Capillarity is the phenomena of fluid rise against a solid vertical wall.   For our research, we consider bounded cases of intermediate corner   angles ( p/2\u3c a + g \u3c   p/2+2 g ), where g is the   angle of contact and 2a is the wedge angle. The Laplace-Young   Capillary equations are used to determine the rise of the fluid, especially at corners.   While there exist asymptotic expansions for the height rise occurring at the corner   of an intermediate angle, not all coefficients are known analytically. Therefore,   numerical solutions are necessary, even though only a few numerical methods have   been published. We explain our least-squares finite element method used in   determining solutions to the Laplace-Young Capillary equations, and then give our numerical results

Dupuis, Genevieve

Flores, Jessica

English

Rose-Hulman Institute of Technology: Rose-Hulman Scholar

Rose-Hulman Undergraduate Mathematics Journal Volume 10 Issue 1 Article 5 Numerical Solutions for Intermediate Angles for the Laplace-Young Capillary Equations Genevieve Dupuis University of Notre Dame, gdupuis@nd.edu Jessica Flores University of Puerto Rico Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Dupuis, Genevieve and Flores, Jessica (2009) "Numerical Solutions for Intermediate Angles for the Laplace-Young Capillary Equations," Rose-Hulman Undergraduate Mathematics Journal: Vol. 10 : Iss. 1 , Article 5. Available at: https://scholar.rose-hulman.edu/rhumj/vol10/iss1/5 Numerical Solutions for Intermediate Angles of the Laplace-YoungCapillary EquationsGenevieve DupuisUniversity of Notre DameJessica FloresUniversity of Puerto Rico at HumacaoAbstractCapillarity is the phenomena of fluid rise against a solid vertical wall. In this paper, we considerbounded cases of intermediate corner angles (π/2 < α + γ < π/2 + 2γ) , where γ is the angle of contactand 2α is the wedge angle. The Laplace-Young Capillary equations are used to determine the rise ofthe fluid, especially at corners. While there exist asymptotic expansions for the height rise occurringat the corner of an intermediate angle, not all coefficients are known analytically. Therefore, numericalsolutions are necessary, even though only a few numerical methods have been published. We explainour least-squares finite element method used in determining solutions to the Laplace-Young Capillaryequations, and then give our numerical results.1 IntroductionThe solutions of the Laplace-Young Capillary equations, u(x, y), give the height rise of liquid at a givenpoint in Cartesian coordinates (x, y) in the domain Ω. In terms of polar coordinates, u(r, θ) is equivalent tothe height rise at a point (r, θ), with the conversion x = rcosθ, y = rsinθ. On domains containing a wedgeangle, much attention has been given to developing asymptotic solutions to the height at the corner point[7], [9], r = 0. The angle 2α measures the wedge, where θ = 0 bisects the wedge angle. The angle given bythe contact of the liquid against the vertical wall is γ (see Figure 1). Corner angles are classified into threedifferent categories: small, intermediate, and large; previously determined analytic solutions are given belowfor each case.1) Small angles: 0 < α < π/2 − γAs the solution for the small angles approaches the corner, the height rise, u(r, θ), is unbounded [5]:u ∼ cosθ − (a2 − sin2θ)1/2ar+ O(r3),where a = sinα/cosγ and θ = ±α.2) Intermediate angles: π/2 − γ < α < π/2 + γThe analytic solutions that have been determined for intermediate angles are bounded and in the formof a power series expansion [5]:u ∼ u0 −rcosθ(a2 − 1)1/2 +· · · .3) Large angles: π/2 + γ < α < πHeight solutions for large angles remain bounded, but the slopes approaching r = 0 are not. An analyticalsolution to the large angles is not yet given and knowledge of the far field condition is required to determineu [7].1Figure 1: α and γ2 Laplace-Young Capillary Equations and Analytical SolutionsSolving the Laplace-Young equations gives the height rise of fluid at a given point. These equations are asfollows:∇ ·(∇u√1 + |∇u|2)= κu in Ωn ·(∇u√1 + |∇u|2)= cos γ on ∂Ω,(1)where κ is the capillary constant of the liquid. κ > 0 indicates positive gravity, κ < 0 indicates negativegravity, and κ = 0 indicates zero-gravity scenario. The unit vector normal to the boundary ∂Ω is denotedby n.The focus in this paper is the intermediate corner angles, and the asymptotic power series solution forthese bounded solutions is as follows:u(r, θ) = u0 + ru1(θ) + r2u2(θ) + r3u3(θ) + ... =i∑i=0riui(θ).Here, u0 is given to be the height rise of the solution at r = 0 and u1(θ) =cosθ(a2−1)1/2, which determinesthe slope to the solution as r → 0 and θ = ±α. The coefficients of this power series, ui(θ), are known interms of u0. However, since u0 is not known analytically, the only coefficient truly known is u1(θ) becauseit does not depend on θ. Thus, once u0 is determined, the remainder of the power series solution is knownanalytically[8]. Note that this power series is the same solution given in the introduction.3 Numerical Methods3.1 Rescaling Laplace-YoungIn solving the Laplace-Young equations, a rescaling of the equations to capillary unit lengths (κ) simplifiesthe problem. Taking the original form of the Laplace-Young equations with a solution of uold , and making2a substitution of uold = Lunew, where L = 1/κ, (1) becomes the following equations:∇ ·(∇Lunew√1 + |∇Lunew|2)= κLunew in Ωn ·(∇Lunew√1 + |∇Lunew|2)= cos γ on ∂Ω.(2)Where ∇Lunew occurs, the substitution (in Cartesian coordinates) xold = Lxnew and yold = Lynew is used,noting that L is a constant:∇Lu =(LuxLuy)=(∂Lu/∂Lx∂Lu/∂Ly)=(uxuy)= ∇u.It follows that ∇Lunew = ∇unew and (1) becomes the rescaled Laplace-Young equation:∇ ·(∇u√1 + |∇u|2)= u in Ωn ·(∇u√1 + |∇u|2)= cos γ on ∂Ω.(3)The scale for the solutions on Ω is now given in capillary length units, and this form of the Laplace-Youngequations will be used throughout the rest of the paper.3.2 LinearizationSince (3) has nonlinear terms associated with it, a linearization of the equations will be done with a currentapproximation of the unknown, such thatb ≈ u.An initial value of b must be assigned, and b = 0 works as a first approximation. Now, we consider the seriesof linear problems from∇ ·(∇u√1 + |∇b|2)= u in Ωn ·(∇u√1 + |∇b|2)= cos γ on ∂Ω.(4)For simplification of notation, letA = A(∇b) = 1√1 + |∇b|2,and A(∇b) will be the known approximation of the nonlinear system. The system can then be written as{∇ · (A∇u) − u = 0n · (A∇u) − cosγ = 0. (5)Using the product rule and the property ∇ · (∇) = ∆, (5) becomes{A∆u + ∇A · ∇u − u = 0n · (A∇u) − cosγ = 0.3In addition to these equations, define U = ∇u as a new unknown to approximate. Also true is that ∇×∇ = 0,and therefore another equation can be added: ∇ × U = 0 in the domain Ω. Combining all equations, thePDE system now looks likeA∇ · U + ∇A · U − u = 0∇× U = 0 in Ω∇u− U = 0,(6)with a boundary condition ofn · (AU) − cosγ = 0 on ∂Ω.With A = A(∇b), the system is now linear.3.3 Least-Squares Finite ElementWe seek to find a solution to (6) by the least-squares finite element approach. To do this we define thefollowing least-squares functional:G(U, u) = ||A∇ · U + ∇A · U − u||2 + ||∇ × U ||2 + ||∇u− U ||2,for all u ∈ V = {v ∈ H1(Ω) : n · ∇v = cos γ} and U ∈ U = {V ∈ H1(Ω) : n · V = cos γ}. MinimizingG(U, u) over U × V yields a symmetric positive definite variational problem. When U and V are replacedwith appropriate finite element spaces, the problem becomes a large positive definite system of algebraicequations which can be solved efficiently. For more details on the least-squares finite element approach,please see [3].We thus solve a sequence of linear problems of the form (6), where each iteration takes b = uold andu = unew. As we iterate, the solution tends toward that of (3), the original nonlinear problem.4 Convergence of Solutions4.1 Sobolev SpacesWe begin our discussion of convergence by defining our notation for Sobolev Spaces [1]. Let W k,p(Ω) denotethe standard Sobolev Space defined by integers k and p, whereu ∈ W k,p ⇔ (∑|α|≤k|Dαu|p)1/p ≡ ||u||k,p < ∞.When p = 2, we simplify the notation by introducing the spacesHk(Ω) = W k,2(Ω),and use the norm || · ||k = || · ||k,2.Consider an illustration of this notation in the one dimensional case: If a single-variable function iscontained in the Sobolev Space H0(Ω), the following condition holds:∫ 10f(t)2dt < ∞.To be contained in the Sobolev space H1(Ω), a function, f , and its first derivative, f ′, must be boundedin the interval [0, 1]:4∫ 10(f(t)2 + f ′(t)2)dt < ∞.And likewise, this can be generalized such that if a function is contained in the Sobolev Space Hn(Ω) if∫ 10(f(t)2 + f ′(t)2 + . . . + f(n)(t)2)dt < ∞.With these definitions, it can be seen that the Sobolev Spaces embed into each other. When a functionis contained in H2(Ω) for example, it is also contained in H1(Ω) and H0(Ω). Similar definitions hold for themulti-variable case that we present.4.2 Convergence of the Least-Squares SolutionNow, convergence of the least-squares solution is consistent with the convergence of the functional. Previousresults from Cai et al [3] have been given to confirm functional convergence, and showing that the givenmethod fits into the case outlined will prove the convergence of the solutions. To begin, the form of theproblem must be given as follows:{∇ · (A∇u) − u = f in Ωn · A∇u = 0 on δΩ.Therefore, the Laplace-Young equations must be modified into this form. To do so, take the originalsystem:{∇ · A∇ũ − ũ = 0 in Ωn · A∇ũ = cosγ on δΩ (7)and consider the related problem{∆u0 = 0 in Ωn · A∇u0 = cosγ on δΩ,(8)which has a solution u ∈ Hk(Ω) that is unique up to a constant, where k < π/2 + 1. [4]. And from this, (7)similarly has a solution ũ ∈ Hk(Ω). Note: α can be chosen sufficiently small in order to find a sufficientlylarge k.Let u = u0 − ũ, then∇ ·A∇u − u = ∇ ·A∇(u0 − ũ) − (u0 − ũ)= ∇ ·A∇u0 − u0≡ f.Notice that f ∈ Hk−2(Ω) since there are two derivatives of u. And on δΩ,n ·A∇u = n · A∇u0 − n · A∇ũ = 0.Combining these two equations, a system matching the required form is created:{∇ · (A∇u) − u = f in Ωn · A∇u = 0 on δΩ. (9)5Now that the system given has the correct form, assumptions about the problem must be satisfied[3].Assumption A0:∫Ωu = 0 so that a Poincare inequality holds. It is known that ũ satisfies∫ũ = 0, andwe may choose u0 ∈ Hk(Ω)/R so that∫u = 0 as well.Assumption A1: Ω is bounded, open, connected, convex, and piecewise C1,1. We consider only suchdomains.Assumption A2: The boundary of Ω is composed of Dirichlet and Neumann parts. The boundaryconditions in equation (9) are of Neumann type.Assumption A3: A is C1,1.Recall the linearized form of the Laplace-Young equations:{A(∇b)∇ ·U + ∇A(∇b) ·U − u = 0n · (A(∇b)U) − cosγ = 0, (10)where A(∇b) = 1√1+|∇b|2and b ≈ u is the old solution used to linearize the system.A Sobolev space embeds into a continuous space in Rn under the following conditions from [4]Wm,p(Ω) ↪→ C0,1(Ω),wherea) m ≥ 0 and 1 ≤ p ≤ ∞ where m and p are integers,b) np+ 1 < m,c) Ω is open, connected, Lipschitz boundary.Therefore,if u ∈ Wm+j,p(Ω), then∑|α|≤j|Dαu| ∈ Wm,p(Ω) ↪→ C0,0(Ω),which means that u ∈ Cj,1(Ω) for any j ∈ N.More specifically, for the Laplace-Young equations, n = 2, p = 2, and j = 1. Choose some δ > 0, so thatwhen ∇b ∈ Hm+1(Ω), it embeds into C1,1(Ω) for m = 2 + δ. Thus,b ∈ H4+δ(Ω) ⇒ ∇b ∈ H3+δ(Ω) ↪→ C1,1(Ω).Lemma 4.1 Let A(f) = 1√1+|f|2. If f ∈ C1,1(Ω), then A(f) ∈ C1,1(Ω).Proof. Since f ∈ C1,1(Ω) for a given f , then ∇f ∈ C0,1(Ω). Thus, it must be shown that A(f) and ∇A(f)are contained in C0,1(Ω). Since the function A is continuous, and the composition of continuous functions isalso continuous, then it follows that A(f) ∈ C0,1(Ω). Now, ∇A(f) = −f∇f(1+f2)3/2, which is also a compositionof continuous functions, and therefore also continuous and in C0,1(Ω).Note: u0 ∈ Hk(Ω) = H4+δ(Ω) for 4 + δ < π2α + 1. Solving for the domain angle, α < π2(3+δ) . Therefore,a choice of α < π6 if δ ∈ R or α < π8 if δ ∈ N.6Let Th be a regular triangulation of the domain, Ω, and assume there exist two finite element approxi-mation subspaces defined over Th:Uh ⊂ U and Vh ⊂ V.The finite element approximation to the problem is determining (Uh, uh) ∈ Uh ×Vh such thatG(Uh, uh) ≤ G(Vh, vh) ∀ (Vh, vh) ∈ Uh × Vh. (11)Uh and Vh are defined as piecewise linear finite element spaces:Vh = {v ∈ C0(Ω) : q|k ∈ P1(K) ∀ K ∈ Th, v ∈ V}andUh = {V ∈ C0(Ω)n : vl|k ∈ P1(K) ∀ K ∈ Th, V ∈ U},where P1(K) is the space containing polynomials with degree no more than one. Note that extending theresults is easy to show for finite element approximations with higher-order.Theorem 4.2 Given the form of the problem and A0 − A3. Assume that the solutions (U, u) of (9) arecontained in H1+δ(Ω)n+1 for some δ ∈ [0, 1]. Let (Uh, uh) ∈ Uh × Vh be the solution of (11). Then,‖u − uh‖1,Ω + ‖U − Uh‖1,Ω ≤ Chδ(‖u‖(1+δ),Ω + ‖U‖(1+δ),Ω),where C does not depend on h, u, or U .Proof. Using the results of the assumptions given and showing the form of the problem matches, the prooffollows directly from [3].Therefore, functional convergence of the solution works to confirm the convergence to the correct desiredsolution of the Laplace-Young equations.5 Numerical ResultsNumerical results are given for intermediate angles and the domain used in computation is defined as:Ω = {(r, θ); 0 < r < R,−α < θ < α} .Table 1 demonstrates the convergence of the numerical slope to analytical solutions. As the discretization,Th, of the domain is further refined with increasing values of n, the slope of the solution becomes arbitrarilyclose to the analytical solution. If h is defined as the width of each triangle along ∂Ω in Th, then n = 1/h.Near the corner, a finer discretization is used with 2n divisions, and further away there are only n divisionsso that there is more focus near r = 0. An illustration of the triangulation of Ω is given in Figure 2 for avalue of n = 16.Table 1 shows this trend in the case where α = π/3 and γ = π/4. Recall that the slope of the solutionas r → 0 is given in the power series solution byu1 =cosθ(a2 − 1)1/2 ,and for our specific case, θ = α = π/3 and γ = π/4. Therefore, the desired analytical slope is u1 =√2/2 ≈ 0.707107. Table 1 shows the error in the slope is reduced approximately linearly, giving confidencein the method and our approximation to u0.7Figure 2: Triangulation of Ω, n = 16n slope error in slope height8 -0.568429 0.138678 1.2508616 -0.652611 0.054496 1.237532 -0.693487 0.013620 1.2237364 -0.701674 0.005433 1.21642128 -0.703394 0.003713 1.2145............∞√2/2 ≈ 0.707107 0.0 unknownTable 1: Slope Convergence: α = π/3 and γ = π/48γ α = π/3 α = π/440 1.3692 n/a45 1.2145 n/a50 1.0683 1.5486255 0.928488 1.3014660 0.792122 1.0899465 0.65801 0.89468570 0.525264 0.70850775 0.393391 0.52772980 0.262018 0.35024885 0.130943 0.174659Table 2: Heights: n = 128In Table 2, various combinations of α and γ are given, along with the numerical results for their heights.In all cases, n = 128 and similar analysis of the convergence of the numerical slopes to the analytical slopeswas confirmed.We would like to briefly give acknowledgements to those who assisted us in our research at Wabash CollegeSummer Institute of Math. Thank you to our advisor of this paper, Chad Westphal, Wabash College. Finally,we would like to thank Yasunori Aoki and Hans De Sterck, University of Waterloo and the National ScienceFoundation, grant DMS-0755260.References[1] R. Adams and J. Fournier, Sobolev Spaces, Second Ed., Academic Press, 2003.[2] Yasunori Aoki, Numerical Solutions for the Young-Laplace Capillary Equations, 2008.[3] Zhiqiang Cai, Thomas A. Manteuffel, and Stephen F. McCormick. First-Order System Least-Squares forSecond-Order Partial Differential Equations: Part II, Society for Industrial and Applied Mathematics,1997.[4] P.G. Claret, The Finite Element for Elliptic Problems, Society for Industrial and Applied Mathematics,2002.[5] P. Concus and R. Finn, On a Class of Capillary Surfaces, J. Analyse Math, 1970.[6] Robert Finn, Equilibrium Capillary Surfaces, Springer-Verlag, 1986.[7] J.R. King, J.R. Ockendon, and H. Ockendon, The Laplace-Young Equation Near a Corner, Q.J. Me-chanics and Applied Mathematics, 1999.[8] E. Miersemann. Asymptotic Expansion at a Corner for the Capillary Problem. Pacific J. Maths, 1988.[9] J. Norbury, G.C. Sander, and C.F. Scott, Corner Solutions of the Laplace-Young Equations, OxfordUniversity Press, 2004.[10] J. Norbury, G.C. Sander, and C.F. Scott, Computation of Capillary Surfaces for the Laplace-YoungEquations, Oxford University Press, 2005.[11] D. Siegel, Height Estimates for Capillary Surfaces, Pacific J. Maths, 1980.9

Numerical Solutions for Intermediate Angles for the Laplace-Young Capillary Equations

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Numerical Solutions for Intermediate Angles for the Laplace-Young Capillary Equations

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