Constant Mean Curvature Surfaces of Revolution versus Willmore Surfaces of Revolution: A Comparative Study with Physical Applications

Transcription

1 Constant Mean Curvature Surfaces of Revolution versus Willmore Surfaces of Revolution: A Comparative Study with Physical Applications by Thanuja Paragoda,B.Sc. A Thesis In Mathematics and Statistics Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of Master of Science Approved Dr. Magdalena Toda Chair of Committee Dr. Eugenio Aulisa Dr. Roger Barnard Dr. Giorgio Bornia Dr. Akif Ibragimov Dean of the Graduate School May, 2014

2 c 2014, Thanuja Paragda

3 ACKNOWLEDGEMENTS I would like to express my profound appreciation and gratitude to my advisor Dr. Magdalena Toda. She continuously helped me by offering advice, guidance and instruction. Without her help, this thesis would not have been possible. Since the day that I applied to Texas Tech University for my graduate studies, she has helped me in numerous ways. Since the moment I was first confronted with difficulties in doing research, she has been like a mother and a friend to me. I would also like to thank my committee members, Dr. Eugenio Aulisa, Dr. Roger Barnard, Dr. Giorgio Bornia, and Dr. Akif Ibragimov. They also helped me by providing advice, guidance and instruction. In addition, I would like to thank everyone in the Department of Mathematics and Statistics at Texas Tech University, for offering me the opportunity to pursue a Master s degree and a PhD degree in Mathematics with financial support. I would also like to acknowledge Prof. Nalin De Silva, Dr. Kumudu Mallawarachchi and other faculty members from the Department of Mathematics, University of Kelaniya, Sri Lanka. I am also grateful to Bhagya Athukorallage for helping me with several numerical calculations and graphs in this thesis, and to Janitha Gunathilake for her suggestions in writing the thesis. Also, special thanks should go to Janaki Sugathadasa who encouraged me in many ways, and therefore contributed to my success over the past two years. Last, but not least, I would like to thank my loving parents, sisters and friends in Sri Lanka, USA and elsewhere for their support and encouragement. ii

4 TABLE OF CONTENTS Acknowledgements ii Abstract iv List of Figures v 1. Introduction Delaunay CMC surfaces and their applications in liquid bridge problems Introduction to Delaunay CMC surfaces Roulette of the conics: Parabola, Ellipse, and Hyperbola Undulary as the roulette of an ellipse, and the corresponding unduloid Nodary and nodoids Catenaries and catenoids Mathematical models for the liquid bridge between two vertical plates Energy minimization approach to model the liquid bridge Profile curve for a rotationally symmetric liquid bridge: undulary Willmore surfaces and their applications in elastic membranes (including biological membranes) Numerical results using COMSOL multi-physics software Conclusions and further research directions Bibliography iii

5 ABSTRACT This thesis studies some special types of surfaces of revolution and their real world applications. The main two cases hereby considered are the constant mean curvature surfaces of revolution (also called Delaunay surfaces) and Willmore s surfaces of revolution, respectively. We first present some geometric results on Delaunay CMC surfaces which correspond to certain classes of ordinary differential equations. We present the original construction of Delaunay surfaces, based on roulettes of conics, after which we characterize these geometric objects as solutions to specific ODEs. We present a few physical models of Delaunay surfaces arising as liquid bridges between two vertical walls - which are proved to be unduloidal surfaces, by using Calculus of Variations. We numerically computed the profile curves of these surfaces and provided some numerical models for them. By contrast, we studied Willmore surfaces as minimizers of the Willmore energy (or bending energy). In particular, we have studied some Willmore surfaces of revolution which come in as solutions to BVP problems consisting of the Willmore equation, together with some special Dirichlet type boundary conditions. With help from Dr. Eugenio Aulisa and Mr. Bhagya Athukorallage, I have provided some numerical computations for the profiles of these surfaces, using COMSOL Multiphysics. Willmore surfaces of revolution have lots of application in the real world, such as elastic biological membranes. At the microbiological level, a model of such Willmore surfaces of revolution is provided by the beta barrels arising from secondary structures in proteins (beta sheets configured as a rotationally symmetric model). iv

6 LIST OF FIGURES 2.1 Roulette of an ellipse E. F 1 and F 2 are the foci of E. l is the locus of the focus F 1, and T is the local tangent line of l The unduloid Cross section of a nodoidal surface The catenoid Liquid drop in between two parallel plates. Distance between the two plates is L Undulary profiles for h = π. Each profile curve is obtained by considering two revolutions of the ellipse, which is defined by the corresponding ɛ value The mean curvature and the profile curve f(x) in the domain[ 1, 1] The mean curvatures in the domain[ 1, 1] when α = 1 : 1 : The profile curves in the domain[ 1, 1] when α = 1 : 1 : The mean curvatures in the domain[ 1, 1] when α = 1 : 10 : The profile curves in the domain[ 1, 1] when α = 1 : 10 : The mean curvatures in the domain[ 1, 1] when α = 1 : 20 : The profile curves in the domain[ 1, 1] when α = 1 : 20 : v

7 CHAPTER 1 INTRODUCTION This work represents a comparative study of two families of special surfaces of revolution. n two separate sections, we study Delaunay surfaces and Willmore surfaces of revolution, and some of their applications. We model the first family via liquid bridges between walls, and the second family via elastic membranes. The first chapter presents a series of basic results on Delaunay surfaces and their geometry. It presents these surfaces as surfaces of revolution whose profile curves represent roulettes of conics. We first present the general parametrization of surfaces of revolution and find an expression for the constant mean curvature using the coefficients of first and second fundamental forms. Further, we examine the roulettes of conics as characterized by certain ordinary differential equations which describe the motion of one of the foci of a conic. We present the differential equations that are satisfied by the undulary, nodary and catenary, which respectively represent the roulettes of the ellipse, hyperbola and parabola. After revolving these roulettes around a line, we obtain the Delaunay surfaces known as unduloids, nodoids and catenoids, respectively. In the second chapter, we discuss some real-world applications of Delaunay surfaces, namely, liquid bridges between vertical plates as first studied by Thomas Vogel, which are Delaunay surfaces. We then present some numerical models for these surfaces (liquid bridges) - which were obtained using MATLAB and COMSOL. These models naturally arise as critical points of a potential-wetting energy of a liquid bridge, and they are obtained via classical methods of Calculus of variations. We numerically computed the profile curves of these surfaces and provided some graphical models for them. By contrast, we studied Willmore surfaces as minimizers of the Willmore (bending) energy. In particular, we have studied some Willmore surfaces of revolution which come in as solutions to BVP problems consisting of the Willmore equation, together with some Dirichlet boundary conditions. With help from Dr. Eugenio Aulisa and Bhagya Athukorallage, I have provided some numerical computations for the profiles of these surfaces, using COMSOL Multiphysics. 1

8 Willmore surfaces of revolution have lots of application in the real world, such as elastic biological membranes which are topologically cylinders. At the microbiological level, a model of such Willmore surfaces of revolution is provided by the beta barrels arising from secondary structures in proteins (beta sheets configured as a rotationally symmetric model). Note that the two aformentioned classes of surfaces have the catenoids and the spheres in common. However, little attention has been given to the families of Delaunay surfaces and Willmore surfaces in general, respectively, until a decade ago, and each of these two topics now represents a fast-developing field with lots of mathematical, physical and biological applications. 2

9 CHAPTER 2 DELAUNAY CMC SURFACES AND THEIR APPLICATIONS IN LIQUID BRIDGE PROBLEMS 2.1 Introduction to Delaunay CMC surfaces Constant mean curvature (CMC) surfaces have played a prominent role in differential geometry. In 1841, Charles Eugéne Delaunay introduced a way of constructing rotationally symmetric CMC surfaces in R 3, by proving that a surface of revolution in R 3 is a CMC surface if and only if its profile curve is the roulette of a conic. A surface of revolution in R 3 is generated by revolving a given profile curve about a line in the plane containing this given curve [18]. Basically Delaunay surfaces are the surfaces of revolution with constant mean curvature. Delaunay delineated these surfaces in Euclidean space explicitly as surfaces of revolution coming from roulettes of the conics [6]. In the appendix of [6], M. Strum characterized these surfaces variationally as they formulated the isoperimetric principle for the unduloid and nodoid which we will describe later in this thesis. Thomas Vogel published a series of articles regarding liquid bridges between vertical walls, which represent an important application of surfaces of revolution with constant mean curvature (i.e. Delaunay surfaces) [22, 23]. Let us consider a surface of revolution given by the parametrization σ(u, v) = (u, f(u) cos v, f(u) sin v), (2.1.1) where u belongs to an open interval (α, β) of the real line, f(u) is a real-valued smooth function, and v belongs to the interval (0, 2π). Definition 1. The first and second fundamental forms [12] of the surface patch σ(u, v) are defined as (2.1.1) I = E du 2 + 2F dudv + G dv 2 and II = e du 2 + 2f dudv + g dv 2, (2.1.2) 3

10 with the corresponding coefficients E = σ u σ u, F = σ u σ v, G = σ v σ v, e = σ uu N, f = σ uv N, g = σ vv N. (2.1.3) Here, N represents the outward unit normal to the surface, N = σ u σ v σ u σ v. (2.1.4) By differentiating the parametric relation σ(u, v), with respect to the parameters u and v, we get σ u (u, v) = (1, f (u) cos v, f (u) sin v), σ v (u, v) = (0, f(u) sin v, f(u) cos v). By using the above definition, we obtain the expressions for the coefficients of the First and Second Fundamental Forms (ref (2.1.3)) with the unit normal N = E = 1 + f (u) 2, F = 0, G = f(u) 2, f f e =, f = 0, g = 1+f 2 1+f 2, f 2 (f, cos v, sin v). Next, with the aid of the above calculated coefficients values, the mean curvature H of the surface [12] can be easily calculated using the relation H = Eg 2F f + Ge. (2.1.5) 2(EG F 2 ) After some simplifications, we obtain H = ff f 2 2f (1 + f 2 ) 3. (2.1.6) The following theorem is a well-known consequence of equation (2.1.6). 4

11 Theorem 2. [19]: A surface of revolution M parametrized by x(u, v) = (u, f(u) cos v, f(u) sin v) has constant mean curvature if and only if the function f(u) satisfies f 2 ± 2af 1 + f 2 = ±b2, where a and b are positive constants. It is a remarkable result found by Delaunay that this differential equation became a paramount approach to constructing all surfaces of revolution of constant mean curvature. For a proof of this result, one can consult [19] Roulette of the conics: Parabola, Ellipse, and Hyperbola Consider the path generated by a focus of a conic that is moving along a straight line. The trace of the focus is called the roulette of the corresponding conic (refer figure2.1). In [16, 8], the authors derive the equations for the roulettes of a parabola, ellipse, and hyperbola respectively. The resulting roulettes are named the catenary, undulary, and nodary, and surfaces that are generated by revolving these curves about a fixed axis are the catenoid, unduloid, and the nodoid respectively. A detailed derivation of the equations of the aforementioned roulettes is given in [8]. However, we focus our attention to the roulette generated by an ellipse, and present the derivation of that given in [8, 19, 21]. Note that limiting cases of unduloids include both the cylinders and a sequence of surfaces. Note: In 2010 Ivalo M. Mladenov presented a new class of axially symmetric surfaces which generalizes Delaunays unduloids and provides solutions of the shape equation is described in explicit parametric form. This class provide the first analytical examples of surfaces with periodic curvatures studied by K. Kenmotsu, and leads to some unexpected relationships among Jacobian elliptic functions and their integrals [7]. 5

12 2.2 Undulary as the roulette of an ellipse, and the corresponding unduloid A detailed derivation of the equations of the aforementioned roulettes are given in [8]. In this section, we provide the actual generation of an undulary as a roulette of an ellipse. We present the derivation given in [8, 19, 21]. Let E be an ellipse with a > b, where a and b are the lengths of the semi-major and semi-minor axes. Consider the motion of E on a straight line s l, and let the path of the focus F 1 be l. O is the contact point of E with s l. Let F 1 O 1 and F 2 O 2 be perpendicular to line s l. Further, assume F 1 and F 2 have the Cartesian coordinates (x, y) and ( x, ỹ), respectively. Let T be the local tangent line of the curve l that passes through the point (x, y), and assume φ be the angle between the x axis and T. Figure 2.1. Roulette of an ellipse E. F 1 and F 2 are the foci of E. l is the locus of the focus F 1, and T is the local tangent line of l. By the definition of an ellipse, we have F 1 O + F 2 O = 2a, (2.2.1) and F 1 O 1 F 2 O 2 = yỹ = b 2. (2.2.2) The last relation is also known as the pedal property for an ellipse. 6

13 Considering the elementary properties of an ellipse and the properties of the locus described by its focus, Delaunay deduced the formulas as follows in y + ỹ = 2a cos φ. (2.2.3) Note that, according to (2.2.2) ỹ = b2 y. Thus, (2.2.3) may be expressed in the form y + b2 y = 2a cos φ. (2.2.4) We assume the curve l is parametrized by the arc length s. Then dx ds = cos φ, (2.2.5) and hence, we obtain: ds dx = 1 + ( ) 2 dy. (2.2.6) dx Finally, after substituting the previous two results into (2.2.4), and simplifying, we obtain the equation 2ay 1 + y 2 y2 = b 2, (2.2.7) which represents a differential equation corresponding to an undulary. Note that this equation is a particular case of the general differential equation obtained at the end of the previous section (see the four differential equations based on ± signs). This shows that the undulary (as a roulette of an ellipse) will generate a CMC surface of revolution when rotated around a line. This surface is called unduloid. The profile curve of an unduloid (that is, undulary) has a parametrization [6] of the 7

14 following type t x(t) = b2 du a (1 + e cos u) 1 e 2 cos 2 u, 0 1 e cos t y(t) = b 1 + e cos t, (2.2.8) where a and b with a > b, represent the semi-major and semi-minor axes of an ellipse respectively, and focus and eccentricity of the ellipse are ɛ and e. Moreover, ɛ and e are given by the equations: ɛ 2 = a 2 b 2, e = ɛ a. Proposition 3. The parametrization of the undulary satisfies the following differential equation for the case ɛ = 1: ( ) 2 dy = 4a2 y 2 dx (y 2 + ɛb 2 ) 1. 2 Proof. The proof is immediate. Note that the differential equation in the statement is equivalent to the equation 2ay 1 + y 2 y2 = b 2, (2.2.9) which characterizes the undulary. Note that epsilon, hereby was introduced to allow a comparison with the analogous differential equation describing a nodary (see Proposition 4). Equation (2.2.9) is a particular case of the ODE presented in theroem 2. The surface of revolution described by an undulary profile is called an unduloid (sometimes spelled onduloid ), and has constant nonzero mean curvature. 8

15 Figure 2.2. The unduloid 2.3 Nodary and nodoids In differential geometry, the locus of a focus of a hyperbola as the point of contact rolls along a straight line in a plane forms the curve which we call the nodary. Then a nodoid is a surface of revolution with constant nonzero mean curvature obtained by rolling a hyperbola along a fixed line, tracing the focus, and revolving the resulting nodary curve around the line [19]. Figure 2.3. Cross section of a nodoidal surface. Let the hyperbola be given by the equation x 2 a 2 y2 b 2 = 1, where a > b > 0. The parametrization of the corresponding nodary [6] is t x(t) = b2 cos u du a (e + cos u) e 2 cos 2 u, 0 e cos t y(t) = b e + cos t. (2.3.1) Proposition 4. The parametrization of the nodary satisfies the following 9

16 differential equation for the case ɛ = 1: Proof. The proof is immediate. ( ) 2 dy = 4a2 y 2 dx (y 2 + ɛb 2 ) 1. 2 We are now briefly presenting the standard nodoid (see, e.g., [20]). Let (x, y, z) be the usual rectangular coordinates for R 3. Consider a Delaunay nodoid with the x axis as its rotation axis, and with constant mean curvature H = 1. Let (x(t), z(t)), t R be a parametrization of the profile curve of the nodoid in the xz plane with z(t) > 0, and so the surface can now be parametrized by D(t, θ) = (x(t), z(t) cos θ, z(t) sin θ), t R, θ [0, 2π). (2.3.2) Suppose further that the parameter t is chosen to make the mapping D(t, θ) conformal with respect to the coordinates (t, θ). Let t = a and t = b be values at which the nodoid achieves two adjacent necks, that is, z(t) has local minima at both t = a and t = b equal to the neck radius r. Conformality implies that the first fundamental form is ds 2 = ((x ) 2 + (z ) 2 )dt 2 + z 2 dθ 2 = ρ 2 (dt 2 + dθ 2 ), with ρ 2 = (x ) 2 + (z ) 2 = z 2. The second fundamental form is then given by dσ 2 = 1 z (x z z x )dt 2 + x dθ 2, and so the coordinates (t, θ) are curvature line coordinates, that is, the coordinates are isothermic. Furthermore, the mean curvature H = 1 implies 2z 3 z x + x z zx = 0. 10

17 Note: In 2005, Wayne Rossman gave two numerical methods for computing the first bifurcation point for Delaunay nodoids. With regard to methods for constructing constant mean curvature surfaces, they concluded that the bifurcation point in the analytic method of Mazzeo-Pacard is the same as a limiting point encountered in the integrable systems method of Dorfmeister-Pedit-Wu [20]. 2.4 Catenaries and catenoids To generate the catenary as a roulette of a conic, we start from the parabola x(t) = a sinh 1 (t), (2.4.1) y(t) = a 1 + t 2, (2.4.2) where a is the focal length (the distance from the vertex to focus) and the focus of this parabola is at the point (0, a). Eliminating the parameter t gives us the usual equation for the catenary and using (2.4.1) and (2.4.2), we have y = a cosh ( ) x, a dy dt dx dt = = at 1 + t 2, a 1 + t 2, which further implies dy dx = t, ( ) 2 dy = t 2. dx (2.4.3) 11

18 Since y 2 = a 2 + a 2 t 2, (2.4.4) one will obtain Proposition 5. The parametrization of a standard catenary satisfies the following differential equation: ( ) 2 dy = y2 1. (2.4.5) dx a2 The standard parameterization of a catenoid (as a surface of revolution whose profile curve is a catenary) is the following: ( v x(u, v) = a cos u cosh, ( a) v y(u, v) = a sin u cosh, a) z(u, v) = v, (2.4.6) where u and v are real parameters, and a is a non-zero real constant. The principal curvatures κ 1, κ 2 of the surface are given by κ 1 = κ 2 = Note that H = 0, and the Gauss curvature 1 a cosh 2 (v/a). 1 K = a 2 cosh 4 (v/a). The catenoid is a well known minimal surface with many important applications to the real world. It can be deformation-retracted to a helicoid (see for example, [13, 1]), meaning that the helicoid and catenoid belong to the same associated family of surfaces (1-parameter family of deformations). 12

19 The following is a well known and very important classical result which can be found in [11]. Theorem 6. A surface of revolution M, which is a minimal surface, is contained in either a plane, or a catenoid [11]. Figure 2.4. The catenoid As a side note, we would like to mention that in 2010 Masato and Taku introduced a physics experiment that involves using a soap film to form a catenoid. Using that soap film they created catenoids between two rings and characterized the catenoid in situ while varying the distance between the rings [13]. This is an appropriate experiment which combines theory and experimentation. 13

20 CHAPTER 3 MATHEMATICAL MODELS FOR THE LIQUID BRIDGE BETWEEN TWO VERTICAL PLATES In this study, we investigate the shape of the tear meniscus that forms around a contact lens. Since the tear film thickness is much smaller than the radius of the cornea and the contact lens, one may neglect the curvature of both the contact lens and the cornea, and treat them as flat surfaces. Thus, to make this problem amenable to analysis, we consider a meniscus of a liquid bridge that forms between two vertical plates. Both modeling and the stability of this type of problem are extensively studied in [22, 23, 24]. Authors used the Young-Laplace equation or energy minimization approach to model the liquid meniscus. In this case, both methods result in a nonlinear differential equation, which describes a constant mean curvature (CMC) surface. One approach is to model these CMC surfaces as Delaunay surfaces: Catenoids, Unduloids, and Nodoids. However, due to the self-intersections, modeling a liquid surface via a nodoid is unphysical [3]. 3.1 Energy minimization approach to model the liquid bridge In this section, we consider a liquid drop, which is trapped between two vertical plates, and model the profile curve of the drop using a Calculus of Variations approach. In this method, a formula for the profile curve that formed between the liquid-air interface is obtained by minimizing the total potential energy of the drop while imposing a volume constraint [5, 15, 23]. The total potential energy of a liquid is mainly composed of three type of energy forms: (i) surface energy of a liquid surface, which is proportional to the surface area of the liquid-air interface (free surface), (ii) wetting energy that arises due to the contact area of solid-liquid interface, and (iii) gravitation potential energy, which we will neglect here. A detailed discussion of these three types of energy terms can be found in [9]. In this study, we consider a rotationally-symmetric liquid drop, and hence the latter energy type is neglected in our analysis. Consider a liquid drop which is trapped between two vertical plates, and the profile of the drop has the equation z = f(x) with respect to the configuration of the 14

21 Cartesian coordinate system, which is on the left plate (plate 1) (refer 3.1). Note that the continuity of the drop implies that f(x) > 0 on the interval [0, L]. Let the distance between the vertical plates be L, and assume the shape of the solid-liquid contact area on the plates to be circles with radii f(0) and f(l), respectively. The relative adhesion coefficients of the liquid with the plates 1 and 2 are β 1 and β 2. Figure 3.1. Liquid drop in between two parallel plates. Distance between the two plates is L. Thus, under the absence of gravity, the total energy E of a rotationally symmetric liquid drop may be written in the following form [22, 9]: E = L 2πγf(x) 1 + f 2 (x) dx γβ 1 πf(0) 2 γβ 2 πf(l) 2. (3.1.1) 0 In (3.1.1), γ denotes the surface energy per unit area of the liquid; β i is the relative adhesion coefficient between the i th wall and the liquid. The integral term represents the surface energy of the liquid drop, and the last two terms denote the wetting energy of the drop. We wish to minimize the energy E given in (3.1.1) 15

22 subjected to the volume constraint L 0 πf(x) 2 dx = V 0, (3.1.2) where V 0 denotes the volume of the liquid drop. Thus, the new energy functional Ē that includes the volume constraint (3.1.2) is Ē = L 0 2πγf(x) ( L 1 + f 2 (x) dx γβ 1 πf(0) 2 γβ 2 πf(l) 2 + λ πf(x) 2 dx V 0 ). 0 (3.1.3) Here, the Lagrange multiplier λ is an unknown constant. We consider the variation of Ē (δē) with respect to the drop radius (capillary surface height) f(x) and the meniscus height at the end points: f(0) and f(l). L δē = 2πγ which simplifies to f 2 (x) δf dx + 2πγ 2πγβ 2 f(l)δf(l) + 2πλ δē = 2πγf(0) (β πγ L 0 L 0 L 0 f f δf x dx 2πγβ 1 f(0)δf(l) 1 + f 2 f(x) δf dx, (3.1.4) ) ( f (0) δf(0) + 2πγf(L) 1 + f 2 (0) ( λ γ f f 2 (x) d ( dx ff 1 + f 2 (x) f (L) 1 + f 2 (L) β 2 ) δf(l) )) δf(x) dx. (3.1.5) The necessary condition for the energy minimization [10, 17] is that δē = 0. Thus, 16

23 we have the following system of equations that reads λ γ f f 2 (x) d ( ) ff dx 1 + f 2 (x) f (0) β f 2 (0) = 0 in [0, L], (3.1.6) = 0 at x = 0, (3.1.7) f (L) 1 + f 2 (L) β 2 = 0 at x = L. (3.1.8) Let the value of the contact angles of the liquid meniscus with the plates be θ 1 and θ 2, and assume they are rotationally invariant. Hence, we have the same contact angle values along the periphery of the contact circles. We observe that f (0) = cot θ 1 and f (L) = cot θ 2. Then the relative adhesion coefficients β 1 and β 2 in (3.1.7) and (3.1.8) may be expressed as β 1 = cos θ 1 and β 2 = cos θ 2. (3.1.9) Finally, simplifying (3.1.6) results in λ γ = ff f 2 1, x [0, L]. (3.1.10) f(1 + f 2 ) 3/2 Equation (3.1.10) represents the liquid surface of the drop in terms of its profile curve f(x), and in the next section, we further observe that the right hand side of this equation relates to the mean curvature of the liquid surface. Since Lagrange multiplier λ and the surface energy per unit area of the liquid γ are constants, (3.1.10) leads us to an equation of the form: λ γ = ff f 2 1 = 2H. (3.1.11) f(1 + f 2 ) 3/2 Remark that we have just proved that the rotational bridge surface is a CMC surface (surface with constant mean curvature). This result, which has not been explicitly proved in the literature, can be further formulated as: Theorem 7. Rotational liquid bridges between vertical walls, which minimize the surface and wetting type energies represent rotationally CMC surfaces with H 0. 17

24 Remark 8. Multiplying (3.1.11) through f [19], rearranging the terms, and integrating with respect to the x variable, one may obtain 2Hff + f [(1 + f 2 ) ff ] (1 + f 2 ) 3 = 0, ( ) f + H(f 2 ) = 0, 1 + f 2 ( ) f + Hf 2 = 0, 1 + f 2 f 1 + f 2 + Hf 2 = C 1, f 2 + f H 1 + f 2 = C 2, (3.1.12) where C 1, C 2 are constants. Observe that (3.1.12) has the same form as in Theorem Profile curve for a rotationally symmetric liquid bridge: undulary As we observed in Section 3.1, surfaces that minimize energy under a volume constraint have a constant mean curvature, and lead to a nonlinear differential equation [4]. In particular, our mathematical model (see (3.1.10)) has a non-zero mean curvature value for its surface. In [14], the author found and solved a complex nonlinear differential equation, which describes roulettes (profile curves of CMC surfaces) of different types of conics. For a nonzero CMC surface, the author analytically obtained a one-parameter family of profile curves in terms of the constant mean curvature H, which yields to [14, 16] ϕ(s; H, B) = ( s Here, B and H are real numbers B sin(2ht) 1 + B 2 + 2B sin(2ht) dt, 1 + B2 + 2B sin(2hs) 2 H However, as stated in [12], (3.1.13) does not have a direct geometrical interpretation of the profile curves. In the same paper, parametrization for the ). (3.1.13) profile curve of an unduloid, which is called the undulary, is given in terms of the 18

25 elliptic functions and formulas for the length, surface area, and volume of appropriate parts presented. Here the parameterization for an undulary is given in terms of two real free parameters: a and c. where ( µt x(t) = af 2 π ) ( µt 4, k + ce 2 π ) 4, k, (3.1.14) z(t) = m sin µt + n, (3.1.15) µ = 2 a + c, k2 = c2 a 2, m = c2 a 2, n = c2 + a 2, (3.1.16) c and F (ϕ, k) and E(ϕ, k) are the elliptic integrals of the first and second kind, respectively. Furthermore, authors show that the appropriate variation of the parameters a and c yield the following four Delaunay surfaces: unduloids, nodoids, spheres, and cylinders. The above parameterizations (3.1.14) and (3.1.15) create a rotationally symmetric surface with the mean curvature H = 1 a + c. In the present analysis, we recall the parametrization of an undulary that was originally introduced by Delaunay (see [2]) and presented as (2.2.8) in our paper. Consider again the standard ellipse x 2 a + y2 = 1, (3.1.17) 2 b2 where, a and b represent the semi-major and semi-minor axes, respectively. Assume a > b. Let the focus and the eccentricity of the ellipse be ɛ and e. As in Section 2.2, ɛ and e are given by the following equations: ɛ 2 = a 2 b 2, (3.1.18) e = ɛ a. (3.1.19) 19

26 By using (3.1.18) and (3.1.19), the parametrization (2.2.8) can be rewritten as x(t) = a2 ɛ 2 a t 0 du, (1 + ɛ cos u) 1 ɛ2 cos a a 2 u 2 y(t) = a 2 ɛ 2 a ɛ cos t a + ɛ cos t. (3.1.20) In [23], the author considered an unduloid generated by an ellipse of arc length 2π. On the other hand, in the present work, we extend Vogel s study by considering an unduloid that is generated from an ellipse of arc length 2h, where h is the distance between the two vertical plates. Thus, the length of the semi-major axis, a, of the ellipse is expressed as a function of both ɛ and h (see Lemma 9). Then, using (3.1.20) together with (3.1.21), we numerically obtain undulary profiles for different ɛ values (see Figures 3.2(a) and 3.2(b)). π. For the numerical example, the distance between the plates (h) is assumed to be Lemma 9. We may express a = a(ɛ, h) [23], which is given by the equation a(ɛ, h) = h π + ɛ2 4 π h + O ( ɛ 3 π 3 h 3 ). (3.1.21) Here, h is the distance between the vertical parallel plates, and ɛ = a 2 b 2. Proof. Since 2h represents the arc length of the ellipse defined in (3.1.17), we have 2h = 2π a2 ɛ 2 cos 2 θ dθ, = 0 2π 0 ( ) 2 ɛ cos θ a 1 dθ. (3.1.22) a Using the power series representation of 1 ( ) ɛ cos θ 2, a (3.1.22) may be rewritten in 20

27 the following form: ( ) 2 ɛ cos θ 1 = 1 ɛ2 cos 2 θ a 2a 2 ɛ4 cos 4 θ 8a 4 ( ) 5 ɛ cos θ + O. a By substituting the above relation into (3.1.22) and integrating the definite integral, we obtain h π = a ɛ2 4a + O(ɛ4 ). (3.1.23) Neglecting the higher order terms and solving the resulted quadratic equation, the length of the semi-major axis a may be expressed as a function of ɛ. a(ɛ) = h ( ) ɛ2 π 2 = h ( ) 2π h 2 π + ɛ2 π ɛ 3 4 h + O π 3 h 3 (3.1.24) In our analysis, we showed that the shape of the liquid meniscus has a constant mean curvature. This result is due to the neglecting gravitation potential energy term from the total energy of the liquid drop. This assumption is satisfied, if the volume of the liquid drop is sufficiently small. Furthermore, if the second variation of the total potential energy is nonnegative, a stable liquid bridge is formed. Stability criteria for a liquid bridge in between two plates are extensively studied in [23, 24]. In [23], the author proved that for sufficiently large volumes a stable drop exists that has a profile curve with no inflection point. Theorem 10. [23] If θ 1 + θ 2 π, then the family of profile curves without inflection remains stable as volume decrees from infinity, at least until either dv dh changes sign or an inflection point appears on the boundary. The latter condition states that all profiles in the family are stable except for the limiting profile that has f = 0 at an endpoint. Note that, in general, the profile of an undulary may have zero, one, two, and four inflection points. For fixed ends, unduloids are stable only if their critical length equals the single period [3, 15]. In our analysis, we are interested in tear film whose thickness varies in the interval [40 60] µm. Thus, the parameter h ɛ. Hence, the surface of the meniscus is an unduloidal patch with no infection points. 21

28 2.5 2 ǫ =0 ǫ =0.2 ǫ =0.4 ǫ =0.6 ǫ =0.8 ǫ = y(t) x(t) (a) h = π and ɛ = 0.2 (b) h = π and ɛ = 0 : 0.2 : 1 Figure 3.2. Undulary profiles for h = π. Each profile curve is obtained by considering two revolutions of the ellipse, which is defined by the corresponding ɛ value. These profile curves are computed using MATLAB. I am grateful to my colleague Bhagya Athukorallage for his help in obtaining these representations. He first obtain numerical values x(t) and y(t) for the undulary parametrization(see eq.(3.1.20)) for different t values and then plot a graph x(t) versus y(t) in the case of ɛ = 0.2 and h = π which is shown in the part(a) of the above graphs. Next, for the same h value by varying ɛ from 0 to 1 with increment 0.1, our profile curves are obtained for the corresponding ɛ value. 22

29 CHAPTER 4 WILLMORE SURFACES AND THEIR APPLICATIONS IN ELASTIC MEMBRANES (INCLUDING BIOLOGICAL MEMBRANES) In this chapter we first review the basic concepts regarding Willmore surfaces. The liquid bridge model in the previous section was based on surface and wetting energies. By contrast, this model of elastic membranes is based on Willmore (bending) energies. There are significant similarities, but also differences, between Willmore surfaces of revolution (elastic surfaces) and the CMC unduloids modeled by liquid bridges. Let us consider a smooth immersed surface r : M R 3. The Willmore energy functional is defined by the bending energy W (r) := H 2 ds (4.0.1) r(m) where H = (k 1 + k 2 )/2 denotes the mean curvature of R := r(m). This functional is a model for the elastic energy of thin shells or biological membranes. In these applications one is usually concerned with minima, or more generally with critical points of the Willmore functional. The corresponding surface R has to satisfy the Willmore equation R H + 2H(H 2 K) = 0 (4.0.2) on R, where R denotes the Laplace- Beltrami operator on R and K its Gauss curvature. A derivation of this Euler-Lagrange equation (called Willmore equation) can be found in [28]. A solution of (4.0.2) is called a Willmore surface [26]. In recent years, the Willmore functional and the associated L 2 -gradient flow, the so-called Willmore flow, have played a great role in surface theory. The corresponding L 2 -gradient flow (called Willmore flow) is given by the geometric evolution problem t x(t) = M(t) H(t)n(t) + H(t)( S(t) (H(t))2 )n(t) (4.0.3) 23

30 which defines a family of surfaces M(t) for t 0 from a given initial surface M(0). Here S(t) denotes the shape operator on M(t), n(t) the normal field on M(t), and. the Frobenius norm on the space of endomorphisms on the tangent bundle T M(t) [27]. Further, we introduce some geometric terminology. Let f : [ 1, 1] (0, ) be a smooth function. We consider the surface generated by the graph of f, the parametrization of which is given by (x, ϕ) r(x, ϕ) = (x, f(x) cos ϕ, f(x) sin ϕ). (4.0.4) Here, we consider x = x 1 as first and ϕ = x 2 as second parameter. The first and second fundamental forms, and the normal to the surface of revolution, are given as follows: (g ij ) = (L ij ) = v(x, ϕ) = ( (1 + f (x)) f(x) 2 ), ( ) 1 f (x) 0, 1 + f (x) 2 0 f(x) f (x) (f (x), cos ϕ, sin ϕ). 2 We use the sign convention that the mean curvature H is positive if the surface is mean convex and negative if it is mean concave with respect to the interior normal v. The mean curvature and Gauss curvature are given respectively by f (x) H = 2(1 + f (x) 2 ) + 1 3/2 2f(x) 1 + f (x), (4.0.5) 2 f (x) K = f(x)(1 + f (x) 2 ). (4.0.6) 2 The Laplace-Beltrami operator on the surface of revolution acts on smooth functions h as follows 24

31 g h = 1 g = 2 i ( gg ij j h) (4.0.7) i,j=1 ( )) 1 f(x) f(x) ( x 1 + f (x) f (x) xh, 2 where g ij are the entries of the inverse of g ij. The Willmore functional of the surfaces Γof revolution is given by W (Γ) = H 2 ds = π Γ ( ) 1 f(x) 1 + f (x) f 2 (x) f(x) 1 + f 2 (1 + f (x) 2 ) (x) 2 dx 3/2 Theorem 11. [27] For every α > 0, there exists a smooth function f C ([ 1, 1], (0, )) such that the corresponding surface of revolution solves the Dirichlet problem for the Willmore equation in ( 1, 1), R H + 2H(H 2 K) = 0 (4.0.8) f(±1) = α, H(±1) = 0. For a proof of this theorem, see[27]. Consider the Willmore-type equation(4.0.8) with the conditions f(±1) = α, H(±1) = 0. where R H = 1 f 1 + f (x) 2 ( d dx f dh 1 + f (x) 2 dx ). (4.0.9) In (4.0.8), H and K are the mean curvature and the Gauss curvature of a surface, respectively. 25

32 Moreover, using (4.0.5) and (4.0.6) K can be expressed in terms of H as follows: ( ) 2 K = f 1 H 1 + f (x) 2 2f. (4.0.10) 1 + f (x) 2 By using (4.0.9) and (4.0.10), we rewrite (4.0.8) to get ( ) d f dh + 2H(H 2 K)f 1 + f dx 1 + f (x) 2 dx (x) 2 = 0 (4.0.11) which implies ( ) d f dh dx 1 + f (x) 2 dx +2Hf ( ) ) 1 + f (x) (H f 1 H 1 + f (x) 2 2u = 0, 1 + f (x) 2 ( ) d f dh dx 1 + f (x) 2 dx ( + 2H (H 2 )f 1 + f (x) 2 2H + Observe that (4.0.5) can be expressed as the divergence form ( ) d f dx 1 + f (x) 2 = 2H ) 1 f = f (x) 2 (4.0.12) 1 f 1 + f (x) 2 (4.0.13) 4.1 Numerical results using COMSOL multi-physics software Rearranging the equations (4.0.12) and (4.0.13), one can obtain: ( ) d f dh 1 2H dx 1 + f (x) 2 dx f = 2H 1 + f (x) ( ) 2 d 1 df = 2H dx 1 + f (x) 2 dx ( (H 2 )f ) 1 + f (x) 2 2H, 1 f 1 + f (x) 2, 26

33 which can be rewritten in the matrix form ( f 0 1+f f 2 ) f f 1+f 2 f = 0 0 2H ( (H 2 )f 1 + f (x) 2 2H 1 2H f 1+f (x) 2 (4.1.1) ( ) H Here, f = f Consider the interval [ 1, 1]. We numerically compute the profile curve f(x) with the boundary conditions: f(±1) = α and H(±1) = 0. (4.1.2) ). In order to obtain profile curves we use COMSOL Multiphysics software with the boundary value conditions given above. At the endpoints of the domain [ 1, 1], we consider that the mean curvature of the surface is constant which equals to zero. If we consider f(±1) = α and H(±1) = β then α and β are the parameters which are the Dirichlet boundary conditions of (4.0.8) and in this case β = 0. Since (4.1.1) is in the form of coefficient form PDE 1 the diffusion coefficients can be found without effort and then we plot the graphs as follows. Table 4.1. COMSOL model builder Parameters x 1 = 1, x 2 = 1,α = 1, β = 0 Variables NL (1 + f 2 x), fnlf NL c 11 = f/nl c 22 = 1/NL, a 21 = 2, f 2 = 1/(fNL) a 11 = 2 ( 2 H + 1/(fNL) + (H 2 ) fnl) K = 2/(fNL) (H 1/(2 fnl)) 27

34 Figure 4.1. The mean curvature and the profile curve f(x) in the domain[ 1, 1]. On this graph the blue line represents the mean curvature and green line represents the profile curve f(x) when the parameter α = 1. Now we vary the parameter α from 1 to 5 with step size 1. Then we obtain the following graphs for H and f(x). When α = 2 we notice that the mean curvature is zero and f(x) is Figure 4.2. The mean curvatures in the domain[ 1, 1] when α = 1 : 1 : 5 28

35 Figure 4.3. The profile curves in the domain[ 1, 1] when α = 1 : 1 : 5 nonzero in the domain [ 1, 1] (thus corresponding to a catenary/catenoid). Now we consider the case α = 1 : 10 : 50 and plot the following graphs. Now we Figure 4.4. The mean curvatures in the domain[ 1, 1] when α = 1 : 10 : 50 consider the case α = 1 : 20 : 100 and compute the following graphs. 29

36 Figure 4.5. The profile curves in the domain[ 1, 1] when α = 1 : 10 : 50 Figure 4.6. The mean curvatures in the domain[ 1, 1] when α = 1 : 20 :

37 Figure 4.7. The profile curves in the domain[ 1, 1] when α = 1 : 20 :

38 CHAPTER 5 CONCLUSIONS AND FURTHER RESEARCH DIRECTIONS In this work, we have studied two important classes of surfaces of revolutions which arise from physical and real-world applications, but have been introduced in a purely mathematical manner: Delaunay surfaces and Willmore surfaces of revolution. Their intersection contains the spheres and the catenoidal surfaces. We presented Charles Delaunay s original construction of CMC surfaces of revolution, based on roulettes of conics, after which we characterized these geometric objects as solutions to an ordinary differential equation which was given it its most general form, whose three particular cases correspond to the undulary, catenary and nodary profiles. We presented a few physical models of Delaunay surfaces arising as liquid bridges between two vertical walls - which were proved to be unduloidal surfaces, by methods of Variational Calculus. We numerically constructed (using MATLAB) the profile curves of these surfaces and provided some graphical models for them. By contrast, we studied Willmore surfaces as minimizers of the Willmore (bending energy) energy. In particular, we have studied some Willmore surfaces of revolution which come in as solutions to BVP problems consisting of the Willmore equation, together with some Navier - Dirichlet type boundary value conditions. With help from Dr. Eugenio Aulisa and Bhagya Athukorallage, I have provided some numerical computations for the profiles of these surfaces, using COMSOL Multiphysics. Willmore surfaces of revolution have lots of application in the real world, such as elastic biological membranes which are topologically cylinders. At the microbiological level, a model of such Willmore surfaces of revolution is provided by the beta barrels arising from secondary structures in proteins (beta sheets configured as a rotationally symmetric model). In this case, we converted the original Willmore equation and the mean curvature expression to a Divergence-form equation in order to compute the mean curvature and profile curves by using COMSOL. We noticed that for certain values of u at the endpoints 1 and +1, which depend on the given data, the mean curvature H of the corresponding profile f becomes zero, which implies that the Willmore surface becomes a minimal surface in that case. 32

39 I am currently performing research on the mathematics of beta-barrels, which can be represented as Willmore surfaces of revolution. A thorough study on when beta barrels can be approximated by minimal surfaces is in progress. Past models of beta sheet and beta barrels considered three major geometric models: a). the one-sheeted hyperboloidal model; b). the strophoidal model; c). the catenoidal model. Through my collaborative research while at Texas Tech University, I am currently able to propose the model of generalized Willmore surfaces of revolution as a unique model of beta barrels, based on their physical properties and their bending energy. This model is meant to replace the prior models a). and b). which appear in a few papers from the microbiology literature of secondary structures for proteins. The third model, c)., is appropriate only for the case when the Willmore solution represents a minimal surface, which is not the case for beta barrels in general. 33

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