Floating Drops. Ray Treinen. January 26, University of Toledo. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

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1 Floating Drops Ray Treinen University of Toledo January 26, 2007 Ray Treinen (University of Toledo) Floating Drops January 26, / 32

2 Overview Introduction and basic surfaces Theory for ODE method for the floating drop. Numerical results for the ODE method for the floating drop. Floating drops and functions of bounded variation. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

3 Equations The equilibrium shape of the interface between liquid and air can be given in terms of the Young-Laplace equation ( ) u = κu 1 + u 2 u is the height of the fluid interface κ = ρg/σ is the capillary constant ρ is the density of the fluid σ is the surface tension of the fluid surface g is the acceleration due to gravity This can be seen as twice the mean curvature is proportional to height. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

4 Equations Assuming the solution is radially symmetric, the Young-Laplace equation can be written as a system: dr dψ = r cos ψ κru sinψ du dψ = r sinψ κru sinψ, r is the radial coordinate ψ is the inclination angle of the generating curve, measured up from horizontal Ray Treinen (University of Toledo) Floating Drops January 26, / 32

5 Equations Assuming the solution is radially symmetric, the Young-Laplace equation can be written as a system: dr dψ = r cos ψ κru sinψ du dψ = r sinψ κru sinψ, r is the radial coordinate ψ is the inclination angle of the generating curve, measured up from horizontal This holds on any portion of the solution curve that does not contain an inflection point Ray Treinen (University of Toledo) Floating Drops January 26, / 32

6 Equations Assuming the solution is radially symmetric, the Young-Laplace equation can be written as a system: dr dψ = r cos ψ κru sinψ du dψ = r sinψ κru sinψ, r is the radial coordinate ψ is the inclination angle of the generating curve, measured up from horizontal This holds on any portion of the solution curve that does not contain an inflection point The solution to this system may be extended past a vertical point Ray Treinen (University of Toledo) Floating Drops January 26, / 32

7 Equations One further form of the differential equations: dr ds du ds dψ ds = cos ψ = sinψ = κu sinψ. r s is the arclength of the curve Ray Treinen (University of Toledo) Floating Drops January 26, / 32

8 Equations One further form of the differential equations: dr ds du ds dψ ds = cos ψ = sinψ = κu sinψ. r s is the arclength of the curve The solution to this system may be extended past both vertical points and inflection points Ray Treinen (University of Toledo) Floating Drops January 26, / 32

9 Basic Surfaces Capillary Tube Fluid Ray Treinen (University of Toledo) Floating Drops January 26, / 32

10 Basic Surfaces Annular Capillary Surface Fluid Fluid Ray Treinen (University of Toledo) Floating Drops January 26, / 32

11 Basic Surfaces Sessile Drop Fluid Ray Treinen (University of Toledo) Floating Drops January 26, / 32

12 Basic Surfaces Pendent Drop Fluid Ray Treinen (University of Toledo) Floating Drops January 26, / 32

13 Basic Surfaces 0.5 Pendent Drop Fluid Ray Treinen (University of Toledo) Floating Drops January 26, / 32

14 Basic Surfaces Unbounded Liquid Bridge Fluid Ray Treinen (University of Toledo) Floating Drops January 26, / 32

15 Floating Drops: Theory for ODE method Consider three fluids, fluid 1 finite, and fluid 0 and fluid 2 occupy the remainder of an infinite reservoir. Let the densities satisfy ρ 0 < ρ 1 < ρ 2. Given the surface tensions σ 01,σ 02,σ 12, then corresponding capillary constants are defined by κ 01 = (ρ 1 ρ 0 )g/σ 01, κ 02 = (ρ 2 ρ 0 )g/σ 02, κ 12 = (ρ 2 ρ 1 )g/σ 12. Minimization of energy implies that the three surfaces satisfy the equations Mu = κ 01 u + λ/σ 01, Mv = κ 12 v λ/σ 12, Mw = κ 02 w M is the mean curvature operator λ is a constant to be determined Ray Treinen (University of Toledo) Floating Drops January 26, / 32

16 Floating Drops: Theory for ODE method Floating Drop Configuration Ray Treinen (University of Toledo) Floating Drops January 26, / 32

17 Floating Drops: Theory for ODE method Force balance implies 0 γ 01,γ 02,γ 12 π, and ( σ 2 γ 02 = π arccos 01 + σ12 2 σ2 02 2σ 01 σ 12 γ 01 = π arccos ( σ σ 2 02 σ2 01 2σ 12 σ 02 Fix the radius r at which the three surfaces meet Let ψ be the inclination angle v at r. ), ). Ray Treinen (University of Toledo) Floating Drops January 26, / 32

18 Floating Drops: Theory for ODE method Match the top and the bottom surfaces, u and v, at r by eliminating the Lagrange multiplier λ. Define U and V by u = U λ/κ 01 σ 01 and v = V + λ/κ 12 σ 01. So that MU = κ 01 U, MV = κ 12 V. u = v at r = r implies λ = [Ū + V] κ 01 σ 01 κ 12 σ 12. κ 01 σ 01 + κ 12 σ 12 Ray Treinen (University of Toledo) Floating Drops January 26, / 32

19 Floating Drops: Theory for ODE method Match the top and the bottom surfaces, u and v, at r by eliminating the Lagrange multiplier λ. Define U and V by u = U λ/κ 01 σ 01 and v = V + λ/κ 12 σ 01. So that MU = κ 01 U, MV = κ 12 V. u = v at r = r implies λ = [Ū + V] κ 01 σ 01 κ 12 σ 12. κ 01 σ 01 + κ 12 σ 12 The difference of u and w at r is F( ψ) = κ 12σ 12 V κ01 σ 01 Ū κ 01 σ 01 + κ 12 σ 12 w. Vary ψ. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

20 Floating Drops: Theory for ODE method Theorem (Elcrat, Neel, Siegel) For each r, there is a drop in which the three surfaces meet at radius r. Theorem (Elcrat, Neel, Siegel) Suppose that 0 < γ 02 < π/2. Then there is a drop of volume V for every prescribed V > 0. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

21 Floating Drops: Numerical results for ODE method Light Drop, unbounded region rho 0 = 0, rho 1 = 8, rho 2 = 20, sigma 01 = 25, sigma 02 = 13, sigma 12 = height(u) radius(r) Ray Treinen (University of Toledo) Floating Drops January 26, / 32

22 Floating Drops: Numerical results for ODE method Light Drop, bounded container rho 0 = 0, rho 1 = 17, rho 2 = 20, sigma 01 = 17, sigma 02 = 15, sigma 12 = height(u) radius(r) Ray Treinen (University of Toledo) Floating Drops January 26, / 32

23 Floating Drops: Numerical results for ODE method 0.6 Light Drop, annular container rho 0 = 0, rho 1 = 17, rho 2 = 20, sigma 01 = 15, sigma 02 = 23, sigma 12 = height(u) radius(r) Ray Treinen (University of Toledo) Floating Drops January 26, / 32

24 Floating Drops: Numerical results for ODE method Heavy Drop, unbounded container 0.06 r =0.030, rho 0 = 0, rho 1 = 45, rho 2 = 10, sig 01 = 20, sig 02 = 12, sig 12 = height(u) radius(r) Ray Treinen (University of Toledo) Floating Drops January 26, / 32

25 Floating Drops: Numerical results for ODE method Heavy Drop, unbounded container, with a neck r =0.050, rho 0 = 0, rho 1 = 14, rho 2 = 10, sig 01 = 13, sig 02 = 20, sig 12 = height(u) radius(r) Ray Treinen (University of Toledo) Floating Drops January 26, / 32

26 Floating Drops: Numerical results for ODE method Heavy Drop, unbounded container, with four necks r =0.050, rho 0 = 0, rho 1 = 14, rho 2 = 10, sig 01 = 13, sig 02 = 20, sig 12 = height(u) radius(r) The Heavy Floating Drop is not unique. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

27 Floating Drops and Functions of Bounded Variation Let Ω R 3 be bounded, open, and Lipschitz. Let E 0,E 1,E 2 Ω be such that E i E j = for i j and E i = Ω. These are the three fluids. Let σ 01,σ 02,σ 12 the surface tensions between each fluid. Take σ ij 0 for i,j 3. Let σ 03,σ 13,σ 23 be the wetting energy of each fluid with the wall. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

28 Floating Drops and Functions of Bounded Variation Definition The energy functional is F(E 0,E 1,E 2 ) = 2 ) (α i Dχ Ei + σ i3 χ Ei dh 2 + ρ i g zdv. Ω Ω E i i=0 The first term measures the total variation of the characteristic function of the set: think of this as the boundary of the set. { } Dχ Ei = sup χ Ei divφ φ Cc 1 (Ω; R3 ), φ 1 Ω Ω Ray Treinen (University of Toledo) Floating Drops January 26, / 32

29 Floating Drops and Functions of Bounded Variation Definition The energy functional is F(E 0,E 1,E 2 ) = 2 ) (α i Dχ Ei + σ i3 χ Ei dh 2 + ρ i g zdv. Ω Ω E i i=0 The first term measures the total variation of the characteristic function of the set: think of this as the boundary of the set. { } Dχ Ei = sup χ Ei divφ φ Cc 1 (Ω; R3 ), φ 1 Ω Ω α 0 = 1 2 (σ 01 + σ 02 σ 12 ) α 1 = 1 2 (σ 01 + σ 12 σ 02 ) α 2 = 1 2 (σ 02 + σ 12 σ 01 ) Ray Treinen (University of Toledo) Floating Drops January 26, / 32

30 Floating Drops and Functions of Bounded Variation Definition The energy functional is F(E 0,E 1,E 2 ) = 2 ) (α i Dχ Ei + σ i3 χ Ei dh 2 + ρ i g zdv. Ω Ω E i i=0 The first term measures the total variation of the characteristic function of the set: think of this as the boundary of the set. { } Dχ Ei = sup χ Ei divφ φ Cc 1 (Ω; R3 ), φ 1 Ω Ω α 0 = 1 2 (σ 01 + σ 02 σ 12 ) α 1 = 1 2 (σ 01 + σ 12 σ 02 ) α 2 = 1 2 (σ 02 + σ 12 σ 01 ) The second term measures the trace on the boundary of the domain. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

31 Floating Drops and Functions of Bounded Variation Definition The energy functional is F(E 0,E 1,E 2 ) = 2 ) (α i Dχ Ei + σ i3 χ Ei dh 2 + ρ i g zdv. Ω Ω E i i=0 The first term measures the total variation of the characteristic function of the set: think of this as the boundary of the set. { } Dχ Ei = sup χ Ei divφ φ Cc 1 (Ω; R3 ), φ 1 Ω Ω α 0 = 1 2 (σ 01 + σ 02 σ 12 ) α 1 = 1 2 (σ 01 + σ 12 σ 02 ) α 2 = 1 2 (σ 02 + σ 12 σ 01 ) The second term measures the trace on the boundary of the domain. The third term is the gravitational energy. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

32 Floating Drops and Functions of Bounded Variation Lemma σ ij 0 for i,j = 0,1,2, i j if and only if F >. Lemma (Massari) Denote the Lipschitz constant of Ω by L and let α i 0 and σ ij 0 and then F is lower semi-continuous. σ ij 1 + L 2 σ i3 σ j3 i,j = 0,1,2, Ray Treinen (University of Toledo) Floating Drops January 26, / 32

33 Floating Drops and Functions of Bounded Variation We seek to minimize F in K = {(E 0,E 1,E 2 ) E i = Ω,E i E j =,i j, E i = v i, v i = Ω } where v i are the prescribed volumes of each fluid and E i are sets of finite perimeter. Theorem Let L be the Lipschitz constant for Ω and let α i 0 and σ ij 0 and σ ij 1 + L 2 σ i3 σ j3 i,j = 0,1,2. Then there exists (E 0,E 1,E 2 ), where E i are sets of finite perimeter, that minimizes F over K. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

34 Floating Drops and Functions of Bounded Variation Next minimize F in K, the elements of K that are radially symmetric. Theorem (Elcrat and Treinen) Let L be the Lipschitz constant for Ω and let α i 0 and σ ij 0 and σ ij 1 + L 2 σ i3 σ j3 i,j = 0,1,2. Then there exists (E 0,E 1,E 2 ), where E i are sets of finite perimeter, that minimizes F over K. Note that the minimizers here are symmetric about the vertical axis. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

35 Floating Drops and Functions of Bounded Variation Definition For any compact set K, the energy functional is F K (E 0,E 1,E 2 ) = 2 (α i i=0 K Dχ Ei + ρ i g K χ Ei zdv ). Definition A solution to the minimization problem is the triple (E 0,E 1,E 2 ) if F K (E 0,E 1,E 2 ) F K (A 0,A 1,A 2 ) for any (A 0,A 1,A 2 ) with i (K A i ) = K and E i A i K. The set E 1 is to have fixed volume v 1 and is what we consider to be the drop. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

36 Floating Drops and Functions of Bounded Variation Theorem (Elcrat and Treinen) For a given volume v 1 and any compact set K there exists (E 0,E 1,E 2 ), radial, that minimizes F K over all radial (A 0,A 1,A 2 ) with i (K A i) = K and E i A i K within the slab R 2 ( T,T). This follows using a somewhat long diagonalization process with minimizers on cylinders with increasing radii. Ray Treinen (University of Toledo) Floating Drops January 26, / 32

37 The End Thank you for listening! Ray Treinen (University of Toledo) Floating Drops January 26, / 32

38 References Elcrat, Alan, Kim, Tae-Eun and Treinen, Ray, Annular capillary surfaces, Arch. Math. (Basel), Elcrat, Alan, Neel, Robert, and Siegel, David, Equilibrium configurations for a floating drop, J. Math. Fluid Mech., Elcrat, Alan and Treinen, Ray, Numerical results for a floating drop, Discrete and Continuous Dynamical Systems Supplements, Elcrat, Alan and Treinen, Ray, Floating drops and functions of bounded variation, Preprint available. R. Finn. Equilibrium Capillary Surfaces New York: Springer-Verlag, Giusti, Enrico, Minimal surfaces and functions of bounded variation, Birkhäuser Verlag, Ray Treinen (University of Toledo) Floating Drops January 26, / 32

39 References Massari, U., The parametric problem of capillarity: the case of two and three fluids, Astérisque, Treinen, Ray, A study of floating drops. Ph.D. thesis, Wichita State University, Treinen, Ray, Continuing annular capillary surfaces, Preprint available. Vogel, Thomas I., Symmetric unbounded liquid bridges, Pacific J. Math., Wente, Henry C., The stability of the axially symmetric pendent drop, Pacific J. Math., Ray Treinen (University of Toledo) Floating Drops January 26, / 32