Chairperson Dr. Sarah Kieweg

Transcription

1 FLUID DYNAMICS OF NON-NEWTONIAN THIN FILMS: SQUEEZING FLOW, FILM RUPTURE AND CONTACT LINE INSTABILITY BY Md. Rajib Anwar 06 Submitted to te graduate degree program in Bioengineering and te Graduate Faculty of te University of Kansas in partial fulfillment of te requirements for te degree of Doctor of Pilosopy Cairperson Dr. Sara Kieweg Co-cairperson Dr. Kyle Camarda Dr. Sara Wilson Dr. P. Scott Hefty Dr. Xuemin Tu Date Defended: June 7 t, 06

2 Te dissertation committee for Md. Rajib Anwar certifies tat tis is te approved version of te following dissertation: FLUID DYNAMICS OF NON-NEWTONIAN THIN FILMS: SQUEEZING FLOW, FILM RUPTURE AND CONTACT LINE INSTABILITY Cairperson Dr. Sara Kieweg Co-cairperson Dr. Kyle Camarda Date Approved: June 3 t, 06 ii

3 ABSTRACT HIV is a global pandemic tat affects millions of lives every year. Vaginal gel formulation as been touted as an effective strategy to figt sexually transmitted diseases (STD) by acting as a pysical barrier, and also by delivering active parmaceutical agents at te intended location. Microbicidal gels can act as a low cost, self-administered metod to prevent HIV and oter STD s. Te overall long term goal of our researc group is to rationally design tese delivery veicles by optimizing teir reological properties for teir specific applications. In tat regard, tere as been an ongoing effort in our group to matematically model te flow of tin films in order to understand te dynamics and stability of tese films, so tat we can understand/predict te beavior of te gel during teir applications. In my dissertation, I ave used tin film lubrication teory to develop matematical models of (a) boundary elasticity-driven squeezing flow of yield stress fluids, (b) rupture/dewetting of Ellis-type sear-tinning liquid films, and (c) contact line instability of Ellistype sear-tinning liquid films. Our investigations sowed tat yield stress can improve te control over te spreading caracteristics of liquids, and also possibly aid in te retention of liquid in place. We ave demonstrated an optimization framework tat can be used to estimate te reological properties of microbicidal gels for different combinations of tissue elasticity, initial bolus volume, and deployment time. Numerical models of te rupture and dewetting of Ellis-type tin films sowed tat te sear-tinning reology could accelerate te rupture process wic could create dry spots earlier tan Newtonian fluids. Dry spots are usually unwanted for bot functional and aestetic purposes. Numerical models of contact line instability ave sown tat te sear-tinning reology increases te finger growt rate for vertical inclines but sows a more complex beavior for flatter inclines. For flatter inclines, te most unstable mode sifts to larger wavelengt but te finger growt rate is dependent on te degree of sear-tinning. Fluids wit low degree of sear-tinning suppress te fingering but fluids wit te igest degree of sear- iii

4 tinning sow an increased growt rate despite te sift in te most unstable mode to larger wavelengt. Tis beavior possibly occurs due to te decrease of apparent viscosity far beyond contact line, especially for fluids wit a narrow range for te Newtonian plateau. Lubrication teory can be applied in te modeling of numerous biological and industrial applications. Te models developed ere are not limited to any specific application, and can easily be modified to study oter applications tat involves non-newtonian fluids. iv

5 ACKNOWLEDGEMENTS I am extremely grateful to Dr. Sara Kieweg, and Dr. Kyle Camarda for teir support and guidance wic as made tis work possible. I am indebted to tem for successfully completing tis researc work. I would like to express my deepest tankfulness to my supervisor, Dr. Kieweg for elping me to overcome te difficulties associated wit my researc and for er valuable advice all along my PD. I express my deepest gratitude to Dr. Sara Wilson, Dr. P. Scott Hefty and Dr. Xuemin Tu for being in my committee. I am also tankful to my colleagues and friends, Bin Hu, Brock Rougton, Tora Witmore, Zouair Talbi, and Kyle Boone for teir support. I am grateful to Bin Hu for elping me wit my researc and providing valuable suggestions about tis work. I would like to tank my parents and my wife, Farana Abedin for supporting me all along my PD and giving me te inspiration to successfully complete te steps of my PD degree. I would also like to tank Dr. Kam Ng, Dr. Dennis Percak and Dr. Bradley Coltrain for teir mentoring and guidance during my internsip at Eastman Kodak. I ave learned a lot about interfacial fluid dynamics at Kodak, and applied some of te numerical tecniques learned tere in my dissertation researc. I would also like to acknowledge my gratefulness to Denise Bridwell for elping me wit various administrative processes wile I was a PD student at te University of Kansas. Tis work was supported by te researc grant from National Institutes of ealt (NIH), Grant # R/R33 AI08697 (from te National Institute of Allergy and Infectious Diseases) v

6 CONTENTS Abstract Contents iii vi. Introduction.. Overview and objective..... Significance of proposed researc in microbicide development 3.3 Review of relevant literature 4.4 Tin film (Lubrication) approximation 7.5 Innovation.. 9. Matematical Modeling of Tissue Elasticity and Gravity-Driven Squeezing Flow of Microbicidal Gels and Optimization of Reological Properties Introduction 0. Problem formulation..3 Results and discussion 7.3. Parametric study in dimensionless form Gel property estimation for optimal coating performance Conclusion Matematical Modeling of Rupture and Dewetting of Ellis-type Sear-Tinning Liquid Films Introduction 5 3. Problem formulation Result and discussion D simulation.. 34 vi

7 3.3. 3D simulation Conclusion Matematical Modeling of Contact Line Instability of Gravity-Driven Flow of Ellistype Sear-Tinning Liquid Films Introduction 4 4. Problem formulation Travelling waves and linear stability analysis D nonlinear simulation Result and Discussion Linear stability analysis Single mode perturbation and comparison wit LSA Random multimode perturbation Conclusion Conclusions and Future work Summary Future work... 6 Appendix A: Regression models for coating lengt and percentage yielding. 64 Appendix A: Derivation of evolution equation for gravity and boundary elasticity driven squeezing flow of Herscel-Bulkley fluid.. 65 Appendix A3: Derivation of evolution equation for rupture of Ellis-type sear-tinning liquid films Appendix A4: Derivation of evolution equation for gravity-driven tin film flow of Ellistype sear-tinning liquid 73 vii

8 LIST OF FIGURES Figure.. A scematic representation of te fluid bolus in vaginal canal (drawing is not to scale). Only alf plane is sown due to symmetry. Te vaginal introitus at te canal opening is towards te bottom. Tis simplified two dimensional model assumes tat te vaginal canal is perfectly aligned wit te x-axis. Placing a constant volume bolus distends te tissue surfaces. Tis serves as te initial state of our model. Flow starts immediately due to te combined effect of gravity and boundary squeezing pressure from te elastic tissue.. 3 Figure.. Effect of wall elasticity on te spreading of bolus. EG ratios used for simulation are, 5, 5 and 5. Plots are sown at t = 0,, 0 and 00. Te value of Bi = 0 and n = used for all simulations. At low EG, te flow is mostly downward due to te dominance of gravity. Higer EG increases te coating lengt and also causes noticeable upward flow. 6 Figure.3. Effect of yield stress on te spreading of bolus. Bi ratios used for simulation are 0, 0., 0.5 and. Plots are sown at t = 0,, 0 and 00. Te value of EG = 0 and n = used for all simulations. Increase in te yield stress (increase in Bi) reduces te coating lengt 7 Figure.4. Dimensionless sear stress contours at t =0 for different Bi ratio = 0. and 0.5. Te value of EG = 0 and n = used for bot simulations. Te tick contour line indicates te yield surface below wic we ave te non-yielding plug layer 8 viii

9 Figure.5. Effect of yield stress on coating lengt and te percentage yielding area of fluid over dimensionless time. Te value of EG = 0 and n = used for all simulations. Te yielding area was calculated by subtracting te plug area from te total fluid area and expressed as a percentage of te total area. 0 Figure 3.. Example plot of dimensionless viscosity as a function of sear stress for Ellis fluid Figure 3.. Scematic of a sinusoidal initial perturbation of te interface of a tin liquid film on a solid substrate.. 3 Figure 3.3. Example of a numerical simulation of te rupture and dewetting process of a Newtonian tin film. Rupture is defined as te point were te film tickness reaces tat of te precursor layer tickness, and dewetting is te process of subsequent ole growt. Te plot sows several snapsots of te film profile as a function of time. Simulation was performed on a grid wit x = Figure 3.4. Te rupture time versus wavelengt for Newtonian and different cases of Ellistype sear tinning fluid. Sear-tinning as negligible effect on te preferred mode of instability. Te grid size for simulation was Δx = Figure 3.5. Comparison of rupture process for Newtonian and Ellis-type sear-tinning fluid. Sear-tinning accelerates te rupture process. Te grid size for simulation was Δx = ix

10 Figure 3.6. Example of 3D rupture and dewetting process of Newtonian and Ellis-type tin films via snapsots at different times. Simulation was performed in a domain of (L x L y ) = (50 50) wit grid size x = 0.5 and y = Figure 3.7. Minimum film tickness as a function of time from te 3D simulation. 39 Figure 4.. Scematic of te cross section of a tin film flowing down an incline due to te action of gravity.. 43 Figure 4.. Plot of dimensionless viscosity as a function of dimensionless sear stress for Ellis fluid Figure 4.3. Growt rate curve of Newtonian tin films for D = 0, and D =. Te markers represent te growt rate at te most unstable wavenumber. Precursor layer tickness of b = 0. was used... 5 Figure 4.4. Te growt rate μ(q) computed from long time beavior of solutions of linear PDE in Equation 4.9 for D = 0, and λ =,3. In eac case, sear-tinning cases β = 0.,, 0 were compared wit te corresponding Newtonian case. Te markers represent te growt rate at te most unstable wavenumber. Inset figures sow a magnified view of te same results. Precursor film tickness b = 0. was used for all te cases.. 53 Figure 4.5. Evolution of a single finger for a sear tinning tin film wit D = 0, β = 0 and λ =. Single mode perturbation was applied by making te transverse direction of te x

11 simulation domain L y equal to te wavelengt wic was calculated from te most unstable wavenumber of LSA result (q = 0.46, l w = π q = 3.7).. 54 Figure 4.6. Comparison of growt rates from single mode perturbation simulation to te growt rate predicted by linear stability analysis Figure D simulation of random multimode perturbations for D = 0 and λ =. Te Newtonian case is compared wit sear-tinning cases of β = and 0. Simulations were performed on a L x L y = [60 96] domain. Please note tat te time of te snapsot for β = 0 case is at t = 0, wereas te oter two cases are at t = 40 (because at t = 40, te finger would be longer tan te simulation domain for β = 0).. 56 Figure D simulation of random multimode perturbations for D = and λ =. Te Newtonian case is compared wit sear-tinning cases of β = and 0. Simulations were performed on a L x L y = [60 96] domain xi

12 LIST OF TABLES Table..Statistical indicators for regression models Table.. Estimation of required reological properties for acieving target coating and maximizing retention for specific tissue elasticity at a given time. Possibility of multiple feasible solutions are demonstrated ere 3 Table 4.. A summary of te comparison between te most unstable wavelengt obtained from LSA, and te caracteristic finger wavelengt obtained from 3D nonlinear simulation. 58 xii

13 Capter INTRODUCTION. Overview and Objective Study of te dynamics of tin film flow is critical for designing effective anti-hiv microbicides tat are intended to be used in a complex biopysical environment. Tin film flows or coating flows ave been studied for various biological, geological and industrial applications suc as microbicidal drug delivery [-6], lung surfactants [7], synovial fluids [8], tear films [9, 0], mud flow [], lava flow [], industrial paint/coating flows [3-5], etc. Te study of tin Newtonian liquid films (e.g. spreading, rupture, contact line instability) as been te subject of extensive investigation for many years [6-8]. However, non-newtonian reology (seartinning, yield stress) plays an important role in te overall flow dynamics and instability of non- Newtonian fluids. Tere is a definite gap in knowledge in terms of understanding te beavior of non-newtonian tin film dynamics and instability. Tese gaps inder te design of effective delivery veicles for microbicide deployment. Our long term goal is to use te knowledge of fluid mecanics and polymer reology to simulate te pysical beavior of drug delivery systems in complex environments, and to apply tis knowledge in designing delivery veicles containing STI preventive and/or terapeutic agents. Te objective of tis dissertation is to investigate te effect of non-newtonian reology on tree

14 different scenarios involving tin films: squeezed film flow, film rupture/dewetting, and contact line instability. Te overall ypotesis of tis project is tat non-newtonian reology of tin films can be used as important design variables for te development of microbicide gels, and matematical modeling of non-newtonian tin film dynamics can elp in understanding te beavior of te delivery veicles in-vivo. Te rationale for te proposed researc is tat, once te dynamics of non-newtonian tin films are clearly understood, we can make informed decisions about designing te delivery veicles. We accomplised our objectives troug studies performed under 3 specific aims:. Matematical modeling of tissue elasticity and gravity-driven squeezing flow of microbicidal gels and optimization of reological properties. Te approac involved developing a partial differential equation describing te tissue elasticity and gravity-driven flow of non-newtonian microbicidal gels and optimizing te spreading and retention of a gel on epitelial surfaces. Te ypotesis was tat reological properties of gels can be numerically optimized to acieve optimal coverage and retention beavior. Tis is important because it indicates tat yield stress of gels can improve te retention of te gel on te vaginal epitelial surface.. Matematical modeling of rupture and dewetting of Ellis-type liquid tin films. Te approac involved developing a matematical model of tin film rupture/dewetting of sear-tinning fluids and numerically solving te problem to get te rupture time, and investigating te effect of sear-tinning reology on te preferred mode of instability. Te ypotesis was tat sear-tinning of a liquid will cause a different rupture time tan Newtonian tin films and ence affect te function of te film as a pysical barrier. 3. Matematical modeling of contact line instability of gravity-driven flow of seartinning Ellis liquid films.

15 Te approac involved developing a matematical model for contact line instability of seartinning fluid films and exploring te effect of sear-tinning (via Ellis reological model) on contact line instability troug linear stability analysis (LSA) and nonlinear simulation. Te ypotesis was tat te sear-tinning beavior of a fluid will likely affect te contact line instability due to te cange in capillary ridge eigt and/or apparent viscosity cange.. Significance of Proposed Researc in Microbicide Development Te impact of te expanding HIV/AIDS pandemic is devastating. Tere were approximately 34 million people living wit HIV in 0 including.5 million new infections in 0 alone [9]. Studies designed to determine te effect of treating sexually transmitted infections (STIs) from patogens suc as C. tracomatis and N. gonorreae, on HIV transmission rates support tat STI prevention and management is an effective strategy [0, ]. Clinical trials of a vaginal gel formulation of anti-retroviral drug Tenofovir exibited te potential of topically applied gel for te prevention of HIV transmission in women []. Microbicidal gels could be a self-administered and inexpensive metod to prevent transmission of HIV and oter STIs in women. In addition to acting as delivery veicles for anti-hiv and oter anti-std agents, te microbicidal gels could also act as pysical barriers between patogens and biological tissues. Factors suc as extent of coating and its retention on te epitelial surface of te lower female reproductive tract (LFRT) play a vital role in te effectiveness of te gel against transmission of HIV and STD [3]. Te microbicide development strategy (MDS) as identified gaps in knowledge for microbicide development wic includes a lack of knowledge in te veicle design [4]. Veicle design is a problem wit multiple facades including gel reology optimization, biocompatibility, user acceptability and anti-sti activity. Tis project focuses on studying te dynamics of fluid tat may be encountered in a microbicidal gel deployment situation using te principles of interfacial and tin film flow. Previous researc as sown tat typical polymeric gel formulations exibit 3

16 sear-tinning beavior and may also ave a yield-stress [5-7]. Despite te advances in te area of matematical modeling of tin film flow, tere is still lack of knowledge in te field of non- Newtonian liquid film flow. Our contribution ere is to investigate te effect of non-newtonian reology on te dynamics and stability of tin films. Te contribution is significant because it is expected to provide us wit strong matematical tools to predict te dynamics of non-newtonian liquid films. Tis knowledge can be applied to te design of effective delivery veicles. Due to te numerous applications of tin films in biological and industrial applications, tese computational models will also inform researcers involved in oter fields (especially in micro-printing and coating applications) about te important consequences of non-newtonian reology on tin film dynamics. Tese models will also serve as platforms for building more complicated models by including detailed contact line pysics, evaporation, complex reology, surface patterning/flexibility etc..3 Review of Relevant Literature Tin film flows can be matematically described by using te tin film lubrication approximation wic enables us to neglect te inertial effects [8]. Tere as been a progressive development in te study of tin films for various problems involving termocapillary flow, coating flow, pase cange at interface, film drainage etc. Te majority of te studies involving tin film flow ave been conducted on Newtonian fluids. Te Carreau, power-law, and Ellis models are some popular constitutive equations tat can be used to describe te sear-tinning beavior of non-newtonian fluids. Bingam (Newtonian wit yield strengt) and Herscel-Bulkley (seartinning wit yield strengt) constitutive equations are commonly used to represent fluids aving yield strengt. 4

17 Tin film spreading and retention depends on te reological properties of te fluid. Tin film teory as been applied in te study of vaginal drug delivery systems by several researcers in te recent past. Kieweg and Katz et al. performed experimental and numerical studies on microbicidal gel deployment under constant squeezing force and concluded tat squeezing force, gel consistency, sear-tinning beavior and yield stress are strong determinants of te coating performance of gels [3, 5, 6]. Szeri et al. developed a matematical model involving wall elasticity to demonstrate te effect of te compliant vaginal wall on te deployment dynamics of microbicidal gels and indicated te important role tat vaginal biomecanics played in gel deployment [4]. Tasoglu et al. sowed tat yield stress could improve te long time retention performance of a Carreau-like fluid [6]. Tese studies were clear indicators tat fluid spreading and retention could be rationally controlled by optimizing te reological properties of te fluid. However, tere is a definite lack of any squeezing flow model of yield-stress fluid tat describes te sort term deployment dynamics of microbicide gels. To close tis gap, we ave (a) developed a matematical model for understanding te effect of elasticity of te tissue and yield-stress reology of te fluid on te spreading of microbicidal gels, and (b) demonstrated an optimization framework to estimate gel reological properties for a given tissue elasticity in order to acieve bot optimal tissue coverage and maximum retention. Researc on te spontaneous rupture and dewetting of tin films as been motivated by various biological and industrial needs [9-33]. In tin films were surface tension plays an important role, a small perturbation of te interface sape can grow and eventually result in te rupture of te film. Rupture is te process were te film tickness approaces zero at a finite time point. Te subsequent spatial growt of te minimum eigt, wic forms dry spots in te liquid film is known as dewetting. Early teoretical works ave predicted te rupture time and lengt-scale of preferred mode of instability of tin films of Newtonian fluids [7, 34]. Te most unstable mode is defined as te wavelengt at wic te initial disturbance grows most rapidly, 5

18 and is te most likely initial spacing between mounds or ridges for a randomly imposed perturbation. However, a number of tese studies used a simplified model of disjoining pressure (te dynamic pressure arising from solid/fluid interaction at molecular level) tat allowed te model to represent te film evolution only up to te point of film rupture. Later works ave used a more complex model of disjoining pressure tat allowed te modeling of te dewetting process followed by rupture [35]. Te growt rate of initial disturbance as been sown to be proportional to surface tension and te contact angle, and inversely proportional to te film tickness and viscosity in Newtonian fluids [35]. Te wavelengt of te most unstable mode (preferred mode) of instability as been sown to be proportional to te film tickness and inversely proportional to te contact angle [35]. Researcers ave investigated te rupture/dewetting beaviors of evaporating films [7, 36], films wit trapped nano-bubbles [37], and colloidal suspension [38]. Evaporation (condensation) leads to faster (slower) growt of initial disturbance [7]. Presence of nano-scale bubbles as also been sown to accelerate te rupture process [37]. Dewetting models of evaporating colloidal suspension as been developed to explain te self-pinning of receding contact line wic is related to te formation of rings of dried particles, also known as te coffee ring effect [38]. Most of te above studies ave considered a Newtonian fluid beavior. However, te reological beavior of many industrial and biological fluid exibit sear-tinning beavior. To address tis gap in knowledge, we ave addressed te rupture and dewetting beavior of Ellistype fluids tat exibits a low sear rate Newtonian plateau along wit sear-tinning beavior at ig sear [39]. Effects of sear-tinning on contact line instability of tin liquid films ave not been fully caracterized yet. Contact line instability wic causes te formation of fingers downstream as been widely studied for Newtonian fluids by various researcers [6, 40-4]. Random perturbation at te contact line can cause te moving front to corrugate, and tis beavior is a function of bot surface tension and inclination angle [43]. Formation of fingers at te moving front is undesirable 6

19 since it may lower te quality of coating or negatively affect te barrier properties by leaving dry spots. Terefore, contact line instability is very important in any coating application. Yield-stress as been sown to suppress contact line instability [44]. Our group as already performed numerical investigation on te effect of power-law reology on contact line instability and te results sowed tat sear-tinning suppresses fingering instability [45]. However, te power-law model predicts an unrealistic infinite viscosity at low sear rates and tis beavior may ave an impact on te contact line instability. Terefore, we ave cosen to subsequently study te contact line instability penomena using te Ellis fluid model wic as a more robust representation of sear-tinning fluids troug te inclusion of a low sear rate Newtonian plateau. To strengten our current knowledge on te effects of non-newtonian reology on contact line instability, we ave developed matematical models of te spreading of Ellis fluids, and investigated te effect of sear-tinning on contact line instability by performing linear stability analysis and non-linear simulation. Collective evidence reviewed in tis section strongly supports te conclusion tat tere is a definite lack of knowledge in terms of matematically explaining te beavior of non-newtonian tin film flows. Tis knowledge is important as it will elp in te development and optimization of various industrial processes, te understanding of biological penomena and in our effort to develop microbicidal delivery veicles of non-newtonian fluids..4 Tin Film (Lubrication) Approximation Lubrication flow is a common penomenon observed in our day to day life suc as application of eye drops or microelectronic printing. Here, te basis of tin film flow is discussed. Please see reference [8] for furter details. A viscous fluid between two rigid boundaries (z = 0 and z = (x, y)) wit a steady flow is considered ere. Let U be te orizontal flow speed and L 7

20 be te orizontal lengt scale for te flow. We start wit te incompressible Navier-Stokes equation, and te continuity equation: u t (.) u u p g u u 0 Here, ρ is density, g is gravitational acceleration, η is dynamic viscosity, u is te velocity vector, and p is pressure. In tis discussion, a two dimensional flow is considered to be one dimensional wen te longitudinal dimension is significantly longer tan te transverse dimension as given below. L (.) Since tere will be no slip condition at te boundaries (z = 0 and z = ), ten u will vary by an order of U over a distance of order in te z-direction. Terefore, te longitudinal gradients of u, u z and u z will be of te order U and U respectively. Similarly, te orizontal gradients of u, u x and u x will be of te order L U and U L respectively and tey are weaker compared to tat in te longitudinal direction. Considering te above equation, te viscous term can be approximated as sown below: u z u (.3) Here, is te kinematic viscosity. Te order of magnitude of te two components of te two terms in te equation of motion can be estimated as given below: 8

21 U ( u ) u (,, ) (.4) L L u z U (,, ) L (.5) is of te order Due to te incompressibility of te fluid, te continuity equation is u 0. Since, w z U L, w is of te orderu L. Taking into account tese estimates, ( u. ) u can be neglected if te following equation is satisfied. UL ( ) L (.6) From te above equation, it can be seen tat te conventional Reynolds number Re UL does not need to be low. Te tin film teory is based on tat te inertial force is eiter comparable or lower tan te viscous force. Even wen te Reynolds number is large, te condition in te above equation can be satisfied, allowing te viscous force to dominate if only L is low enoug..5 Innovation Tis work is innovative and original because (a) it models te squeezing flow dynamics of yield-stress fluid, (b) it investigates te rupture instability of Ellis-type tin films for te first time, and (c) it investigates te contact line instability of Ellis-type sear-tinning fluid films for te first time. Overall, tis knowledge will elp us to acieve predictable coating beavior in important biological and industrial applications. Te researc outcomes are significant to microbicide development as well because tey will aid in te design of effective delivery veicles. 9

22 Capter MATHEMATICAL MODELING OF TISSUE ELASTICITY AND GRAVITY-DRIVEN SQUEEZING FLOW OF MICROBICIDAL GELS AND OPTIMIZATION OF RHEOLOGICAL PROPERTIES Te contents of tis capter was publised in Md Rajib Anwar, Kyle V. Camarda, and Sara Kieweg (05). Matematical model of microbicidal flow dynamics and optimization of reological properties for intravaginal drug delivery: role of tissue mecanics and fluid reology. Journal of Biomecanics, 48(9), Introduction Recent clinical trials of te vaginal gel formulation of te anti-retroviral drug Tenofovir ave sown te efficacy of topically applied microbicidal gels against HIV transmission to women [] Microbicidal gels could provide us wit a self-administered and low cost strategy to figt HIV and oter sexually transmitted infections (STIs). A microbicidal gel is deployed as a delivery veicle of anti-hiv and oter anti-sti agents, and it is also used to act as a pysical barrier between te patogens and te biological tissue. Te efficacy of a microbicidal gel depends on te extent of coating and retention on te epitelial surface of te lower female reproductive tract (LFRT) [3] 0

23 Te coating mecanism of a microbicidal gel is complex in nature and driven by te combined effect of pysical mecanisms in te LFRT and te reological properties of te gel. Gravity and te squeezing force arising from te elasticity of vaginal tissue are pysical mecanisms tat play important roles in te coating mecanism. Variation of te elasticity of vaginal tissue can cange te extent of coating of tese gels due to different magnitudes of te resulting squeezing force. Tese polymer gels usually exibit sear-tinning beavior and may also ave a yield stress [5-7]. Certain important anatomical and pysiological features of te vaginal environment, suc as ph, volume and composition of vaginal fluid, sould be kept in mind wile designing an effective microbicidal gel. Te long term goal of our researc is to develop a framework for rational design of microbicide gels wit active ingredients. As a step towards acieving tat goal, we ave used squeezing flow mecanics and polymer reology to develop a framework for optimizing te gel reological properties to acieve optimal spreading and retention wic may be furter extended later by incorporating target properties suc as biocompatibility and activity against STIs. Studies on tin film flows for biological applications are growing in number [8]. Most common biological applications include lining of pleural surfaces of te lung [7], flow of synovial fluid between joints [8], and tear film between te eye and eyelid [9, 0]. Tin film teory as been applied in te study of vaginal drug delivery systems by several researcers in recent past. Kieweg and Katz et al. performed experimental and numerical studies on microbicidal gel deployment under constant squeezing force and concluded tat squeezing force, gel consistency, sear-tinning beavior and yield stress were strong determinants of te coating performance of gels [3, 5, 6]. Szeri et al. developed a matematical model involving wall elasticity to demonstrate te effect of compliant vaginal wall on te deployment dynamics of microbicidal gels and indicated te important role tat vaginal biomecanics played in gel deployment [4]. Tasoglu et al. sowed tat yield stress could improve te long time retention performance of a Carreau-

24 like fluid [6]. Te goal of tis study is (a) to develop a matematical model for understanding te effect of elasticity of te tissue and yield-stress reology of te fluid on te spreading of microbicidal gels, and (b) to demonstrate an optimization framework to estimate gel reological properties for a given tissue elasticity in order to acieve bot optimal tissue coverage and maximum retention.. Problem Formulation In tis section, we develop te nonlinear partial differential equation tat describes te evolution of te fluid/solid interface as a function of time and space, (x, t). Figure. sows te scematic of te flow conditions of te two dimensional (-D) model i.e. one dimensional (-D) spreading, were te fluid is spreading due to te gravity and an elastic response from te vaginal tissue. Tis model considers a omogenous tissue compartment wic represents te entire vaginal tissue (including epitelium, smoot muscular layers, and te underlying lamina propria), as well as furter surrounding/underlying structures wic may contribute to te closing pressures exerted on te gel in te vagina (e.g. nearby bladder wall). Te flow domain is symmetric about te center axis since we are considering only vertical flow (along x-direction) in tis study. In tis approximation, te portion of te vaginal axis sown in Figure. is aligned wit te x-axis, and gravity is downill along te x-axis. In a uman, te orientation of te vaginal axis wit respect to te gravity depends on bot te location along te s-saped vaginal axis and te orientation of te woman. Te axis orientation in tis model allows consideration of te full range of te impact of gravity on a gel.

25 Figure.. A scematic representation of te fluid bolus in vaginal canal (drawing is not to scale). Only alf plane is sown due to symmetry. Te vaginal introitus at te canal opening is towards te bottom. Tis simplified two dimensional model assumes tat te vaginal canal is perfectly aligned wit te x-axis. Placing a constant volume bolus distends te tissue surfaces. Tis serves as te initial state of our model. Flow starts immediately due to te combined effect of gravity and boundary squeezing pressure from te elastic tissue. Since te tickness of te gel bolus is muc smaller tan te extent of te spreading, te tin film lubrication approximation [8] allows us to neglect te convective terms in te equations of conservation of linear momentum. Te Herscel-Bulkley constitutive law wic is used to represent te reology of a yield stress fluid is as follows: 3

26 4 z u z u m n zx 0 (.) Here τ zx is te sear stress, τ 0 is te yield stress, u is te velocity in te x-direction, m is te consistency and n is te power-law index. Te elasticity of te vaginal wall is incorporated in te pressure term of te model. A linear elastic tissue beavior is considered wic assumes tat te vaginal wall is omogeneous and isotropic, and te pressure p acting on te fluid boundary due to te tissue elasticity is directly proportional to strain in te distensible tissue. Te pressure for a deformation (x, t) is given by p = E = E (x, t)/t [4]. Here, E is te elastic modulus of te tissue, T is te tickness of te tissue layer and = (x, t) T is te strain in te tissue. We apply a no-slip condition at te fluid-solid interface and zero sear stress at te symmetry line. By using te above pressure formulation, tin film approximation, and Herscel-Bulkley constitutive relation, we arrive at te following nonlinear partial differential equation describing te evolution of te fluid-solid interface. 0 n n Y Y g x T E g x T E n n n n m x t n n n n (.) 0, max 0 g x T E Y Here, Y defines te yield surface. See Appendix A for details of te derivation. Te surface, z = Y(x, t) separates a searing flow region (between te yield surface and te vaginal wall) and te adjacent plug region. Tis central plug region is not a true plug, but rater a weakly yielding zone

27 [46]. For a constant volume spreading suc as in our case, a no-flux boundary condition ( x = 0) was applied at te two ends of te spatial domain wic was taken large enoug to ensure tat te fluid would not reac te ends. Previous literature reported tat te extent of gel coating increases wit te increase of E and n, and decreases wit te increase of m and τ 0 [4, 6]. In order to understand te interplay between te tissue's elastic properties and te fluid's yield stress on te deployment dynamics, we use appropriate scaling to create a non-dimensional evolution equation for te fluid/solid interface (x = x L, = H, t = tu L, P = P ρgt and τ xz = τ xz ρgh, were U = H n + (ρg m) n is te caracteristic flow velocity). t n n EG x n n x EG x n Y n n Y n 0 n (.3) Y Bi max, 0 EG x Tis non-dimensional form provides us wit two important dimensionless variables: te yield stress to gravity ratio Bi = τ 0 ρgh, and te tissue elasticity to gravity ratio EG = E ρgl. Here, H and L are te caracteristic lengts along te eigt and flow direction. A dimensional interpretation, along wit approximate dimensional values for tissue elasticity and gel properties, will be discussed in te Results and Discussion section. 5

28 Figure.. Effect of wall elasticity on te spreading of bolus. EG ratios used for simulation are, 5, 5 and 5. Plots are sown at t = 0,, 0 and 00. Te value of Bi = 0 and n = used for all simulations. At low EG, te flow is mostly downward due to te dominance of gravity. Higer EG increases te coating lengt and also causes noticeable upward flow. In te rest of te capter, we first explore te effects of varying tissue elasticity and yield stress by solving te non-dimensional form of te evolution equation for different ranges of tese parameters. Te nonlinear evolution equation was solved by an implicit finite difference sceme wic is similar to te tecnique used in our previously publised researc [-3]. We used first order backward difference in time and second order central difference in spatial derivatives. Newton s metod was used to solve te system of algebraic equations resulting from te discretization. A spatial mes interval dx = 0.0 and an adaptive time-stepping sceme was used to matc our tolerance requirement of e-6 (default time-step size was set to dt = 0.00). We also monitored te volume of fluid since insufficient spatial or temporal resolution resulted in a violation 6

29 of mass conservation. Next, we demonstrated a framework for optimizing te gel reological properties by leveraging te yield stress of te gel for a given tissue elasticity. Optimization metods are described below in te analysis contained in te Results and Discussion section. Figure.3. Effect of yield stress on te spreading of bolus. Bi ratios used for simulation are 0, 0., 0.5 and. Plots are sown at t = 0,, 0 and 00. Te value of EG = 0 and n = used for all simulations. Increase in te yield stress (increase in Bi) reduces te coating lengt..3 Results and Discussion Reological properties of gels and elasticity of tissue are required as input parameters of te matematical model. Based on measured properties of some commercial gels, a reasonable range of reological properties could be considered as following: m = 0-80 Pa-s n, n = and τ 0 = 0-00 Pa [3, 5]. Exact elasticity of vaginal tissue is still unknown and wide ranges of 7

30 elasticity suc as 3-0 kpa [47], and -36 kpa [48] are reported. Te lengt of te vaginal canal can range from 7-0 cm, wit a wall tickness between cm [4, 49]. For our numerical simulation, a constant tissue tickness of T =.5 cm was considered and represented te entire vaginal wall and te effect of surrounding tissues. Te initial bolus sape was parabolic (see Fig..) wit a lengt L = 4 cm and maximum eigt H = 0.3 cm. Tis configuration yields a bolus volume of.6 ml/widt dimension of te vagina. Figure.4. Dimensionless sear stress contours at t =0 for different Bi ratio = 0. and 0.5. Te value of EG = 0 and n = used for bot simulations. Te tick contour line indicates te yield surface below wic we ave te non-yielding plug layer..3. Parametric Study in Dimensionless Form In tis section, a series of numerical simulations were carried out in dimension-less form as described in Equation.3, to investigate ow te tissue elasticity and gel yield stress influence te coating mecanism. Numerical simulations were performed for EG =, 5, 5 and 5 to 8

31 investigate te effect of tissue elasticity. Using our non-dimensional scaling, EG = -5 corresponds to tissue elasticity of approximately -50 kpa. Results sow tat increasing te magnitude of tissue elasticity increases te extent of coating and also causes noticeable upward flow (Figure.). At low values of EG, te flow is mostly downwards due to te influence of gravity. Effect of yield-stress on spreading was investigated by performing numerical simulations for Bi = 0, 0., 0.5 and. Tis range of Bi number corresponds to a dimensional yield-stress of approximately 0-30 Pa. Results sow tat increasing te yield stress (or Bi ratio) decreases te extent of spreading (Figure.3) by reducing te yielding region of te flow domain. Figure.4 sows te sear stress contours at t = 0 for two different Bi values: 0. and 0.5. Te tick contour line sows te yield surface wic separates te yielding region from te plug flow region. Te fluid is yielding in regions were te dimensionless sear stress is iger tan te Bi value. As sown in Figure.4, te yielding region for Bi = 0.5 is muc smaller tan te yielding region for Bi = 0.. We can easily calculate te percentage of te fluid domain tat is yielding at any given time since we know te location of te yield surface and te fluid/solid interface. Figure.5 sows te influence of yield stress on te coating lengt and percentage yielding area over a period of time. Te percentage yielding drops rapidly over time for te iger yield stress fluid and tis beavior can be leveraged to improve te retention time of microbicidal gels. 9

32 Figure.5. Effect of yield stress on coating lengt and te percentage yielding area of fluid over dimensionless time. Te value of EG = 0 and n = used for all simulations. Te yielding area was calculated by subtracting te plug area from te total fluid area and expressed as a percentage of te total area..3. Gel Property Estimation for Optimal Coating Performance Running a series of numerical simulations provides us useful insigt about te transient coating beavior of gels for different tissue elasticity and reological properties. However, we would also like to estimate te reological properties required to acieve an optimal epitelial coverage and maximize retention. In particular, we would like to optimize delivery for a given tissue elasticity in a target deployment time. Maalingam et al. used a response surface metodology, wic is a visual tecnique, to design semi-solid microbicidal gels based on te squeezing flow model of gel spreading between rigid boundaries [50]. Our approac is to estimate te gel reological properties suc tat it minimizes te percentage yielding at te end of a target deployment time wile acieving te target epitelial coating for a given tissue elasticity. 0

33 Te approac involves tree analysis steps: () we performed a set of numerical flow simulations in dimensional form, () we developed nonlinear regression models for coating lengt and percentage yielding by fitting te output data from te numerical flow simulations, and (3) we solved te inverse problem for several cases. First, we created 500 quasi-random test points witin our parameter ranges by using te sobolset function in Matlab. We used te reological property ranges mentioned previously (m = 0-80 Pa-s n, n = and τ 0 = 0-00 Pa) and a range of tissue elasticity E = -50 kpa for creating te model building set. Tese 500 parameter sets were inputs to te numerical simulation in dimensional form given by Equation.. We considered a maximum deployment time of 80 seconds, relevant for te coitally-dependent drug delivery metod were te user would insert te gel prior to te intercourse. Second, we used nonlinear regression to fit te numerical results (for coating and % yielding) to te parameters m, n, τ 0, E and t. A regression model is needed for te subsequent step in matematical optimization. We cose a regression model-based approac because, we were not optimizing te primary variable of te PDE ((x, t)) and te nonlinear problem ad many local optima. Te resulting regression model for coating lengt and percentage yielding is given in Appendix A. Several models containing various combination of te parameters were generated in te R software. Model selection was based on Mallow s C p statistic wic indicates potential over-fitting of model wit increasing number of parameters. Te model wit good coefficient of determination (adjusted R ) and lowest Mallow s C p was selected. A k-fold cross-validation (for k = 0) was performed to calculate te predictive squared correlation coefficient (Q ). Te calculated Q of te proposed models were very close to R, wic indicates tat te models are adequate for predicting te spreading and percentage yielding of te injected gel bolus over time. Te resulting statistical indicators are reported in Table..

34 Tird, te above regression models were used to solve te inverse problem using a nonlinear optimization metod in order to estimate te gel reological properties (τ 0, m and n) tat result in minimum percentage yielding and acieve te target coating lengt for a given tissue elasticity and deployment time. Te optimization problem was formulated as follows: minimize %Yielding s. t. Coating lengt Target lengt m min m m max n min n n max τ 0 min τ 0 τ 0 max E = Given tissue elasticity t = Deployment time Te optimization problem was solved to local optimality by using te optimization software GAMS and applying te CONOPT nonlinear solver [5]. Table. provides an example list of estimated reological properties for different tissue elasticity and deployment times. Two sets of feasible solutions are reported in te table to empasize te possibility of multiple feasible solutions for nonlinear optimization problems. Solution of te problem tat is obtained may vary depending on te initial guess. Moreover, te solver can be forced to find solutions witin a specific range of a particular reological parameter by enforcing a constraint on te maximum or minimum allowable limit. Tese estimations can provide microbicide designers wit a specific set of target reological properties wic can be used to rationally design gels. Results in Table. sow tat te estimated m is reacing towards te lower limit of our model (0 Pa-s n ) wen longer coating lengts at a relatively lower tissue elasticity are required. Tis indicates an important conclusion: a larger initial bolus volume is necessary if longer coating lengt (wit maximized retention) is required for tissues wit lower elasticity.

35 Table.. Statistical indicators for regression models Model name Adjusted R Q p-value Coating lengt <.e-6 % Yielding <.e-6 Table.. Estimation of required reological properties for acieving target coating and maximizing retention for specific tissue elasticity at a given time. Possibility of multiple feasible solutions are demonstrated ere. Target Tissue Deployment Estimated (m, n, 0) Estimated Estimated % lengt (cm) elasticity (kpa) time (sec) (Pa-s n, no unit, Pa) Spreading (cm) Yielding at te end of deployment time (0.0, 0.9, 45.8) (7.0,.0, 43.67) (5.0,.0, ) (50.0, 0.95, 37.09) (0.0,.0, 30.34) (0.0, 0.9, 5.539) (0.0,.0, 9.468) (6.0, 0.95, 6.667) (0.0,.0,.54)

36 .4 Conclusion Te study as presented a model framework wic can be used to design gel formulations and delivery location (along te vaginal axis) for specific groups of women, wose vaginal tissue properties may vary wit parameters suc as age or parity. Importantly, it demonstrates a framework for optimizing more tan one target beavior simultaneously a metod we can extend in te future to a larger set of target gel functions (e.g. biocompatibility or anti-sti activity). Wile using tis type of regression model based optimization, it is always important to ensure tat an appropriate range of reological properties, tissue elasticity, deployment time and initial bolus volume is cosen for te model building set. Our model neglected te effect of boundary dilution. In te future, we would like to investigate weter boundary dilution as any significant effect witin suc a sort time scale of deployment (-5 minutes). Te results of te presented numerical study will also enable us to investigate te effect of viscoelastic properties of te vaginal and surrounding tissue on te microbicidal flow dynamics. Tis model also assumes te tissue of te vaginal and surrounding structures to be omogeneous and isotropic wic is not te case in reality. Future studies may incorporate layers of different tissues (e.g. vaginal epitelium, smoot musculature, and underlying tissues suc as bladder structures) in te surrounding tissue compartment of te model. Terefore, it is crucial tat efforts are made towards developing new instruments so tat vaginal mecanical properties could be measured in vivo, and te impact of te surrounding underlying tissues on te closing pressures of te vaginal tissue could be quantified and modeled. Tese measured properties can ten be used in numerical models to study te deployment dynamics more accurately. 4

37 Capter 3 MATHEMATICAL MODELING OF RUPTURE AND DEWETTING OF ELLIS-TYPE SHEAR-THINNING LIQUID FILMS 3. Introduction In tin films were surface tension plays an important role, a small perturbation of te interface sape can grow and eventually result in te rupture of te film. Rupture is te process were te film tickness approaces zero at a finite time point. Te subsequent spatial growt of te minimum eigt, wic forms dry spots in te liquid film is known as dewetting. Tin coating films are observed in many industrial and natural applications, suc as microscale printing [33], industrial coatings [3], polymer solar cells [30], tear film [3], lung surfactant [9] etc. A microbicidal gel is anoter example of a barrier film, wic is deployed at te lower female reproductive tract as a delivery veicle of active parmaceutical ingredients against HIV and oter sexually transmitted infections, and it is also used to act as a pysical barrier between te patogens and te biological tissue. In most applications, it is desirable to avoid dry spots or uneven coating for bot functional and aestetic purposes. Early teoretical works ave predicted te rupture time and wavelengt of te most unstable mode (preferred mode) of instability of tin films [7, 34]. Te most unstable mode is 5

38 defined as te wavelengt at wic te initial disturbance grows most rapidly, and is te most likely initial spacing between mounds or ridges for a randomly imposed perturbation. However, a number of tese studies used a simplified model of disjoining pressure (te dynamic pressure arising from solid/fluid interaction at molecular level) tat allowed te model to represent te film evolution only up to te point of film rupture. Later works ave used a more complex model of disjoining pressure tat allowed te modeling of te dewetting process followed by rupture [35]. Te growt rate of disturbance as been sown to be proportional to surface tension and te contact angle, and inversely proportional to te film tickness and viscosity in Newtonian fluids [35]. Te wavelengt of te most unstable mode (preferred mode) of instability as been sown to be proportional to te film tickness and inversely proportional to te contact angle [35]. Researcer ave investigated te rupture/dewetting beaviors of evaporating films [7, 36], films wit trapped nano-bubbles [37], and colloidal suspension [38]. Evaporation (condensation) leads to faster (slower) growt of initial disturbance [7]. Presence of nano-scale bubbles as also been sown to accelerate te rupture process [37]. Dewetting models of evaporating colloidal suspension as been developed to explain te self-pinning of receding contact line wic is related to te formation of rings of dried particles, also known as te coffee ring effect [38]. Most of te above studies ave considered a Newtonian fluid beavior. However, te reological beavior of many industrial and biological fluid exibit sear-tinning beavior. In tis study, we use te Ellis reological model wic describes a sear-tinning fluid tat as a low sear-rate Newtonian plateau.[39] In tis capter, we present a matematical model for te evolution of te rupture and dewetting process of Ellis-type sear-tinning liquid films. Te goal of tis capter is to investigate te effect of sear-tinning on (a) te rupture time, and (b) te lengt-scale (wavelengt) of te most unstable (preferred) mode of instability. 6

39 3. Problem Formulation We consider a tin layer of liquid film or droplet on a plane substrate. Te tickness of te liquid film is represented in dimensional form by (x, y, t ) were x and y axes are ortogonal on te plane of te substrate, and t is te time. We assume an Ellis constitutive model for describing te sear-tinning beavior of te tin film [39]: 0 / (3.) Here, η is te viscosity, η 0 is te viscosity at zero sear stress, τ is te sear-stress at wic te viscosity is reduced by alf, and λ is te sear-tinning index. Wen λ =, te liquid is Newtonian, wile for λ >, te liquid is sear-tinning. At a ig sear stress, te Ellis model exibits power-law type sear-tinning beavior. However, te Ellis model as a low sear stress Newtonian plateau. Since ig sear stresses only occur near te contact line, te Ellis model is a particularly interesting candidate for modeling tin film flows of sear-tinning liquid films because it can capture te Newtonian beavior in te low sear-rate regions. Assuming slow flow and, and small slope approximation of te free surface curvature to te Navier-Stokes equations (te tin film lubrication approximation ), and considering an incompressible liquid, a partial differential equation for te film tickness evolution equation can be derived. Applying a no-slip boundary condition at te substrate and assuming zero stress at te air-liquid interface, we arrive at te 3D tin film (i.e. D spreading) equation: t Q Q 3 0 x x Qy y (3.) 7

40 8 F x Q x 3 F y Q y 3 3 y x F Here, σ is te surface tension, and is te small slope approximation of te free surface curvature. See Appendix A3 for details of te derivation. Te term Π is known as te disjoining pressure. Te simplest model for disjoining pressure is given by 3 A (3.3) Here, A is a van der Waals attraction force between two surfaces, known as te Hamaker constant [5]. Tis model of disjoining pressure was used by many researcers to study te finite-time rupture of tin liquid films [7, 34, 37, 53]. However, a problem wit tis model is tat it cannot be continued past te time of first occurrence of film rupture because te tickness becomes negative. Tis problem can be avoided by considering a more detailed model for disjoining pressure, so tat te dewetting process can also be simulated. In tis study, we use te two term disjoining pressure model introduced by Frumkin [54] and Derjaguin [55]: m n C * * (3.4)

41 Here, C, and te exponents n and m are positive constants wit n > m >. Te local disjoining energy density as a single stable energy minimum at te tin precursor layer tickness, =. Wen te tickness of te film reaces tat of te precursor layer, te disjoining pressure witin te precursor layer exactly balances te capillary pressure. Since te disjoining pressure is assumed to depend on only te local interfacial separation, te validity of te expression in Equation 3.4 also requires small slope approximation. Te tickness of te precursor layer is generally small compared to te tickness of te film, and it allows te motion of te apparent contact line by removing te sear-stress singularity. Te exponents (n, m) are selected to (3,) following previously publised literature [35]. Te term C is given by (see reference [35, 56] for details): C n m n m e * n m n m * e cos (3.5) Here, θ e is te equilibrium contact angle. Te wettability is controlled by te constant, C. Small or zero value of te constant indicates good or perfect wetting conditions, wereas iger values indicate partial wetting scenarios. Te dimensionless formulation is developed by considering appropriate scales following reference [35]. Using initial film tickness, 0, caracteristic lengt scale of l c on te substrate plane, and zero sear rate viscosity η 0 as reference quantities, te transformation to dimensionless variables are, x x l, y y l, t t t 0 c c c (3.6) Te caracteristic time is given by 9

42 t c 4 0lc (3.7) Figure 3.. Example plot of dimensionless viscosity as a function of sear stress for Ellis fluid Following Braun et al. [57], te non-dimensional form of Ellis model can be written as: (3.8) Here, β = ((σ 0 /l 3 c ) τ ) λ describes te degree of sear-tinning. A value of β = 0 returns a Newtonian fluid beavior wereas increasing value of β indicates increased sear-tinning. Te oter sear-tinning parameter λ determines te slope of te power-law type sear-tinning region. Higer value of λ indicates a broader Newtonian plateau. Terefore, fluids wit iger value of λ will act more Newtonian-like at low sear stresses. Figure 3. sows beavior of Ellis fluid as a function of sear stress. 30

43 3 Te 3D dimensionless evolution equation (D PDE), wit tese scaling becomes: y Q x Q t y x (3.9) F x Q x 3 F y Q y 3 3 y x F m n C * * 0 * m n l m n C e c Here, C is te dimensionless contact angle parameter tat defines te wettability. We first perform a rupture and dewetting study in D (D PDE) (a) to evaluate te rupture time of sear-tinning fluids compared to Newtonian fluids, and (b) to evaluate te effect of perturbation wavelengt on te rupture time. Te D evolution equation is given by 0 3 F x x x t (3.0)

44 3 F x x For te D numerical simulation, we use an initial condition in te form x, 0 0.cosx L /, were L is te lengt of te domain. Periodic boundary condition is enforced at te two ends of te domain. We investigated te effect of sear-tinning by performing numerical simulations wit = 0.,, 0 and =, 3. We ave compared te scaled rupture time wit tat of te Newtonian fluid to investigate te influence of sear-tinning on unstable film rupture. Figure 3. sows a scematic of te form of te initial sinusoidal perturbation. For te 3D simulation, we use an initial condition in te form x y,0 x, y,, were is a random number between 0.0 and 0.0. Periodic boundary conditions were enforced at te domain boundaries. Te simulation was performed in a domain of (L x L y ) = (50 50). Figure 3.. Scematic of a sinusoidal initial perturbation of te interface of a tin liquid film on a solid substrate. All simulations were performed for precursor layer tickness, 0 = 0.. Second order central difference sceme was used to discretize te spatial variables and time integration was 3

45 performed using Matlab s variable step ODE5s solver wit default tolerance of E-6. Volume of te fluid was monitored as a function of time since insufficient spatial or temporal resolution would violate te conservation of mass. 3.3 Results and Discussion For partially wetting fluid, a value for te contact angle parameter C needs to be selected wic can be used in te model. A linear stability analysis performed by Scwartz et al. predicted a relationsip between te most unstable wavenumber and te contact angle parameter by te following expression for a Newtonian fluid (see [35] for details): m mc* max m 0 q (3.0) Here, q max is te most unstable wavenumber, wic is related to te wavelengt, L by te expression, q max = π L. Using a q max value of 0.707, wic is obtained from linear stability analysis performed for Newtonian fluids by several autors [7, 37, 58] for te simple disjoining pressure model as described in Equation 3.3, we obtain C

46 Figure 3.3. Example of a numerical simulation of te rupture and dewetting process of a Newtonian tin film. Rupture is defined as te point were te film tickness reaces tat of te precursor layer tickness, and dewetting is te process of subsequent ole growt. Te plot sows several snapsots of te film profile as a function of time. Simulation was performed on a grid wit x = D simulation Figure 3.3 sows a representative illustration of te rupture and dewetting process for a Newtonian tin film. As mentioned earlier, te selection of Deraguin disjoining pressure model allows us to continue te simulation after te rupture point (were te film tickness reaces tat of te precursor layer tickness). 34

47 Figure 3.4 sows te effect of wavelengt on te rupture time for Newtonian and Ellis-type tin films. Te rupture point was considered as te time wen te film tickness reaces witin 0% of te precursor layer tickness. It can be seen tat increasing te value of β (i.e. increasing sear-tinning) reduces te rupture time. No significant cange in te most unstable wavelengt is observed from te simulations. However, cases wit λ = displays a muc faster rupture process compared to tat of λ = 3, for all cases of β. Tis is because of te narrower Newtonian plateau in te prior case. At low sear stresses, smaller values of λ acts more sear-tinning. Figure 3.4. Te rupture time versus wavelengt for Newtonian and different cases of Ellistype sear tinning fluid. Sear-tinning as negligible effect on te preferred mode of instability. Te grid size for simulation was Δx =

48 Figure 3.5 sows te rupture process by plotting te scaled minimum film tickness as a function of time for a specific wavelengt, L = (wavenumber, q = 0.707). Te results demonstrate ow te sear-tinning beavior accelerates te rupture process, wic is defined as te film tickness reducing to tat of te precursor layer tickness. Te early time film tickness evolution are very similar for bot Newtonian and sear-tinning fluids. However, te tickness evolution of sear tinning fluid quickly accelerates due to te decrease in te apparent viscosity. As mentioned earlier, a narrower Newtonian plateau (i.e., te smaller λ) leads to faster rupture of te tin film. Figure 3.5. Comparison of rupture process for Newtonian and Ellis-type sear-tinning fluid. Sear-tinning accelerates te rupture process. Te grid size for simulation was Δx =

49 3.3. 3D Simulation Figure 3.6 demonstrates an example of te rupture and subsequent dewetting process of tin films via several snapsots at different times after an initial disturbance in te scale of % of te film tickness is imposed. Te results were consistent wit te D simulation as it sowed an acceleration of te rupture process due to te sear-tinning. As expected, te Newtonian tin film sows te slowest rupture/dewetting process, and te sear-tinning film wit λ = sowed a faster rupture/dewetting compare to λ = 3. Te initial spacing between te resulting ridges are qualitatively similar wic was expected since sear-tinning did not make any significant cange in te most unstable wavelengt in our D simulations. Figure 3.7 plots te minimum film tickness from te 3D simulations. Te beavior of film tickness evolution due to te random perturbation is consistent wit te pattern observed in te D simulations were a single mode perturbation was applied to te initial film tickness. As discusses earlier, narrower Newtonian plateau (smaller λ) led to faster rupture of te tin film for te same value of β = 0. 37

50 Figure 3.6. Example of 3D rupture and dewetting process of Newtonian and Ellis-type tin films via snapsots at different times. Simulation was performed in a domain of (L x L y ) = (50 50) wit grid size x = 0.5 and y =

51 Figure 3.7. Minimum film tickness as a function of time from te 3D simulation. 3.4 Conclusion In conclusion, we ave developed a matematical model of tin film evolution for Ellistype sear-tinning fluid. We ave neglected te effect of gravity and considered partial wetting. Substrate eterogeneity was not considered. Results of rupture/dewetting simulations of Ellistype sear-tinning fluid ave indicated tat sear-tinning reology increases te growt rate of instability but as a negligible effect on te wavelengt of preferred mode of instability. However, te initial conditions in many industrial process (for example in roll to roll type coating process) migt be significantly different. Te searing of te liquid in te processing equipment may reduce te apparent viscosity of te liquid prior to te application of coating, and te apparent viscosity 39

52 migt go up significantly after te application of coating, wic can elp in slowing down of dewetting process, if a liquid wit ig zero sear rate viscosity is used. Experimental investigation of te dewetting process could be used to validate te finding of tis numerical study, following te experimental procedures described in reference [35]. Future study could include Herscel-Bulkley type reological model to study te effect of yield stress on te rupture and dewetting. Yield stress as been sown to suppress fingering instability in gravity-driven film flow [44]. A similar stabilizing effect in te case of free surface instability could potentially be acieved by using yield stress type fluids in micro-printing and coating applications. 40

53 Capter 4 MATHEMATICAL MODELING OF CONTACT LINE INSTABILITY OF GRAVITY-DRIVEN FLOW OF ELLIS- TYPE SHEAR-THINNING LIQUID FILMS 4. Introduction Gravity-driven tin film flow of viscous liquids on solid substrates ave been extensively studied due to its ubiquitous presence in various fields suc as paint/coating flows [59, 60], geological flows [, ], and microbicidal drug delivery [4-6, 3, 6]. Formation of finger-like patterns in tin films wit a moving contact line is a well observed penomena [8]. Contact line instability or fingering instability wic causes te formation of fingers downstream as been widely studied for Newtonian fluids by various researcers [6, 40, 4, 43]. Random perturbation at te contact line can cause te moving front to corrugate, and tis beavior is a function of bot te surface tension and inclination angle [43]. Driven tin films flowing on a substrate form a tick leading edge at te advancing contact line due to te action of te surface tension. Tis tick leading edge, also known as te capillary ridge [], is unstable to perturbations wic are parallel to te advancing contact line [6]. Formation of fingers at te moving front is undesirable since it may lower te quality of coating, or negatively affect te barrier properties by leaving dry spots. 4

54 Huppert experimentally demonstrated te finger formation penomena of tin films flowing down on an incline [8], wic was followed by numerous experimental and numerical studies. Troian et al. performed linear stability analysis of tin film flow down a vertical plane and computed te wavelengt tat produces maximum growt rate of fingers [6]. Bertozzi and Brenner explained te discrepancies between linear stability analysis, and experimentally observed instability [6]. Tey ave pointed out tat transient amplification of small disturbances can occur in cases even wen te linear stability analysis results indicate stability, and if te transient amplification is large enoug to sift te system into a nonlinear regime, it can lead to instability. Kondic and Diez pointed out tat flatter inclines decrease te growt rate of te disturbance, and makes te most unstable wavelengt larger [4]. All te above studies were performed wit a complete wetting assumption. Eres et al. studied instability of partial wetting tin films [63]. Tey ave concluded tat sufficiently large contact angle creates long straigt sided fingers, and for even iger contact angle te fingers break up into a series of droplets. Most of te work reported on contact line fingering instability ave dealt wit Newtonian fluids. However, non-newtonian fluids are encountered in many industrial, biological and geopysical scenarios. Tere as been some publised researc regarding contact line fingering instability of non-newtonian fluids in recent years. Balmfort et al. performed linear stability analysis of viscoplastic fluids, and teir results indicated tat yield stress can suppress te formation of fingers [44]. Numerical and experimental study on viscoelastic fluids ave indicated tat elasticity also tends to ave a stabilizing effect [64, 65]. Sear-tinning as been sown to affect te size of te capillary ridge for power-law tin films, indicating tat sear-tinning may also suppress fingering instability []. Our group ad already performed numerical investigation on te effect of power-law reology on contact line instability, and te results sowed tat seartinning indeed suppressed fingering instability [45]. However, te power-law model predicts an infinite viscosity at low sear rates wereas most real sear-tinning fluids ave a low sear rate 4

55 Newtonian plateau [39]. Tis beavior may ave an impact on te contact line instability. Terefore, we ave cosen to subsequently study te contact line instability penomena using te Ellis fluid model wic as a more robust representation of sear-tinning beavior over a wider range of sear rates troug te inclusion of a low sear rate Newtonian plateau. In tis capter, we examine te stability of te contact line of an Ellis-type sear-tinning tin film. Te approac is linear stability analysis (LSA), and verification by comparing te findings of LSA wit te 3D nonlinear simulation of finger growt due to single mode and random multimode perturbations. Tis approac as been successfully applied in our previous study of power-law fluids [45]. Figure 4.. Scematic of te cross section of a tin film flowing down an incline due to te action of gravity. 4. Problem Formulation We consider a tin film of viscous liquid flowing down on a solid substrate inclined at an angle α wit te orizontal direction. Standard Cartesian coordinate system is considered were x is te direction of flow, y is te transverse direction and z is normal to te substrate. Te liquid 43

56 surface corresponds to z = (x, y, t ) were t is te time. We assume an Ellis constitutive model for describing te sear-tinning beavior of te tin film [39]: 0 / (4.) Here, η is te viscosity, η 0 is te viscosity at zero sear stress, τ is te sear-stress at wic te viscosity is reduced by alf, and λ is te sear-tinning index. Wen λ =, te liquid is Newtonian, wile for λ >, te liquid is sear-tinning. At a ig sear stress, te Ellis model exibits power-law type sear-tinning. However, te Ellis model as a low sear stress Newtonian plateau. Ellis model is a particularly interesting candidate for modeling tin film flows of sear-tinning liquid films because te free surface of tin film flows are stress free, and sear stress is small in regions tat are far from te contact line. We utilize te lubrication teory by dept averaging te liquid velocity over te tickness of te film. Te governing differential equation describing te 3D flow (i.e. D spreading) of an Ellis tin film on an incline is given by: t Q Q 3 0 x x Qy y (4.) Q x 3 g sin g cos x x F Q y 3 g cos y F y 44

57 F 3 / sin cos g g x x g cos y y Here [Q x, Q y ] are mass flux, ρ is te viscosity, g is te acceleration due to te gravity, σ is te surface tension, and is te small slope approximation of te free surface curvature. See Appendix A4 for details of te derivation. For developing te non-dimensional form of te tin film evolution equation, te fluid eigt, is scaled by te tickness of te film far beind te contact line, c. Te spatial coordinates, and time are scaled as (x, y, t) = (x x c, y x c, t t c ). Following Kondic [43], te caracteristic spatial and time scale as been cosen to be: 3 a sin c x c (4.3) 3 a c tc sin c Here, a = σ ρg is known as te capillary lengt wic provides a measure of te lengt scales wen capillary effects become important compared to gravitational ones. Te velocity scale is cosen as U = x c t c. Te capillary number is defined as Ca = ηu σ, wic measures te relative importance of viscous forces to surface tension forces. Following Braun et al. [57], te nondimensional form of Ellis model can be written as: (4.4) 45

58 Here, β = (ρg c sin α τ ) λ describes te degree of sear-tinning. A value of β = 0 returns a Newtonian fluid beavior wereas increasing value of β indicates increased sear-tinning. Te oter sear-tinning parameter λ determines te slope of te power-law type sear-tinning region. Higer value of λ indicates a Newtonian plateau over a wider range of sear stress. Terefore, fluids wit iger value of λ will act more Newtonian-like at low sear stresses. Figure 4. sows beavior of Ellis fluid as a function of te sear stress. Figure 4.. Plot of dimensionless viscosity as a function of dimensionless sear stress for Ellis fluid Wit tese rescaling, te non-dimensional evolution equation is described as: Q t x x Qy y (4.5) Q 3 D x x x F 46

59 Q y 3 D y y F 3 F D x x Dy y Here, D = (3Ca) 3 cot α is te measure of normal component of gravity (D = 0 indicating vertical incline wereas larger D indicating flatter inclines) [6, 43]. All te analysis were performed by considering a completely wetting fluid wit zero equilibrium contact angle, and adopting a precursor film of tickness b to alleviate te contact line singularity [40, 66]. A decrease in te value of b increases te growt rate of disturbance due to te increased eigt of te capillary ridge [6]. We ave used b = 0., following publised literature on Newtonian [43] and power-law [45] tin film models. 4.. Travelling Waves and Linear Stability Analysis To perform linear stability analysis, a travelling wave solution was first determined as a base state. Detailed description of tecniques used for obtaining travelling wave solutions, and linear stability analysis as been reported in previous literatures [6, 43]. Te base state before te onset of instability is a travelling wave solution (x, y, t) = 0 (x Ut) of equation (4.5). Te function 0 (x) satisfies te following equation. 3 3 U0 0 0x 0xxx f D D 0x 0xxx (4.6) Here, te moving reference frame velocity, U, and te constant of integration, f, can be found by matcing te front onto te rest of te solution: x, 0 and x, 0 b, were b = 0. 47

60 is te tickness of te precursor layer. Using tese two matcing conditions, we get te following travelling wave velocity and integration constant. U 3 b 3 b b b (4.7) f b b 3 b b 3 Te y independent PDE governing te sape of te front can be written as: t x 3 D D x xxx 3 x xxx U x (4.8) Te base profile can be obtained by eiter solving te ODE in equation 4.6, or te PDE in Equation 4.8. We ave solved Equation 4.8 numerically to get te base state profile. Once te base state solution, 0 (x) is acieved, we consider a perturbation, ε (x, y, t) to te front 0. Here, 0 and are of O(), and ε. We plug in (x, y, t) = 0 (x) + ε (x, y, t) into te 3D nondimensional PDE in te moving reference frame, and linearize te PDE by keeping only te terms tat are in te order of ε. Next, we replace te solution by its Fourier integral in y by using te superposition principle, (x, y, t) = 0 g(x, t)e iqy dq, were q is te wavenumber wic is related to te spatial period of te perturbation, l w, by q = π l w. Eac q can be considered separately due to te linearity of te system. Terefore, for a given q, we obtain te following equation for g(x, t): 48

61 g t [ x D q g g x 0 xxx g] 4 Dq q g q g xx Ug x 0 0 (4.9) D0 x 0 xxx Here, subscripts x and t indicates partial derivatives. PDE s described in equation 4.8 and 4.9 were solved wit grid spacing, Δx = 0.0. A mes convergence study was performed to establis mes independence of te solution. Second order central difference sceme was used to discretize te spatial variables and time integration was performed using Matlab s variable step ODE5s solver wit default tolerance of E-6. Te perturbation function g(x, t) as an exponential time dependence, g(x, t) = φ(x)e μt due to te omogeneity of Equation 4.9. Here, te quantity μ is known as te growt rate: small perturbations grow and develop patterns if μ is positive, wereas te imposed perturbations disappear if μ is negative. Once g(x, t) for a particular wavenumber, q, is obtained by solving Equation 4.9, te growt rate, μ, can be calculated as: g (4.0) g t 4.. 3D Nonlinear Simulation Full nonlinear 3D simulation (D spreading) were performed on a rectangular domain of size L x L y. Constant flux boundary condition was applied in te longitudinal or flow direction at x = 0 and x = L x, wile periodic boundary conditions were applied in te transverse direction at 49

62 y = 0 and y = L y [4]. For a single mode perturbation of wavelengt l w, te initial state was a step function wit a sligt perturbation of te contact line in te y direction given by x x 0.cos 0 y l (4.) w For simulating a random multimode perturbation, a 50 mode sinusoidal function was used to perturb te contact line: x x 0 50 y A cos i i lw, i (4.) Here, l w,i = L y i, i =,, 50, and L y is te lengt in te transverse direction wic is same for all te simulations. Te amplitudes A i were random numbers between 0. and 0.. All simulations were performed for a precursor film tickness, b = 0., and grid spacing, Δx = 0., Δy = 0.. Second order central difference sceme was used to discretize te spatial variables and time integration was performed using Matlab s variable step ODE5s solver wit default tolerance of E-6. 50

63 Figure 4.3. Growt rate curve of Newtonian tin films for D = 0, and D =. Te markers represent te growt rate at te most unstable wavenumber. Precursor layer tickness of b = 0. was used. 4.3 Results and Discussion 4.3. Linear Stability Analysis Figure 4.3 sows te growt rate curve of Newtonian tin films, and te results matc wit previously publised results by Kondic [43]. Te results also matc wit te Newtonian growt rate results publised by Hu and Kieweg [45]. As can be seen from te figure, flatter inclination angle reduces te growt rate, tigtens te band of unstable mode, and moves te most unstable mode to bigger wavelengts. We present te results of linear stability analysis of Ellis tin films in Figure 4.4. Analysis was performed for vertical (D = 0) and sallow (D = ) inclines. It can be seen tat te growt rate of Ellis-type sear-tinning liquid films are functions of bot te degree of sear-tinning, β, and te parameter for slope of te sear-tinning region, λ. As sown earlier in Figure 4., degree 5

64 of sear tinning β = 0., and 0 was considered as progressively more sear tinning cases. Parameter λ = and 3 were considered for slopes of sear-tinning. Here λ = 3 indicates a steeper slope compared to λ = wic results in a wider Newtonian plateau for te same value of β. For a smaller value of β, sear-tinning is mostly confined near te contact line region wereas for iger values of β, significant sear-tinning can occur far beind te contact line. As sown in Figure 4., for vertical incline (D = 0), te growt rate increases, and te band of unstable mode tigtens wit te increase in te degree of sear-tinning, β for bot λ = and 3. However, te growt rate for λ = 3 is lower due to te presence of wider Newtonian plateau, i.e. at lower values of sear stresses, te fluid wit wider Newtonian plateau acts more Newtonianlike. No appreciable cange in te most unstable mode is observed in eiter case. However, a sligt sift in te most unstable wavenumber towards larger wavelengt is observed for te β = 0 case wile λ =. In comparison, te power-law model for D = 0, predicted a decrease in growt rate wit increased sear-tinning and no cange in te most unstable wavenumber [45]. For a flatter incline (D = ), a sift in te most unstable wavenumber to larger wavelengt was observed in all cases. For a sear-tinning fluid wit wider Newtonian plateau (λ = 3), increasing te degree of sear-tinning decreases te growt rate and sifts te most unstable wavenumber to larger wavelengt. As can be seen from te figure, β = 0 completely stabilizes te contact line. However, for sear-tinning fluids wit a narrower Newtonian plateau (λ = ), ig degree of sear-tinning (large value of β) increases te growt rate despite te sift of te most unstable wavenumber towards larger wavelengt. Te sear-tinning of te fluid far beind te contact line is te probable cause of tis penomena. For igly sear-tinning fluids wit a narrow Newtonian plateau, te apparent viscosity is significantly decreased far beyond te contact line. Tis increases te overall flow velocity of te liquid, wic probably results in te increase in overall growt rate. In comparison, te power-law model for D =, predicted a decrease in growt rate wit increased sear-tinning and no cange in te most unstable 5

65 wavenumber [45]. Tese results igligt te complex beavior of contact line instability of seartinning fluids, wen modeled by Ellis reological model. Figure 4.4. Te growt rate μ(q) computed from long time beavior of solutions of linear PDE in Equation 4.9 for D = 0, and λ =,3. In eac case, sear-tinning cases β = 0.,, 0 were compared wit te corresponding Newtonian case. Te markers represent te growt rate at te most unstable wavenumber. Inset figures sow a magnified view of te same results. Precursor film tickness b = 0. was used for all te cases. 53

66 4.3. Single Mode Perturbation and Comparison wit LSA We performed single finger evolution simulations to compare te growt rate from 3D nonlinear simulation wit te growt rate obtained via linear stability analysis. Te lengt in te transverse direction of te simulation domain were specified as equal to te most unstable wavenumbers from LSA. Perturbations were given following Equation 4.. Figure 4.5 sows te evolution of a single finger over time as a representative case. Figure 4.5. Evolution of a single finger for a sear tinning tin film wit D = 0, β = 0 and λ =. Single mode perturbation was applied by making te transverse direction of te simulation domain L y equal to te wavelengt wic was calculated from te most unstable wavenumber of LSA result (q = 0.46, l w = π q = 3.7). Figure 4.6 quantitatively compares te growt rate of a single finger from 3D nonlinear simulation wit growt rate obtained via LSA. Te finger lengt L is te distance from te finger s tip to root, and is normalized wit te initial finger lengt L 0 imposed by te initial perturbation. 54

67 Te growt rates of LSA matces closely wit te nonlinear simulation at early times. At later times, te finger growt slows down and approaces a lower constant speed since te overall travelling speed for a constant flux configuration is decided by te travelling wave speed. Te results sow tat cases wit iger growt rates deviate earlier from te LSA curve, indicating an earlier transition into te nonlinear regime. Figure 4.6. Comparison of growt rates from single mode perturbation simulation to te growt rate predicted by linear stability analysis Random Multimode Perturbation We ave also investigated te beavior of te contact line to randomly imposed multimode perturbation according to Equation 4., and compared tem qualitatively wit LSA results. Figure 4.7 sows te resulting fingers from te imposed perturbation for D = 0 and λ =. Te Newtonian case is compared wit sear-tinning cases of β = and 0. It is obvious tat te 55

68 growt rate increases (by observing finger lengt) wit increased degree of sear-tinning, and igly sear-tinning fluids develop long and slender fingers very quickly. Figure D simulation of random multimode perturbations for D = 0 and λ =. Te Newtonian case is compared wit sear-tinning cases of β = and 0. Simulations were performed on a L x L y = [60 96] domain. Please note tat te time of te snapsot for β = 0 case is at t = 0, wereas te oter two cases are at t = 40 (because at t = 40, te finger would be longer tan te simulation domain for β = 0). Figure 4.8 sows te resulting fingers from randomly imposed perturbation for D = and λ =. Te Newtonian case is compared wit sear-tinning cases of β = and 0. Te 3D simulations are in good agreement wit te LSA results. Small degree of sear-tinning (β = ) 56

69 seems to ave reduced te growt rate. However, ig degree of sear-tinning ave reduced te number of fingers wic is in agreement wit te sift in te most unstable wavenumber to iger wavelengt in te growt rate plot. Tere is also an increase in te growt of fingers wic, as mentioned earlier, is probably a result of significant sear-tinning occurring far beyond te contact line. Similarly, good qualitative agreements wit LSA results were also found for te λ = 3 cases (not sown ere). Figure D simulation of random multimode perturbations for D = and λ =. Te Newtonian case is compared wit sear-tinning cases of β = and 0. Simulations were performed on a L x L y = [60 96] domain. 57