ICMS Workshop Mixed boundary value problems in the study of superhydrophobic surfaces. Edinburgh, May 2012 Darren Crowdy Imperial College London

Transcription

1 ICMS Workshop 2012 Mixed boundary value problems in the study of superhydrophobic surfaces Edinburgh, May 2012 Darren Crowdy Imperial College London www2.imperial.ac.uk/ dgcrowdy. p.1

2 Superhydrophobic surfaces. p.2

3 Superhydrophobic surfaces Beading on lotus leaves, and preferential slip properties, is due to the special microstructure making up those surfaces. p.3

4 Outline of the talk This talk will present: 1. physical background on superhydrophobic surfaces a hot topic in physics 2. new exact solutions of mixed boundary value problems of no-slip/no-shear type involving both harmonic and biharmonic field equations 3. suggest this area as a topical test-bed for applications of, and extensions to, the Fokas transform method I will suggest lots of (for me) interesting problems.... p.4

5 Background A recurring thread running through my own research interests is multiple connectivity A question I always ask myself is: What is the multiply connected generalization of that formula, result, phenomenon? We have found a very fruitful new analytical approach that allows (often elegant) generalizations of very classical results [D. Crowdy, Geometric function theory: a modern view of a classical subject, Nonlinearity, 21(10), (2008)]. p.5

6 Multiply connected domains Fact 1: Any finitely connected planar domain is conformally equivalent to a multiply connected circular domain call it D ζ [Goluzin, (1969)] δ 2 q 2 δ 1 q 1 δ 3 q 3 unit disc circle centres {δ j } radii {q j } (conformal moduli) This is one canonical class of domains (there are others) We choose to develop theory within this class of domains. p.6

7 Schottky double of a planar domain Fact 2: Any multiply connected planar domain has associated to it a compact (symmetric) Riemann surface of genus equal to the number of holes. It is called the Schottky double Model of a genus-2 compact Riemann surface image from Indra s Pearls, Mumford, Series & Wright. p.7

8 Prime functions Fact 3: Any compact Riemann surface has a (unique) prime function associated to it Example: The genus-0 Riemann surface is the Riemann sphere and the prime function is ω(ζ, γ) = (ζ γ). p.8

9 Modelling the Schottky double of D ζ Interior circles {C j }; generate exterior circles {C j } by reflection C 2 θ 2 (ζ) C 2 C 1 θ 1 (ζ) C 3 C1 C 3 θ 3 (ζ) Mathematically, we identify circular boundaries or glue them together by the action of Möbius maps θ j (ζ) Let F be the exterior of all six circles (in this case). p.9

10 The Schottky-Klein prime function Theorem [Hejhal (1972)]: There is a unique function X(ζ, γ) defined by the properties: (i) X(ζ, γ) is analytic everywhere in F. (ii) For γ F, X(ζ, γ) has a second-order zero at each of the points {θ(γ) θ Θ}. (iii) For γ F, (iv) For j = 1,..., M, lim ζ γ X(ζ, γ) (ζ γ) 2 = 1. X(θ j (ζ), γ) = exp( 2πi(2(v j (ζ) v j (γ))+τ jj )) dθ j(ζ) X(ζ, γ) dζ The Schottky-Klein prime function is ω(ζ, γ) = (X(ζ, γ)) 1/2. p.10

11 Simply and doubly connected cases Simply connected case: This is just ω(ζ, α) = (ζ α) Doubly connected case: When the single interior circle is concentric with radius q so that D ζ is q < ζ < 1, SK prime function is ω(ζ, α) = α C P(ζα 1 ; q) where P(ζ; q) can be represented by an infinite product P(ζ; q) = (1 ζ) (1 q 2k ζ)(1 q 2k ζ 1 ) k=1. p.11

12 Simply and doubly connected cases Simply connected case: This is just ω(ζ, α) = (ζ α) Doubly connected case: When the single interior circle is concentric with radius q so that D ζ is q < ζ < 1, SK prime function is ω(ζ, α) = α C P(ζα 1 ; q) where P(ζ; q) can be represented by an infinite product P(ζ; q) = (1 ζ) (1 q 2k ζ)(1 q 2k ζ 1 ) k=1 [This can be related to first Jacobi theta function ties in to elliptic function theory]. p.12

13 Mappings to canonical slit domains (New) fact: Mappings from circular domains to all the canonical slit domains have elegant representations in terms of ω(.,.) Crowdy & Marshall, Conformal mappings between canonical multiply connected domains, Comput. Meth. Func. Theory, 6, (2006). p.13

14 Circular slit mappings η(ζ;α) η(α;α) α circular domain circular slit domain Pick α in D ζ. Mapping from D ζ to D η in an η-plane is η(ζ; α) = ω(ζ, α) α ω(ζ, ᾱ 1 ), η(α; α) = 0. p.14

15 Radial slit mappings β r(β;α,β)= r(ζ;α,β) α r(α;α,β)=0 circular domain radial slit domain Pick α and β in D ζ. Mapping from D ζ to radial slit domain is r(ζ; α, β) = R ω(ζ, α)ω(ζ, α 1 ) ω(ζ, β)ω(ζ, β 1, r(α; α, β) = 0, r(β; α, β) = ). p.15

16 Special case of radial slit mapping β r ~ (ζ;α,β) α circular domain radial slit domain If α and β both lie on the same boundary circle: r(ζ; α, β) = R ω(ζ, α) ω(ζ, β), r(α; α, β) = 0, r(β; α, β) =. p.16

17 Superhydrophobic surfaces. p.17

18 Superhydrophobic surfaces Note the microstructure! ( pillars ). p.18

19 Superhydrophobic surfaces To model the slip properties of these surfaces, it is natural to think of a shear flow over a mattress of bubbles A mattress is a no-slip surface punctuated with bubbles, or stress-free interfaces, on which slip is allowed The angle θ is the protrusion angle of the bubbles. p.19

20 Unidirectional surfaces: slots/grooves Ou & Rothstein, Phys. Fluids, 17, (2005) In Cassie state there is a liquid-gas interface, suspended between regions of no-slip, on which there is no shear stress Mixed boundary conditions of no-slip/no-shear lead to effective slip over the surface Flows are usually taken to be at zero Reynolds number (Stokes flow), e.g. in microfluidic applications. p.20

21 Slip lengths of superhydrophobic surfaces Slip over such surfaces quantified by notion of slip length For a surface with unidirectional grooves, slip lengths for transverse and longitudinal flow are usually different Experimentalists use slip length as an important characterization Mathematically, whether flow is longitudinal or transverse determines whether it is a harmonic or biharmonic problem. p.21

22 Transverse shear over periodic slots Transverse flow period 2L no slip no shear Consider flat unidirectional bubbles: θ = 0. What is slip length λ? Clearly a function of slot geometry, but how?. p.22

23 Longitudinal shear over periodic slots Longitudinal flow period 2L no slip no shear Consider flat unidirectional bubbles: θ = 0. What is slip length λ?. p.23

24 Philip s classic (1973) paper Philip (1973) found exact solution for shear flow over a periodic array of flat (θ = 0) no-shear slots embedded in a no-slip surface He found λ = 2λ Remember this. p.24

25 General patterned surfaces of slots Philip (1973) only considers a single no-shear slot per period What about surfaces with any number of slots per period window? This question is important in design/optimization issues: opens up the parameter space Important to establish how the geometry of the slots affects slip lengths. p.25

26 General patterned surfaces of slots Longitudinal flow period 2L no slip no shear Now allow any number of no-shear intervals per period window We have to solve a mixed (linear) elliptic boundary value problem. p.26

27 General patterned surfaces of slots Longitudinal flow period 2L no slip no shear In longitudinal case flow is of form u = (0, 0, w(x, y)) where 2 w = 0. p.27

28 General patterned surfaces of slots Longitudinal flow period 2L no slip no shear Let w(x, y) = Im[ γh(z)]; h(z) quasi-periodic across the strip. p.28

29 General patterned surfaces of slots Longitudinal flow period 2L no slip no shear Let w(x, y) = Im[ γh(z)]; h(z) quasi-periodic across the strip h(z) z as z. p.29

30 General patterned surfaces of slots Longitudinal flow period 2L no slip no shear Let w(x, y) = Im[ γh(z)]; h(z) quasi-periodic across the strip h(z) z as z w = γim[h(z)] = 0 on no slip surfaces. p.30

31 General patterned surfaces of slots Longitudinal flow period 2L no slip no shear Let w(x, y) = Im[ γh(z)]; h(z) quasi-periodic across the strip h(z) z as z w = γim[h(z)] = 0 on no slip surfaces w/ y = 0 on no shear slots. p.31

32 Can view h(z) as a conformal slit map z plane h plane h(z). p.32

33 Can view h(z) as a conformal slit map z plane h plane h(z) h(z) z as z. p.33

34 Can view h(z) as a conformal slit map z plane h plane h(z) h(z) z as z h(z) quasi-periodic across the strip. p.34

35 Can view h(z) as a conformal slit map z plane h plane h(z) h(z) z as z h(z) quasi-periodic across the strip Im[h(z)] is constant (zero) on no-slip surfaces. p.35

36 Can view h(z) as a conformal slit map z plane h plane h(z) h(z) z as z h(z) quasi-periodic across the strip Im[h(z)] is constant (zero) on no-slip surfaces Re[h(z)] is constant on no-shear surfaces (by Cauchy-Riemann). p.36

37 General patterned surfaces of grooves Transverse flow period 2L no slip no shear. p.37

38 General patterned surfaces of grooves Transverse flow period 2L no slip no shear 0 = p + 2 u or 4 ψ = 0, (u, v) = ψ (u, v) = 0 on no-slip surfaces pn i + 2e ij n j = 0 on no shear surfaces ψ γy as y. p.38

39 Complex variable approach to Stokes flow It is known that we can write a general biharmonic function as ψ = Im[zf(z) + g(z)] where f(z) and g(z) are analytic in the fluid region The geometry, and far field, of this flow configuration imply g(z) = zf(z). p.39

40 Complex variable approach to Stokes flow It is known that we can write a general biharmonic function as ψ = Im[zf(z) + g(z)] where f(z) and g(z) are analytic in the fluid region The geometry, and far-field, of this flow configuration imply g(z) = zf(z) Moreover, the boundary conditions imply f(z) = h(z) where h(z) is the function described earlier. p.40

41 A single function h(z) solves both problems! It can be shown that both solutions to transverse and longitudinal flow problem can be written in terms of a single function h(z) Transverse problem: Flow is two dimensional in (x, y)-plane. Streamfunction ψ(x, y) is ψ(x, y) = Im[ γ(z z)h(z)] Longitudinal problem: Flow is of form u = (0, 0, w(x, y)) where w(x, y) = Im[ γh(z)] γ is the imposed shear rate. p.41

42 But how to find h(z)? z plane h plane h(z) To find the slip length, we need to find h(z) (ideally, explicitly) Key idea: consider a uniformization. p.42

43 Exponential mapping of these strips Upper-strip maps to interior of unit disc: exp( iπz/l) exp( iπh/l) Domains now resemble circular slit and radial slit domains Idea: Consider a circular domain uniformization. p.43

44 Introduce a uniformizing ζ-plane C 0 α D ζ C 1 C 2 q q 2 B C 1 D E F A δ 1 δ 2 A B C D E F L 0 L 1 L 2 Introduce a preimage ζ plane the upper half semi-disc This region maps to a single period window A point α maps to z = C 0, C 1, C 2 map to L 0, L 1, L 2 respectively. p.44

45 Circular slit map The ζ-domain is mapped to the domain by the circular slit map ω(ζ, α)ω(ζ, 1/α) ω(ζ, α)ω(ζ, 1/α). p.45

46 Radial slit map The ζ-domain is mapped to the domain by the radial slit map ω(ζ, α)ω(ζ, 1/α) ω(ζ, α)ω(ζ, 1/α). p.46

47 Uniformization using circular preimage With H(ζ) = h(z(ζ)), the solution can be parametrized as z(ζ) = il [ ] ω(ζ, α)ω(ζ, 1/α) π log ω(ζ, α)ω(ζ, 1/α) H(ζ) = il [ ] ω(ζ, α)ω(ζ, 1/α) π log ω(ζ, α)ω(ζ, 1/α) + d, where α is a point in the circular region Since h(z) is determined, so is the flow. Explicit solutions! Geometry of strips encoded in α and conformal moduli. p.47

48 Explicit formula for slip lengths From the exact solutions, slip lengths can be obtained It can be shown that the relation λ = 2λ always holds [cf: Lauga & Stone, J. Fluid Mech., (2003)] It is not only true for the single slot case. p.48

49 General patterned surfaces of grooves It can be shown that, for any number of no-shear slots, λ = 2L π log ω(α, 1/α) ω(α, 1/α) Prime function neatly encodes information on the geometry of a large class of superhydrophobic surfaces!. p.49

50 General patterned surfaces of grooves It can be shown that, for any number of no-shear slots, λ = 2L π log ω(α, 1/α) ω(α, 1/α) Prime function neatly encodes information on the geometry of a large class of superhydrophobic surfaces! If we set ω(ζ, γ) = (ζ γ) the previous formula gives. p.50

51 General patterned surfaces of grooves It can be shown that, for any number of no-shear slots, λ = 2L π log ω(α, 1/α) ω(α, 1/α) Prime function neatly encodes information on the geometry of a large class of superhydrophobic surfaces! If we set ω(ζ, γ) = (ζ γ) the previous formula gives λ = c πδ log sec(πδ/2) where δ = c/(2l) (c is slot length, L is period) Philip s (1973) single slot formula! [D.Crowdy, Phys. Fluids, 23, (2011)]. p.51

52 Two slots per period window Now we can introduce new length scales Explore the effect of adding a no-slip defect in a no-shear slot This may be brought about either deliberately or by accident (i.e. contaminants) α period 2 π no shear slots s r r s φ ρ ρ 1/2 1 λ = 2L ( P( α 2 ) π log, ρ) P(α 2, ρ). p.52

53 Slip lengths longitudinal slip length gap width between no shear slots Slit length as function of gap width Note the dramatic decrease in slip length even for a very small gap. p.53

54 Computing the Schottky-Klein prime function I feel no satisfaction in formulas unless I feel their numerical magnitude Lord Kelvin There is an infinite product formula for the prime function: ω(ζ, γ) = (ζ γ) (θ i (ζ) γ)(θ i (γ) ζ) (θ θ i Θ i (ζ) ζ)(θ i (γ) γ) Θ is a Schottky group (excluding identity and inverses) Precise convergence criteria for this formula are not known (there are some sufficiency results e.g. Fuchsian groups; or groups satisfying the Burnside condition basically, circles sufficiently small and well separated ) H. Baker, Abelian functions, 1897 Even if convergent, the infinite product requires many operations, even at low truncation levels, for even moderately high genus Use of the product is slow!. p.54

55 An observation (1 ζ) (1 ρ k ζ)(1 ρ k /ζ) = A ( 1) n ρ n(n 1) ζ n where the constant k=1 n= A = / (1 + ρ 2n ) 2 n=1 n=1 ρ n(n 1). p.55

56 An observation (1 ζ) (1 ρ k ζ)(1 ρ k /ζ) = A ( 1) n ρ n(n 1) ζ n where the constant k=1 n= A = / (1 + ρ 2n ) 2 n=1 n=1 ρ n(n 1) This is the Jacobi triple product identity Two representations for the doubly connected prime function. p.56

57 A new computation of the prime function Crowdy & Marshall have recently devised an alternative numerical construction of the prime function [Crowdy & Marshall, Computing the Schottky-Klein prime function on the Schottky double of planar domains, 7(1), , (2007))] It does not rely on the convergence of an infinite product (or sum) over the elements of a Schottky group The key idea is to generalize the Jacobi triple product identity! Empirical evidence shows that it works and it is fast!. p.57

58 Comparison for a typical 3-connected domain Truncation level X( i, 1) (infinite product) level i level i level i level i level i N 1 X( i, 1) (new method) i i i. p.58

59 Software to compute the prime function We have made available freely downloadable software, based on the new numerical scheme, for computation of the prime function MATLAB M-files are freely available at www2.imperial.ac.uk/ dgcrowdy/skprime. For a description of the algorithm: Crowdy & Marshall, Computing the Schottky-Klein prime function on the Schottky double of planar domains, Comput. Methods Func. Th., 7, (2007) [Special thanks to my current Ph.D student Chris Green for writing these codes]. p.59

60 Future directions The transverse problem involves Stokes flow in a semi-strip. Crowdy & Fokas have extended the Fokas transform method to biharmonic problems in a semi-strip [Crowdy & Fokas, Explicit solutions for the plane elastostatic semi-strip, Proc. Roy. Soc. A, (2004)] So.... p.60

61 Can we solve using the Fokas transform? Longitudinal flow period 2L no slip no shear. p.61

62 Can we solve using the Fokas transform? Longitudinal flow period 2L no slip no shear Unbounded, but ok in this case. p.62

63 Can we solve using the Fokas transform? Longitudinal flow period 2L no slip no shear Unbounded, but ok in this case New ingredient: BC s not just Dirichlet or Neumann, but periodic. p.63

64 Can we solve using the Fokas transform? Longitudinal flow period 2L no slip no shear Unbounded, but ok in this case New ingredient: BC s not just Dirichlet or Neumann, but periodic New ingredient: nature of BC changes on same polygonal edge. p.64

65 Interesting questions: Q1: What is the effect of boundary conditions of periodicity type in the analysis of the global relation?. p.65

66 Interesting questions: Q1: What is the effect of boundary conditions of periodicity type in the analysis of the global relation? Q2: How do we see the Schottky-Klein prime function arising in this approach?. p.66

67 Interesting questions: Q1: What is the effect of boundary conditions of periodicity type in the analysis of the global relation? Q2: How do we see the Schottky-Klein prime function arising in this approach? Q3: Do alternative representations of the slip lengths arise? Q4: If successful, potential to generalize results to other geometries. p.67

68 Negative slip lengths are even possible Negative slip lengths are even possible [Steinberger, Nature, (2007)] No-shear effect of the bubbles does not always increase the slip length Bubbles that protrude too far into the fluid can increase the drag. p.68

69 Superhydrophobic surfaces Explore effect of non-zero protrusion angle, θ 0 To model the slip properties of these surfaces, it is natural to think of a shear flow over a mattress of bubbles A mattress is a no-slip surface punctuated with bubbles, or stress-free interfaces, on which slip is allowed The bubbles can protrude both into the fluid or into the surface The angle θ is the protrusion angle. p.69

70 Transverse shear flow over bubble mattress An analytical formula for the transverse slip length, in the dilute limit, is known to be λ δc = M(θ) = π 2 0 A(s, θ)ds where A(s, θ) = s sinh 2s(π θ) + s sin 2θ [ cos 2θ + ] s sin 2θ cosh sπ + sinh s(π 2θ) sinh sπ It was derived using bipolar coordinates combined with Mellin transforms Davis & Lauga, Phys. Fluids, (2009). p.70

71 Longitudinal shear flow over bubble mattress The longitudinal problem can also be solved explicitly in dilute limit: Crowdy [Phys. Fluids, (2011)] found the explicit result λ δc = N(θ) = π 24 [ 3π 2 4πθ + 2θ 2 ] (π θ) 2 [Theo & Koo [2009] found a phenomenological formula (agrees well)]. p.71

72 Future directions Q1: These are polycircular arc domains: can the Fokas approach to Laplace s equation in a polygon be extended to polycircular arc domains?. p.72

73 Future directions Q1: These are polycircular arc domains: can the Fokas approach to Laplace s equation in a polygon be extended to polycircular arc domains? Q2: Can the Crowdy-Fokas approach to biharmonic equation in a polygon be extended to polycircular arc domains? (corner singularities arise in this case). p.73

74 Future directions Q1: These are polycircular arc domains: can the Fokas approach to Laplace s equation in a polygon be extended to polycircular arc domains? Q2: Can the Crowdy-Fokas approach to biharmonic equation in a polygon be extended to polycircular arc domains? (corner singularities arise in this case) 2 q=0 Q3: Multiply connected Fokas transform?. p.74