Capillary surfaces and complex analysis: new opportunities to study menisci singularities. Mars Alimov, Kazan Federal University, Russia

Size: px

Start display at page:

Download "Capillary surfaces and complex analysis: new opportunities to study menisci singularities. Mars Alimov, Kazan Federal University, Russia"

Magnus Skinner
5 years ago
Views:

1 Capillary surfaces and complex analysis: new opportunities to study menisci singularities Mars limov Kazan Federal University Russia Kostya Kornev Clemson University SC

2 Outline Intro to wetting and capillarity Wetting of shaped fibers. Formulation through matched asymptotic expansion Hodograph method and singular menisci

3 Contact angle and wetting phenomena Droplet forms a finite contact angle when it meets the substrate. This angle depends on surface tensions sl and sg Hydrophobic Hydrophilic Flat substrate Cylindrical fibers 3

tension P=P - P + = (1/R1+1/R) Spherical drop R1=R=R; Hence P=P-P + =

4 Laplace s law of capillarity. Pressure inside the bubble P Pressure outside the bubble P + Surface tension P=P - P + = (1/R1+1/R) Spherical drop R1=R=R; Hence P=P-P + = (1/R1+1/R) = /R R n Liquid cylinder: W.Wick drop of water Scholastic Press NY 1997 P =P - P + = (1/R1+1/R) = /R R1 = R

5 Laplace s law of capillarity and minimal surfaces ir inside and outside no pressure difference P=P - P + = 0 (1/R1+1/R)=0 Praha s minimal surface 5

6 z L Capillary rise: meniscus at the flat plate - density g - gravity (y) : dz h df x d Force balance at the plate in x-direction 0 df df z x -Lgh / - inl + L = 0 h =[(1-sin)/(g)] 1/ Water =0: h=(/g) 1/ ~ 4 mm Meniscus height depends only on the materials properties of the plate/liquid pair!

7 Meniscus on a fiber or can a fish recognize the fiber? Two characteristic length scales: R =(/ g) 1/ R h C The Bond number: = (R/ ) <<1 for micrometer thick fibers hc Rcos ln 1sin ln e 4 E= E Lo LL (1973) The meniscus on a needle - a lesson in matching. J Fluid Mech 13:65-78 The thinner the fiber the smaller the meniscus! Significant difference with the plate!

8 Complexity of shapes of natural and man-made fibers Spiral and grooved fibers. Courtesy of.lobovsky dvanced Fiber Engineering NJ 100m 50m Fibers from carbon nanotubes 50 m 0 m 4 m 300m Yarns from nanofibers produced by electrostatic spinning 8

9 Wetting of shaped fibers JOURNL OF THE ROYL SOCIETY INTERFCE (013) 10 (85)

10 Mechanism of wetting of elliptical fibers L min (minimum elevation) Round cross-section Lmax (maximum elevation) Elliptical cross-section? ir Liquid ir Liquid 10

toward the wider part where the curvature is smaller R c

11 Radii of curvature of sags and bulges on contact line Capillary pressure pushes meniscus from the sharp edges toward the wider part where the curvature is smaller R c R c radius of curvature of the sag.trofimov and V. Vekselman 11

12 Outline Intro to wetting and capillarity as related to fibers Wetting of shaped fibers. Formulation through matched asymptotic expansion Hodograph method and singular menisci 1

13 Capillary rise. Laplace equation in Cartesian coordinates Z H( X Y) 1 1 H H N H 1 X Y N gh 0 1 H 1 H g H 0 Profile of the free surface Normal vector to the free liquid surface Laplace equation of capillarity with the body force = gravity

14 Boundary conditions is the fiber profile in the (XY) - section Introducing the unit normal vector n pointing to the fiber exterior the Young equation reads n ( n n 0) x y y n : Nn cos. B C x H H 1 H nx ny cos. X Y 1 t infinity where the meniscus levels off we have r X Y H : 0

15 Math model in dimensionless variables r( x y) R L L = area fiber fiber y h H / / g n : 1 h 1 h h 0 B C x : 1 1 h h cos n r x y h : 0 Numerics: Orr FM Brown R Scriven LE J. Colloid Interface Sci Pozrikidis C. 010 Computation of three-dimensional hydrostatic menisci. IM J. ppl. Math Surface evolver by K.Brakke

Method of matched asymptotic expansions <<1 Lo LL. 1973 J. Fluid Mech.

16 Method of matched asymptotic expansions <<1 Lo LL J. Fluid Mech Working hypothesis: Two regions two different asymptotics: In the inner region the shape of meniscus is significantly affected by the fiber shape x y O(1) : h x y h x y O i i0 ( ) ( ) ( ) i0 1 i0 h h i0 h Meniscus on an elliptical wire as seen from different angles Outer region Outer region Inner region Inner region Outer region Outer region i0 1 : 1 0 h : 1 cos. n x y O(1/ ) In the outer region and is described by a linear equation h h 0 C. Zhao and V.Vekselman the shape of meniscus is universal r x y : h 0 16

Outer region Inner region x y O(1/ ) Outer region Outer region h h 0 r x y h : 0 General solution: K n Outer region Inner region n n cos n sin o0 h (rφ) = C0K 0 r K r C n D n r n1 "006-01-14 Surface

17 Outer region Inner region x y O(1/ ) Outer region Outer region h h 0 r x y h : 0 General solution: K n Outer region Inner region n n cos n sin o0 h (rφ) = C0K 0 r K r C n D n r n1 " Surface waves" by Roger McLassus - dia.org/wiki/file: _surface_waves.j pg#mediaviewer/file: _Surface_waves.jpg Outer region Observe that all ripples become circular at infinity; same phenomenon is shaping meniscus are the modified Bessel functions constants C n??? In the leading order the shape of meniscus is described by the boxed term. Royal Society Proceedings 014 vol. 470 no

18 Linking two expansions by the first integral Integrate over N h 0 l cos hd where l is the dimensionless fiber perimeter. i o hd hd hd h d h d C \ C C \ C M M M M Inner expansion of the outer expansion specifies C 0 and provides the boundary condition for the inner problem E l cos l cos e h K0 r ln r ln r o0 i [ ] 0 E 0.577

19 Meniscus on an arbitrarily shaped fiber as a problem of minimal surfaces y : 1 h 1 h 0 : 1 1 h h cos n E lcos e r : h( r ) ln r ln B C n x When h is obtained meniscus shape is calculated using the van Dyke construction: meniscus l cos h ( x y) h( x y) K r ln r ln e E 0

20 Outline Intro to wetting and capillarity as related to fibers Wetting of shaped fibers. Formulation through matched asymptotic expansion Hodograph method and singular menisci 0

21 nalogy: D flow of non-newtonian fluids through porous media Entov VM Goldstein RV (1994) Qualitative Methods in Continuum Mechanics. : J 0 h ( J ) J J Continuity equation J h ( J ) ( J ) 1 J ( J) 0 ( J) 0 : Jn cos Non-linear Darcy s equation with flux J E lcos e r : h( r ) ln r ln J t the fiber surface the normal component of the flux is constant y J J z B C 1 1 J x

22 . J 0 The stream function J / y J / x x y J x J y Two equivalent pairs ( J ) J J 0 J J cos J J sin x y ssumption on the functional behavior ( J ) hj ( )

23 . Hodograph transformation for the stream function and meniscus height hj ( ) 1 h 1 h J 1 J J 1 J J J ( J ) Sokolovsky transformation 1 t ln 1 1 J ln J arccosh J Leading to the Cauchy-Riemann equations nd complex potentials h h t t W h i t i Chaplygin relations with the (xy) coordinates dx cos sinht dh sin cosh t d dy sin sinht dh cos cosh t d.

24 J Example: meniscus on a circular cylinder y J i i B B i iq J z B C 1 1 J x 0 t C Physical plane Hodograph plane Plane of complex potential Using the (t) plane one infers: h t 0 implying h C t C 1 t C h 0 W ( h) With the aim of the boundary conditions one reproduces Lo s solution E h cos ln r r cos ln e 4

25 . Meniscus on a oval fiber with = 0 i i B a B i il b j W C 0 t j Hodograph plane t C h C 0 i0 h Plane of complex potential Using `inverse' method of truncated Fourier series : nd follow the grid l W h ae ( ) O 1 : t i where [0 ] j j

26 Singularities of contact line on smooth fibers Royal Society Proceedings 014 vol. 470 no

27 Singularities of contact line on fibers with sharp edges c z y When the model of a locally flat liquid surface becomes incompatible with the contact angle condition? Cross-section O x O x Shaded region corresponds to the solid fiber 7

28 Shaping menisci at the chemically uniform edges Laplace equation in cylindrical coordinates 1 H 1 G H Gr 0 r r r r r Gravity is less important Fiber O Looking at the meniscus from the top H 1 H Gr ( ) 1 r r X H( r ) rf( ) const ( F F )(1 F ) 1F F Similarity solution F( ) cos sin Solution 1

29 . Domains of analyticity sin cos cos( )cos( ) 0 Generalization of the Concus-Finn condition 3 Square of admissible angles where continuous solution exists. Mech. Res.Comm external P. Concus and R. Finn PNS 63 () internal 0 9 (1969) 3

30 NEW SYNGULRITY Illustration of the meniscus shape at the edge of a polypropylene film that has been dipped in a syrup. The arrows on the magnified image indicate a jump of the end point of the contact line along the film edge.

31 Conclusions Profile of menisci on the complex shaped fibers is mostly controlled by capillary and wetting forces: effect of meniscus weight is insignificant for microfibers On elliptical fibers having two different curvatures the contact line sags down and bulges up. The height difference between these two limiting points can be significant. Mathematical model of meniscus on a slender fiber is reduced to a model of a minimal surface supported by a fiber of the given shape Contact line may have singularities even on smooth fibers! Domain of analyticity of the contact meniscus meeting the V- type edges is specified for chemically homogeneous and heterogeneous sides and V-angles. New type of singularity with a finite jump on the contact line was discovered.

Multiphase Flow and Heat Transfer

Multiphase Flow and Heat Transfer ME546 -Sudheer Siddapureddy sudheer@iitp.ac.in Surface Tension The free surface between air and water at a molecular scale Molecules sitting at a free liquid surface against