RESEARCH STATEMENT BHAGYA ATHUKORALLAGE

Transcription

1 RESEARCH STATEENT BHAGYA ATHUKORALLAGE y research interests lie in several areas of applied mathematics. In particular, I recently focused my attention on the mathematical theory of capillary interfaces and on the modeling of biological membranes using geometric PDEs. ethods from calculus of variations, differential geometry, and fluid mechanics are a common ground for the study of these problems. I am interested in approaching challenging real-life problems by considering theoretical, numerical, and even experimental aspects. Broadly speaking, I have a keen interest in solving problems with fascinating physics, challenging mathematics, and significant real-world applications. y research focuses on flows in the presence of a free surface, such as thin liquid films, drops or bubbles. These flows are governed by capillary forces (surface tension and lead to many interesting shapes and phenomena. 1. athematical Theory of Capillary Interfaces and Hysteresis Phenomena The study of capillary interfaces is an important research area due to its prominent roles in soil science, plant biology, and surface physics (self-cleaning surfaces, amongst others. The surface chemistry of a solid substrate is the key factor in determining its wetting behavior; hence, the specific wetting properties can be obtained by modifying the surface chemistry. Due to interfacial molecular interactions, the boundary of a capillary surface behaves in specific ways based on the chemical properties of the liquid, solid substrate, medium surrounding the liquid, and the substrate s smoothness and uniformity (see [6]. It is a well-established fact that the chemical compositions of the solid, the liquid and the presence of physical roughness of the solid substrates lead to a range of stable contact angle values for a given solid-liquid system individually or in combination. Further, any changes in the contact angles follow certain rules in response to changing liquid volume for isotropic surface imperfections of the solid. This phenomenon is called contact angle hysteresis. In particular, for a surface with heterogeneities (either chemical or textural, the equilibrium contact angle, θ, can take any value between the characteristic contact angles: advancing (θ A and receding (θ R. In my dissertation, I study the curvature changes to capillary surfaces whose boundaries are in contact with a solid surface. The angle of contact between the solid and liquid is subject to the contact angle hysteresis. Further, I examine the physical mechanisms that lead to the dissipation of energy: the hysteresis in the contact angle and viscosity of the fluids involved. (a (b Figure 1. (a Contact angle hysteresis effect: advancing and receding contact angles of a liquid drop. (b Plot of the contact angle θ versus contact diameter D for a drop on a solid surface. 1

2 Current Results I derived the equations for the capillary interface and for the flow rules for the change of the contact angles for a capillary geometry at a constant volume by minimizing the Helmholtz free energy functional using calculus of variations. Let Σ s be the domain occupied by the solid, and Ω L (t be the domain filled with liquid at time t. Σ f denotes the interface between the liquid and the gas, and Σ w is the wetted part of the solid. At the contact point, that is Σ f Σ w, one encounters three different phases: solid-liquid (SL, solid-gas (SG, and liquid-gas (LG. Each interface between phases is associated with an interfacial energy, and we denote the corresponding energy densities by γ SL, γ SG, and γ LG, respectively. The angle subtended between the tangential planes to the solid-liquid and liquid-gas interfaces at the three-phase contact line is called the contact angle θ. The Helmholtz free energy of the entire system, which takes into account the surface energies for each interface and the potential energy due to gravity, is (see [7]: (1.1 E = γ LG Σ f + γ SL Σ w + γ SG ( Σ s Σ w + ψ(z, t dv. Here, ( is the measure of (, and ψ(z, t is the potential energy density due to gravity. We assume γ SL is set-valued, that is γ SL [γ SLmin, γ SLmax ]. Let cos θ Y := (γ SG γ SL /γ LG. For an isothermal condition and for a fixed t, the virtual variation of (1.1 with respect to Ω L over the convex set, subject to the volume constraint Ω L (t = V 0 (t yields (1.2 2H + ρgu + λ(t γ LG = 0 on Σ f. In (1.2, H is the mean curvature of the free surface Σ f, and λ(t denotes the Lagrange multiplier associated to the volume constraint. Let Σ f = (x, y, u(x, y. Then, the capillary interface is prescribed by an equation of the form: (1.3 ( u 1 + u 2 = Ω L (t ρgu + λ(t γ LG on Σ f. Due to the contact angle hysteresis effect, multiple capillary interface shapes are possible for the same fixed volume, although these shapes have different capillary pressure (λ values. In [1] and [5], I attempted to fill a gap in the current understanding of the contact angle hysteresis phenomena, by investigating the relationship between capillary pressure and volume. In work done in collaboration with co-authors, while taking into account the hysteresis in the contact angles, I performed a numerical simulation where equation (1.3 is solved while the volume of a capillary geometry first increases and subsequently decreases. Further, I investigated dissipation of energy during this one cycle of capillary action. To make the problem more concrete, I considered two types of symmetric geometries: a liquid droplet on a solid substrate (see [1] and a liquid column formed in a capillary tube (see [5]. Capillary Pressure Hysteresis: (a The principal result of this work showed the hysteretic behavior between the capillary pressure and volume. I also performed an experiment that exhibits the contact angle hysteresis for liquid drop on a solid substrate. By analyzing the data obtained using the droplet images and assuming the radially symmetry of the capillary geometry, equation (1.3 is solved as a two-point boundary value problem. I successfully verified the hysteresis phenomenon in capillary pressure, which is consistent with my numerical predictions. (b I analyzed the energy dissipation in a capillary tube with diameter of 1 mm and 5 mm and calculated the energy losses due to wetting and volume energy terms. Both terms increase with the diameter of the capillary tube, and the hysteresis loss due to wetting energy increases with a diameter of the capillary tube as a percentage of total hysteresis losses. Page 2

3 Viscous Energy Dissipation: The objective of this work was to calculate the viscous energy dissipation for the fluid flow that results from the deformation of the capillary interface while the contact angle varies and the volume remains constant. A liquid and a gas bound between two parallel planar solid surfaces were considered, and solid surfaces were subjected to contact angle hysteresis effect. The fluid flow was modeled by the Navier-Stokes equations, while the Young-Laplace equation characterizes the initial and final capillary surfaces. Suppose the capillary interface, f(x, t, is over a domain [0, L] at time t, and let the initial and final capillary surfaces be f i = f(x, 0 and f f = f(x, that satisfy f (x L (1.4 ρgf(x γ (1 + f 2 (x + λ = 0 in [0, L] with f(x dx = V 0. 3 In the initial domain D(0 := [0, L] [0, f i ] and the final domain D( := [0, L] [0, f f ] posses equal areas. I considered a Newtonian, incompressible fluid with velocity u = u(x, y, t, v(x, y, t, where u and v denote the fluid velocity components in the x and y directions. The fluid flow was analyzed using the Navier-Stokes and the Continuity equations: ( u ρ t + u u = ρg p + µ 2 u and u = 0, together with the boundary conditions u(0, y, t = u(l, y, t = 0. The problem was to solve for the initial velocity of the fluid that results in transforming the domain D(0 to D(. By employing a technique of Fourier series, I solved for the initial fluid velocity distribution that deforms the initial capillary surface to the final surface. I showed the existence of the final capillary surface using the initial velocity field that lies in a certain class. Further, I developed an algorithm for its computation and then calculate the viscous energy dissipation (see [3] and proved the existence of classical solutions to the problem (see [4]. In study (i, I showed that the energy required to overcome the adhesion is obtained from the area of the capillary pressure versus volume hysteresis loop. By comparing the results obtained from projects (i and (ii, I concluded that the hysteresis energy dissipation is significantly greater than that due to viscosity for comparable dimensions. The main conclusion that can be drawn from these results is the importance of contact angle hysteresis in modeling the micro- and nanofluidic systems. Future Perspectives During the past decade, research related to micro- and nanofluidic devices has gained more attention from scientists in various disciplines spanning from bioengineering to medicine and chemistry. These devices involve fluid flows in structures with at least one transversal dimension approaching the micrometer range. In order to get an accurate modelling of the fluid physics, one needs to consider the associated rich variety of the physical phenomena (e.g. electrokinetic effects, fluidstructure interactions, hysteresis in the contact angles of these microfluidic devices. Creating numerical software tools for modelling these coupled systems will help us identify the optimal design parameters and the dominant physical mechanisms and phenomena that need to be accounted for in microscale fluid flows. In soil physics, a hysteretic behavior is observed between the water content of the soil and the corresponding water potential obtained under wetting and drying processes. The key factors that cause this behavior are the ink-bottle effect, resulting from the non-uniformity in geometry of the soil pores, and the contact angle hysteresis in soil-water menisci. Over the past few decades, researchers have tried to understand this possible hysteresis behavior and its impact on soil systems but rarely consider the hysteresis in the contact angles in their models (see [12, 10, 8]. Because of the microscopic nature of soil pore spaces, modelling of imbibition and drainage in soil particles employs similar techniques used in microfluidic systems. It is well known that hysteresis systems are associated with a rate-independent energy dissipation. Thus, studying the energy dissipation in soil-water systems may help us to understand the modelling of hydrological and ecological systems. Page 3 0

4 I recently identified that the capillary pressure versus liquid volume exhibits hysteresis, which further yields an energy dissipation during wetting and drying cycles of a sessile drop and of a liquid column in a capillary tube, respectively (see [1, 5]. In the future, I wish to address how a diameter of a capillary tube affects the energy dissipation due to the capillary pressure. We may gain insight into the energy loss in soil-water systems by understanding how capillary tubes with different diameters contribute to the energy dissipation, since the soil pore spaces can be thought of as a network of capillary tubes. None of the previous studies, however, have considered the energy loss due to the capillary pressure. An important question that can be answered from this project is the significance of energy dissipation by soil-water hysteresis on the global scale. Further, as an extension of this research work, one can model the soil-water hysteresis by taking into account the effect of dynamic contact angles (see [9]. Another research project that I wish to do in the future is to study the dynamic stability of contact lenses. In [2], using a calculus of variations approach, I mathematically modeled a tear meniscus around a symmetric, spherical cap lens that was at static equilibrium. Static equilibrium was analyzed by taking the net force and net moment to be zero. A contact lens is subject to both forces due to the eyelid and the forces that result from the posterior tear film. These forces result in translational and rotational motions of the lens on the cornea and, thus, affect visual acuity. By setting up a Lagrangian energy functional formulation for the contact lens and taking into account the external forces due to eyelids and the posterior tear film, I plan to study the dynamic stability of contact lenses. This work will help us to understand the lens design parameters that yield to a lens with good rotational and translational stability. 2. Geometric PDEs and Generalized Willmore Surfaces Classical differential geometry extensively studies minimal surfaces, which represent critical points of the Dirichlet energy (see [11]. In particular, all minimal surfaces represent Willmore surfaces, which are minimizers of the Willmore (bending energy. y research interests are in surfaces that appear as minimizers of a certain type of energy. Current Results Lately, my interest has focused on an elastic membrane model for beta barrels in protein biology, corresponding to a certain Generalized Willmore type (GWE energy functional. I defined the Generalized Willmore energy to be a combination between the classical bending energy (Willmore energy and the surface energy of the membrane. I investigated the corresponding Euler Lagrange equation, as well as a specific boundary value problem and named these solutions Generalized Willmore surfaces of revolution. The following classical principle represents the starting point of my research on this topic: Theorem: Assume is a compact oriented surface, immersed in R 3. Let E w be the Generalized Willmore energy functional (2.1 E w = (kh 2 + µ ds, where k = 2k c represents double the usual bending rigidity, while µ is the surface energy per unit area. Then, the Euler-Lagrange equation of (2.1 is (2.2 H + 2H(H 2 K ϵ = 0. Here ϵ = µ k and H represent the Laplace-Beltrami operator (acting on H corresponding to the naturally induced metric of the surface immersion map. In the same spirit, I considered surfaces with smooth boundaries and studied the boundary value problem corresponding to Generalized Willmore Energy, which led me to the same Euler- Lagrange equation as above (GWE together with a specific condition on the boundary. This work is naturally connected to a problem in microbiology and genetics. In order to solve the Page 4

5 boundary value problems associated with Generalized Willmore Surfaces (of revolution or more general, I successfully used the COSOL ultiphysics R software, which is based on advanced numerical and finite element methods. I studied and modeled an important 1-parameter family of Generalized Willmore surfaces of revolution, which play a very important role in protein biology. I have provided a characterization on the boundary value conditions which generate catenoidal solutions for the GWE. I am further involved in a project that includes Generalized Willmore Surfaces in space forms (Euclidean, hyperbolic, and spherical. Future Perspectives At present, I am studying a problem that arises by minimizing the Generalized Willmore type energy functional subject to area and volume constraints of a red blood cell membrane. In particular, I consider the summation of the bending and surface energies of a closed membrane surface that encloses a volume Ω. Then the corresponding energy functional reads E = ( kc (H c γ ds λ 1 ( Ω 1 dv C vol λ 2 ( 1 ds C area. Here, k c, c 0, and γ are the elastic modulus, spontaneous curvature, and the surface energy per unit area of the cell membrane, respectively. The equilibrium shapes may be numerically obtained by solving the corresponding Euler-Lagrange equation employing the finite element method. Being able to construct a model for a red blood cell, a study can be performed to analyze the effects of the parameters k c, c 0, and γ on the shape of the cell. An important discovery that can be made from this study is to identify the parameters that are responsible for certain shape parameters. References [1] B. Athukorallage, E. Aulisa, R. Iyer, and L. Zhang. acroscopic Theory for Capillary-Pressure Hysteresis. Langmuir, In press- DOI /la504495c. [2] Bhagya Athukorallage and Ram Iyer. odel of a contact lens and tear layer at static equilibrium. In American Control Conference (ACC, 2013, pages IEEE, [3] Bhagya Athukorallage and Ram Iyer. Energy dissipation due to viscosity during deformation of a capillary surface subject to contact angle hysteresis. Physica B: Condensed atter, 435(0:28 30, th International Symposium on Hysteresis odeling and icromagnetics (H [4] Bhagya Athukorallage and Ram Iyer. On a two-point boundary value problem for the Navier-Stokes equations arising from capillary effect, anuscript submitted for publication in Journal of athematical Fluid echanics. [5] Bhagya Athukorallage and Ram Iyer. Investigation of energy dissipation due to contact angle hysteresis in capillary effect, anuscript submitted for publication in Journal of Physics: Conference Series: International Workshop on ulti-rate Processes and Hysteresis. [6] P. G. de Gennes, F. Brochard-Wyart, and D. Quéré. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York, [7] R. Finn. Equilibrium capillary surfaces. Springer-Verlag, [8] Yixiang Gan, Federico aggi, Giuseppe Buscarnera, and Itai Einav. A particle-water based model for water retention hysteresis. arxiv preprint arxiv: , [9] arkus Hilpert and Avishai Ben-David. Infiltration of liquid droplets into porous media: Effects of dynamic contact angle and contact angle hysteresis. International Journal of ultiphase Flow, 35(3: , [10] William J Likos and Ning Lu. Hysteresis of capillary stress in unsaturated granular soil. Journal of Engineering echanics, 130(6: , [11] Petko I arinov. Stability analysis of capillary surfaces with planar or spherical boundary in the absence of gravity. PhD thesis, The University of Toledo, [12] An-Nan Zhou. A contact angle-dependent hysteresis model for soilwater retention behaviour. Computers and Geotechnics, 49(0:36 42, Department of athematics and Statistics, Texas Tech University, Lubbock, Texas address: bhagya.athukorala@ttu.edu Page 5