independent part of rq r at O(αζ shouldbeaconstant. Giventhatthe no flux condition has to be satisfied at the disc periphery, this constantiszero. In other words, H 3 3 κ = H r H sin θ H θ H κ θ r θ (.38 where the angled brackets with the θ subscripts denote the θ independent (or averaged terms. On substituting the expressions for H, H, κ and H,we obtain { d dr r d dr ( r df } = 644r( r. (.39 dr The above equation is integrated to obtain f (r whilerecognizingthatf (r is finite at origin, f (r asr and rf (rdr =, f (r = r 9r r6 6r 8. (.4 Substitution of H, H and H in O(α 3 termgives, H 3 (r, θ = 5r 3 ( r 4 cos 3 θ + 56r ( r 595r 4 +35r 6 ( r cos θ 5 4

40 We can now assemble the first three terms in α and the O(αζ term, H(r, θ = ( r α 6( r r cos θ { + α 8( r 3 r cos θ r 9r } 3 r6 6r 8 { + α 3 5r 3 ( r 4 cos 3 θ + 56r } ( r 595r 4 +35r 6 ( r cos θ 5 { } + αζ 48r( r (6 3r sinθ + O(α 4. (.4 Results to be presented later show that the above approximation agrees well with the measured profile only for small α. Forlargeα, weresorttonumericalcomputations. Since the surface tension term in polar coordinates complicates the discretization, we found it convenient to recast the equations in the Cartesian coordinate system, H t = φ x x + φ y y (.4 where φ x and φ y are the liquid fluxes in x and y direction, respectively, and are given by φ x = ζ H3 3 φ y = ζ H3 3 ( x H yh and (.43 ( y H + αh3 + xh (.44 3 with ζ Ca ɛ 3 and x +.Sincethefilmthicknessisnotaxisymmetric, all boundary conditions were applied at the disc periphery. Equation (.4 was solved in conjunction with (.43 and (.44 subject to the zero flux condition at the disc periphery which was enforced by setting the thickness to zero at the periphery, H ( r x + y = =. (.45 5

41 Further, since the contact line is assumed pinned, we also set the height to zero at fictitious nodes (Wilson et al.,, (Javier and Kondic, located just outside the disc. This amounts to assuming a zero contact angle, Since we also have H θ r= =,thisimplies H r r= =. (.46 H x = H y = at r x + y =. (.47 Note that in the absence of surface tension, only (.45 is applicable. Finite difference time marching scheme was used to solve the governing equation for thickness H(x, y, t andwassimilartothatofjavierandkondic(javierand Kondic,. The numerical domain was a square defined by x and y. Nodes on and outside the periphery of disc define the boundary nodes. The details of discretization are given in Appendix A. For a given value of α and ζ, the initial condition for the dimensionless film thickness profile was specified as a section of a sphere, { } H(t =,x,y=h R R (x + y (R h max where h max is the dimensional height at the center of the disc and R is the radius of the sphere. With a known thickness profile and the corresponding flux calculated from (.43 and (.44, (.4 was solved to determine the height at the next time step at all the nodes. The predicted thickness profile was compared at different times and the steady state was achieved when the profile became time independent. Simulations were performed for varying grid sizes and time step values to check grid and time-step dependency of the solution. The solution was grid independent after, nodes with a maximum non-dimensional time step of.. 6

42 .4 Results and Discussion.4. Without surface tension Figures.5(a and.5(b compare the non-dimensional film thickness contours obtained fromthe method ofcharacteristics (.3 with those obtained numerically by solving (.4 without the surface tension term. Recall that (.3 requires the knowledge of Φ(q topredictthecontours. Here,weobtainedΦ(q fromthenu- merical results (without surface tension effects and then used (.3 to locate the constant height contours. It is clear from the figures that the contourspredicted by (.3 show an excellent agreement with the numerical solution obtained in the absence of surface tension. This shows that the numerical procedureissound and predicts profiles in agreement with the analytical solution. As will be shown later, small wiggles (artifacts of numerical error observed in some of the contours predicted numerically were eliminated on inclusion of the surface tension term. (a Eq. (.3 Numerical (.4, no surface tension (b Eq. (.3 Numerical (.4, no surface tension.5.5 y.5 H =.7 H =.9 H =.85 H = x y.5 H =.94 H =.79 H =.9 H = x Figure.5: Constant thickness contours for (a α =.58, (b α =.9. Figure.6(a compares the quadratic thickness profile (Φ(q determined from (.5 with the numerical predictions for α =.9.Thequadraticprofileagrees well with the numerically determined profile though the location of maximum thickness (q max determinedusingthequadraticassumptiondifferedslightly from the numerical results. Figure.6(b shows that the contours predictedby(.3 7

43 (a Quadratic profile (.3 & (.5 Numerical (.4, no surface tension (b Eq. (.3 Numerical (.4, no surface tension.5.5 Φ (q = h/h y H = H =.79 H = q = x q = q max H = x Figure.6: Comparison of (a thickness profile (Φ(q predicted by (5 with that obtained numerically ((8, without the surface tension effects and (b constant thickness contours, obtained using the quadratic profile for Φ(q, for α =.9. and (.5 are also in good agreement with the numerical results. This shows that the assumption of a quadratic thickness profile for Φ(q capturestheessential features of the profile for small α (α <.5. However, the predicted quadratic thickness profile deviated from that predicted numerically for larger values of α. These results suggest that (.3 along with the assumed quadratic profile predicts film thickness contours in agreement with those determined numerically ((.4, without the surface tension effects only for small values of α. Recall that the the film thickness profile is indeterminate in the absence of surface tension and so one may wonder as to how our numerical procedure gives a unique solution. We believe that our numerical method with zero surface tension will suffer from numerical diffusion resulting from the discretization error. For example, the discretization of the first order derivative using central difference scheme leads to an error that is proportional to the third order derivative of the dependent quantity. And it is the latter that will play the role of surface tension in deciding a unique profile. As we shall see later, it is indeed fortuitousthat the small discretization errors in the absence of surface tension lead to profiles that are close to those calculated with surface tension term and those measured 8

44 experimentally. Next, we compare the measured profiles with those predicted by (.4. when surface tension is included. With surface tension Film thickness profile obtained numerically (.4, both with and without surface tension effects, and by the asymptotic analysis (.4 are compared with the measured profiles in figures.7,.8 and.9. The maximum error in the measurement was less than 5 % of the average film thickness. The film thickness profiles are plotted along lines of constant x and y. Theprofilesalongy =-.4,and.4show more liquid in the left half of the disc, where gravity and viscous forces oppose each other. It is clear from the figures.7 and.8 that the numerically predicted profiles with and without surface tension differ slightly for small values of α. This is also confirmed by the numerical value of ζ Ca ɛ 3 5 which is small, impling that the surface tension term will be significant only inthepresenceof large curvature variations. Though the thickness profiles obtained on neglecting surface tension effects show small wiggles in some plots, there is an overall good agreement with the experiments. For larger values of α (figure.9, the predicted thickness profile in the presence of surface tension agrees well with the measurements. However, significant deviations can be observed for profiles obtained in the absence of surface tension (e.g. see plots for y =andx =.4. Figures.7 and.8 show that the predicted film thickness profiles using the asymptotic analysis (.4 compare very well with the experimental measurements. However, increase in the value of α, leadstoslightdeviationofthepredictedfilmthickness profiles from the experiments (figure.9. For low values of α (figures.7 and.8, the predicted thickness profiles with and without surface tension are symmetric about x-axis (see plots for x =-.4,,.4 while for larger values (figure.9 asymmetry is observed in the profiles that account for surface tension (see x =-.4. Thissuggeststhattheasymmetry about θ = π is a consequence of the increased surface tension effect and cannot be obtained from a balance of viscosity and gravity alone. This is further supported 9

45 .5 y =.5 x = H (= h/h.5 H (= h/h x.5.5 y.5 x =.4.5 x =.4 H (= h/h.5 H (= h/h y.5.5 y.5 y =.4.5 y =.4 H (= h/h.5 H (= h/h x.5.5 x Figure.7: Comparison of numerical results with (solid line and without (dashed line surface tension effects with experiments ( alongdifferentconstantx and y lines. The dotted line represents the asymptotic solution. Here, the results are shown for α =.9,h =.3mmandCa =6.56,whichcorrespondstorotation rate of 4 rpm, liquid volume of 5 ml and µ =35Pa sona7cmdiameterdisc. 3

46 .5 y =.5 x = H (= h/h.5 H (= h/h x.5.5 y.5 x =.4.5 x =.4 H (= h/h.5 H (= h/h y.5.5 y.5 y =.4.5 y =.4 H (= h/h.5 H (= h/h x.5.5 x Figure.8: Comparison of numerical results with (solid line and without (dashed line surface tension effects with experiments ( alongdifferentconstantx and y lines. The dotted line represents the asymptotic solution. Here, the results are shown for α =.58, h =.3mmandCa =8.9whichcorrespondstorotation rate of rpm, liquid volume of 5 ml and µ =35Pa sona7cmdiameterdisc. 3

47 .5 y =.5 x = H (= h/h.5 H (= h/h x.5.5 y.5 x =.4.5 x =.4 H (= h/h.5 H (= h/h y.5.5 y.5 y =.4.5 y =.4 H (= h/h.5 H (= h/h x.5.5 x Figure.9: Comparison of numerical results with (solid line and without (dashed line surface tension effects with experiments ( alongdifferentconstantx and y lines. The dotted line represents the asymptotic solution. Here, the results are shown for α =.6, h =.3mmandCa =4.45whichcorrespondsto rotation rate of rpm, liquid volume of 5 ml and µ =35Pa sona7cmdiameter disc. 3

48 by the sin θ term appearing in O(αζ termin(.4. Further,computationsshow that asymmetry about θ = π is observed for α>.9. It should also be noted that for large values of α (α >., liquid volume could not be conserved with, nodes due to the increasing height gradients. For example, a steady state solution with volume conservation was obtained for α =.6onlywhenthenumberof nodes was increased to,5. Given the long computation times required for large α, werestrictedoursimulationstoα<.. The plots for y =infigures.7,.8and.9showthatthelocationofmaximum thickness moves towards center of the disc with decreasing α for a given liquid volume (or average thickness h. This is because of the increased viscous force that can carry more liquid towards the right half of the disc resulting in a more symmetric thickness profile with negligible curvature variations. Prediction of the maximum volume supported by a rotating disc Experiments were performed to determine the maximum volume supported by the disc for varying rotation rates and for liquids of viscosity 5 Pa sand35 Pa s. It was found that the value of α evaluated at the maximum volume was constant and approximately equal to.3 for all the experiments (figure.. Interestingly, the analytically predicted value of α max (.85 from (.3 and (.5 is close to the observed result. In other words, the maximum supported µωr volume is V max.7 5.Thisisreminiscentofasimilarresultobtainedfor ρg the flow of viscous liquid coating the outside surface of a rotating cylinder (Moffatt, 977 where the maximum supported volume is once again characterized by a balance between gravity and viscous forces with the maximum volume per unit µωr length of the cylinder (of radius R obtained as Despitetheagreement ρg between analytical prediction and observation, it should be recalledthat(.3 and (.5 predict symmetric profiles about θ = π while the observed profiles were highly asymmetric. 33

49 .4 35 Pa s 5 Pa s.3 α max Ω (rpm Figure.: Values of α for maximum possible liquid volume supported by a rotating disc of diameter 7 cm as a function of rotation rate for two different viscosity liquids..5 Conclusions We have studied both theoretically and experimentally the flow of Newtonian viscous liquids coating a vertical rotating disc in the creeping flow regime. A lubrication analysis accounting for gravity, viscosity andsurfacetensionresulted in a time evolution equation for the film thickness. The analysis predicted contours of constant height to be circular and symmetric about the horizontal axis. The constant height contours predicted by the analytical solution compared well with those obtained experimentally for small liquid volumes orhighrotationrates. However, for obtaining a unique solution, the surface tension effects need to be included. An asymptotic analysis accounting for surface tension resulted in an approximate solution. For moderate liquid volumes (.9 <α<., the experimentally observed asymmetry in the thickness profiles was captured on inclusion of the surface tension term in the time evolution equation. Experiments also showed that the maximum liquid supported by a rotating disc under creeping flow conditions is characterized by α ρgh µωr.3. 34

50 Chapter 3 Ring shaped structure in free surface flow of non-newtonian liquids Abstract We report the formation of a ring like structure (donut shaped in the flow of particular non-newtonian liquids coating a rotating vertical disc. Experiments were performed with a known volume of the liquid and at varying rotationrates such that the inertial effects were negligible. Liquid injected on the rotating disc initially coats the surface uniformly, which then redistributes itself such that at steady state a significant amount collected into a circular ring, off center with the axis of rotation. Beyond a critical rotation speed and for a given liquid volume, the ring formation did not occur. The ring formation was not observed in the case of Newtonian liquids. Rheological measurements along with the shear rate calculations suggest that the shear thinning nature of the liquids may play a role in the observed ring formation. 35

51 3. Experiments It is well known that particular non-newtonian liquids show adiverserangeof interesting flow phenomena(bird et al., 987. In this work, we report a unique flow pattern of particular non-newtonian liquids coating a rotating vertical disc. Our experiments were performed on a vertical rotating disc shown in the figure.. Varying liquid volumes were selected so that the liquid coated the disk completely with the contact line pinned at the circumference of the disc. The latter was ensured by fabricating discs with sharp edges. 3.. Experiments with shampoo Acommerciallyavailableshampoo,Sunsilk R (from Unilever PLC was chosen for the experiments. The steady shear viscosity and the first normal stress difference for shampoo were measured at 5 o Cand3 o Conaconeandplategeometryof cone angle. rad and plate diameter 5 mm (figure 3.. The shampoo is shear thinning over the measured range but does not exhibit any significant normal stress differences for shear rates less than s. The storage and loss moduli of the shampoo (figure 3.(a were measured on the same geometry. Figure 3.3 presents the viscosity variation with the shear stress for stress ramp experiments performed on the shampoo at 5 o Cand3 o Contheconeandplategeometry. In all experiments, the disc surface was wetted mildly with a moist cloth prior to liquid injection to facilitate easy spreading of the liquid. Detailed experiments were performed with the shampoo on 7 and 9 cm diameter discs at liquid volumes varying from 4 ml to 7 ml and rotation rates from to 4 rpm. The Reynolds numbers, Re ρωrh µ,wasnegligibleinallcases. Here,ρ and µ are the density and the low shear viscosity of the liquid, Ω is the rotation rate of the disc, R is the radius of the disc and h is the average film thickness. Photographs in figure 3.4 show the dynamics of ring formation for 6 ml of shampoo on a 9 cm diameter disc rotating at rpm in the clockwise direction. Here, photographs are shown after, 3, 4 and 7 minutes of complete injection of 36

52 Viscosity (Pa.s Viscosity of shampoo at 5 Viscosity of shampoo at 3 Viscosity of dispersion at 8 N for shampoo at 5 N for shampoo at 3 N for dispersion at 8 o C o C o C o C o C o C N (Pa..... Shear Rate (/s - Figure 3.: Steady shear viscosity and first normal stress difference (N measured on a cone and plate geometry for shampoo and colloidal dispersion. 37

53 (a G and G (Pa 3 G at 5 o C G at 5 o C G at 3 o C G at 3 o C Freq (rad/s (b 3 G at 8 o C G at 8 o C G and G (Pa Freq (rad/s Figure 3.: Small angle oscillatory shear measurements for (a shampoo and (b colloidal dispersion. 38

54 o 5 C o 3 C Viscosity (Pa.s.. Shear stress (Pa Figure 3.3: Stress ramp experiment for shampoo. liquid on the rotating disc. It can be seen from the figure that the liquid starts collecting on the left half of the disc and forms a faint ring like structure within -3 minutes of the start of the experiment. A small blob of liquid appears close to the periphery of the disc on the left half of the ring. After 7minutes,theflow reaches steady state with the ring becoming distinct. It is important to note that the position of the ring (and the blob does not change in the laboratory frame of reference. Ring formation was observed on both the discs. Figure 3.5 shows photographs at different rotation rates ( - 3rpmafter minutes of complete injection (steady state for 6 ml of shampoo on a 9 cm diameter disc. It is clear from the figure that the location of the ring is the same irrespective of the rotation rates though the ring appears diffuse with increasing rotation rate. Also, the shampoo blob moves slightly upward at higher rotation rates. Experiments were also performed at a fixed rotation rate with varying volumes. Figure 3.6 shows the ring formation at rpm for shampoo at volumes of 4, 5 and 6 ml on the 9 cm diameter disc. The ring appears more distinct at 5and6mlthanatthelowervolume. Experimentsconductedoverarangeof 39

55 (a (b fluid blob (c (d Figure 3.4: Dynamics of ring formation for 6 ml of shampoo coating a 9 cm diameter disc rotating at rpm after (a min, (b 3 min, (c 4 min and (d 7 min at 3 o C. The arrow shows the direction of rotation. rotation speeds and liquid volumes suggest that for a given disc diameter and rotation speed there exists a critical volume below which the ringisnotformed. For example, no ring was formed when the 9 cm diameter disc rotating at rpm was coated with liquid volumes less than 4 ml. This observation is in line with the trend observed in the images of figure 3.6 where the ring becomes more distinct at higher liquid volumes. Also, beyond a critical rotation speed and for a given liquid volume, the ring formation did not occur. For example, for the 9cmdiameter disc at rotation rates greater then 4 rpm no ring was formed. Film thickness profile was measured at steady state along y =forashampooexperiment(at 3 o C performed on a 7 cm diameter disc rotating clockwise at rpm for5mlof shampoo volume. Figure 3.7 shows the nondimensional film thickness (H profile, rendered dimensionless by the average film thickness (h =.78 cm. The plot 4

56 (a (b (c Figure 3.5: Ring formation on a 9 cm diameter disc for 6 ml of shampoo at rotation rates of (a rpm, (b rpm, and (c 3 rpm at 3 o C. The arrow shows the direction of rotation. shows steep variation in the film thickness values on the left half of the disc where it rises sharply from H(x = = to a maximum value H(x =.75 =.8. The thickness next falls equally rapidly to H =.75 at x =.6 followedbya more gradual decrease to H =.4 atx =.. For higher values of x, thefilm thickness again increases to H =.5 atx =andthendecreasesmonotonically to H =atx =. Itshouldbenotedthatwhiletheringformationincaseof the shampoo was robust and steady, drying could effect the profile after about an hour when the set-up was exposed to the ambient conditions. Consequently, reliable film thickness measurements could only be done along oneoftheaxisin agivenexperiment. It is interesting to compare qualitatively the film thickness distributionfor Newtonian liquids with the present results. In both the cases agreateramount 4

57 (a (b (c Figure 3.6: Ring formation on a 9 cm diameter disc rotating at rpmforshampoo volumes (a 4 ml, (b 5 ml and (c 6 ml at 3 o C. The arrow shows the direction of rotation. of liquid collects on the half where gravity and viscous forces oppose each other (left half for our experiments. But while there is only one maxima from where the film thickness decreases monotonically, a ring like structure is formed in case of the non-newtonian liquid. Since the shampoo is a water based solution, experiments were repeatedwith the disc enclosed inside an acrylic cylinder to eliminate drying effects. Here too, the liquid collected into a ring on the left half of the disc confirming the absence of drying effects on the flow. Interestingly, the flow could be switched from the ring state to the no ring state (and back by changing the ambient temperature during the course of the experiment. For example, the ring disappeared when the ambient temperature was reduced from 3 o Cto5 o Cfor9cmdiameterdiscrotatingatrpmand 4

58 supporting 6 ml of shampoo. The ring reappeared when the temperature was raised back to 3 o C. This reconfirms the absence of drying effects on the ring formation and suggests that the phenomenon is sensitive to small changes in the rheology of the liquid. We also observe the ring formation phenomenon with other shear thinning liquids. These include an aqueous colloidal dispersion of latex particles and an aqueous solution of CTAB and sodium salicylate and will be discussed next. 3.. Experiments with an aqueous colloidal dispersion of latex particles Experiments were performed with an aqueous colloidal dispersion, Dur-O-Set C335 R (from Celanese Inc., on the same geometry. The dispersion contained % (w/w Polyvinyl Acetate Homopolymer in water. The steady shear viscosity and the first normal stress difference for the Dur-O-Setdispersionwere measured at 8 o Conaconeandplategeometryofconeangle.radandplate diameter 5 mm (figure 3.. The despersion also shows shear thinning behaviour over the measured rang and do not exhibit any significant normal stress differences for shear rates less than s. The storage and loss moduli for the despersion (figure 3.(b were measured on a 4 mm parallel plate geometry. Figure 3.8 shows the ring formation obtained after minutes of complete injection for 6 ml of colloidal dispersion on a 9 cm diameter disc rotating at rpm in the clockwise direction. The ring formation was robust and highly reproducible Experiments with an aqueous solution of CTAB and sodium salicylate An aqueous solution of.5 M cytyltrimethylammoniam bromide (CTAB and.5 M sodium salicylate exhibits a viscosity that is comparable to the shampoo. The rheology of the solution is reproduced from the study of Won-Jong and Yang (, which was measured on an ARES rheometer (cone and plate geometry of 43

59 5 mm in diameter with the cone angle of.4 rad. Figure 3.9 shows the steady shear viscosity of the same concentration solution over the measured rang of shear rates. It is clear from the figure that solution shows constant viscosityatlowshear rates (till.7 sec andexhibitsshearthinningbehaviorathighershearrates. The oscillatory measurement data are plotted in figure 3.. An aqueous solution of.5 M CTAB (Sigma and.5 M sodium salicylate (Aldrich was prepared. Experiments performed with this solution showed ring formation phenomenon similar to that observed with the shampoo and the colloidal dispersion. An initial uniform coating of the liquid collected into ring shaped structure within 5 minutes of complete injection. The film thickness profile was measured at steady state along y =foranexperiment(at3 o C performed on a9cmdiameterdiscrotatingclockwiseat.5rpmfor.5mlofliquid volume. Figure 3. shows the nondimensional film thickness profile, rendered dimensionless by the average film thickness (h =.39cm. Asinthecaseoftheshampoo experiment, the plot shows a steep variation in the film thickness values on the left half of the disc where it rises sharply from h(x = = to a maximum value h(x =.7 =.75. The thickness then falls rapidly to h =.atx =.57 and stays constant till x =-.9. Forhighervaluesofx, thefilmthicknessagain increases to h =.48 at x =andthendecreasesmonotonicallytoh =at x =. Itshouldbenotedthatwhiletheringformationincaseofthe shampoo and the CTAB - sodium salicylate mixture was robust and steady, drying could effect the profile after about an hour when the set-up was exposed to the ambient conditions. Consequently, reliable film thickness measurements could only be done along one of the axis (such as those shown in figures 3.7 and Shear thinning hypothesis for the ring formation As the full velocity field for the Newtonian liquids is known itmaybeinstructiveto evaluate the shear field to get an insight into the ring formation phenomenon observed in particular non-newtonian liquids. Under the lubrication approximation 44

60 (h R theshearratecanbeapproximatedby [ γ = ( ux ( ] uy + z z (3. where the corresponding velocity gradient in the Cartesian coordinates are given by, u x z = p (z H and (3. ( x u y p z = y + α (z H. (3.3 On neglecting the surface tension terms the dimensionless shear rate for the Newtonian liquids is obtained as, γ = α (H(x, y z (3.4 Thus for a given x and y, theshearrateismaximumatthesurfaceofthedisc. Further, for z =,thehighestshearrateoccursatthelocationofthemaximum thickness. This suggests that in case of shear thinning liquids, such as those reported here, the thinning of the liquid in regions of large thickness may drain the liquid to other sections of the disc resulting in the ring shaped topography. For the clockwise rotation of disc, majority of the liquid collects on the left half of the disc where the gravitational and viscous forces oppose eachother. While the liquid close to the disc gets carried over from the left half to the right half of the disc, the liquid close to the liquid-air interface (or away from the disc surface falls back on the left side resulting in recirculation and consequently higher shear rates. This is further supported by (3.4 where large film thickness on the left side of the disc (for clockwise rotation lead to high shear rates in those regions. This causes a reduction in liquid viscosity leading to the depletion of liquid from the recirculating regions and the formation of the ring shaped structure. The proposed mechanism is supported by the photographs in figures 3.5 and 3.6 where increased rotation rate at a fixed volume and a decreased liquid volume at afixedrotation rate lead to diffused rings. To correlate the results of the shampoo experiments with the proposed mechanism we calculated the value of α for power law liquids (see chapter 4 for the 45

61 expression of α for power law liquids for all the experiments. The values of m and n were determined by curve fitting the power-law model to the shampoo rheology data (figure 3.(a. Figure 3.(b and 3.(c plot the average film thickness as afunctionofα for experiments performed on shampoo at 5 o Cand3 o Con7and 9 cm diameter discs. Here,the closed symbols represent the experiments where ring formation was observed while the open symbols represent thosewithoutring formation. It is clear from the figure that for a given temperature there is a critical value of α above which the ring formation was present. This is in agreement with our earlier observation of a minimum liquid volume and maximum rotation speed for the ring formation. At 3 o C, the critical α is.5 while at 5 o Cit is.4. Further, it should be recalled that the flow could be switched from the ring state to the no ring state by decreasing the ambient temperature. This observation is also in line with the above analysis since the ring formed for the 9 cm diameter disc rotating at rpm and supporting 6 ml of shampoo (h =.95 cm with the corresponding value of α at 3 o Cbeing.8.Butondecreasingthe temperature to 5 o C, for which the α reduces to., the ring disappeared. It is quite possible that the existence of a critical α may be related to the shape of the viscosity versus shear rate profile for the shampoo. Specifically, the viscosity is constant at both temperatures for shear rates below. s.thusonemaynot expect ring formation at low values of α, whichcorrespondstolowshearrates. The small angle oscillatory shear measurements presented in figures 3.(aand (b and in figure 3. give a measure of the elasticity of the material with the characteristic relaxation time, which is roughly the longest time required for the elastic structures in the liquid to relax, obtained from the inverse of the crossover frequency. In the case of shampoo (3.(a, the relaxation times of.45 s and.8 s at 5 o Cand3 o Crespectivelyaremorethantwoorderslowerthan the typical flow time scale (Ω andaboutanorderofmagnitudesmallerthan the reciprocal of the characteristic shear rate, h. Thus the value of Deborah ΩR number for shampoo is of the order of 3 which shows that elastic effects are negligible in the flow. In the case of CTAB - sodium salicylate solution (3., 46

62 the relaxation time ( s at 3 o Ciscomparablewiththetypicalflowtimescale. Further, the oscillatory shear data for the colloidal dispersion (figure 3.(b are different from that of shampoo and CATB - sodium salicylate solution with G > G over the entire measured frequency range. This shows that the elastic nature of the liquids are significantly different even though the ring formationisobserved in all three cases. This reinforces the proposed shear thinning hypothesis for the ring formation. We also do not expect normal stress differences to cause the ring formation since otherwise the hoop stress generated by the circular streamlines in the left half of the disc would collect more liquid in that region rather than deplete it of fluid. Further, stress ramp experiment performed on shampoo shows gradual decrease in the viscosity (figure 3.3 which suggests absence ofyieldstressinthe shampoo. 3. Conclusions In conclusion, we identify a new phenomenon of ring formationwhenacertainclass of non-newtonian liquids coat a vertical rotating disc. Such aphenomenonisnot observed for Newtonian liquids with comparable viscosities. Fluids with both finite (CTAB - sodium salicylate solution and negligible (shampoo and colloidal dispersion elastic effects showed the ring formation phenomenon, confirming the absence of the elastic effects on the ring formation. Also, the fluidsweredevoid of yield stress suggesting that yield stress is not the cause for the ring formation. We have proposed a qualitative explanation for the ring formation based on the shear rate dependent viscosity. In the next chapter, we present the lubrication approximation for shear thinning liquids. The time evolution equations were derived for two different shear thinning rheological models, power law and Ellis model, to verify our proposed hypothesis. Best fits to the shampoo viscosity data were obtained for both the models and the time evolution equations forthefreesurface were solved numerically. 47

63 Figure 3.7: Measured nondimensional film thickness profile along y =fora shampoo experiment (at 3 o C on a 9 cm diameter disc rotating clockwise at rpm for 5 ml of shampoo volume (h =.78 cm Figure 3.8: Ring formation on a 9 cm diameter disc rotating at rpmafter min for 6 ml of aqueous colloidal dispersion at 8 o C. 48