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

2

• Intro to wetting and capillarity

• Wetting of shaped fibers. Formulation through matched asymptotic expansion

• Hodograph method and singular menisci

Outline

Contact angle and wetting phenomena

3

Droplet forms a finite contact angle when it meets the substrate.

This angle depends on surface tensions , sl and sg

Hydrophobic

Flat substrate

Cylindrical fibers

Hydrophilic

Laplace’s law of capillarity.

P =P - P+ =

(1/R1+1/R2)

= /R2

R2 R1 =

Spherical drop, R1=R2=R; Hence

P=P-P+ = (1/R1+1/R2) = 2/R R

n

Liquid cylinder:

P=P - P+ = (1/R1+1/R2)

W.Wick, A drop of water, Scholastic Press, NY, 1997

Pressure inside the bubble P Pressure outside the bubble P+

Surface tension

5

Air inside and outside, no pressure difference P=P - P+ = 0 (1/R1+1/R2)=0

Laplace’s law of capillarity and minimal surfaces

Praha’s minimal surface

Capillary rise: meniscus at the flat plate

dz A(y)

h

dF dFz

dFx

0

d

x

z L

- density, g - gravity

:

-Lgh2/2 - inL + L = 0

h =[2(1-sin)/(g)]1/2

Water, =0: h=(2/g)1/2 ~ 4 mm

Meniscus height depends only on the materials properties of the plate/liquid pair!

Force balance at the plate in x-direction

Meniscus on a fiber or can a fish recognize the fiber?

Two characteristic length scales:

The Bond number: = (R/ )2,

<<1 for micrometer thick fibers hC

E=0.57721

R, =(/ g)1/2

2R

cos ln 1 sin ln 4E

Ch R e

Lo LL (1973) The meniscus on a needle - a lesson in

matching. J Fluid Mech 132:65-78

The thinner the fiber the smaller the meniscus!

Significant difference with the plate!

8

Spiral and grooved fibers.

Courtesy of A.Lobovsky, Advanced Fiber Engineering, NJ

Complexity of shapes of natural and man-made fibers

Fibers from carbon nanotubes

Yarns from nanofibers produced by electrostatic spinning

100m 50m

50 m 20 m 4 m 300m

Wetting of shaped fibers

9

JOURNAL OF THE ROYAL SOCIETY INTERFACE

(2013) 10 (85) 20130336

10

Mechanism of wetting of elliptical fibers

Round cross-section Elliptical cross-section

minL maxL(maximum elevation) (minimum elevation)

Air

Liquid

Air

Liquid

?

11

Radii of curvature of sags and bulges on contact line

– radius of curvature of the sag cRcR

Capillary pressure

pushes meniscus from

the sharp edges

toward the wider part

where the curvature is

smaller

A.Trofimov, and V. Vekselman,

12

• Intro to wetting and capillarity as related to fibers

• Wetting of shaped fibers. Formulation through matched asymptotic expansion

• Hodograph method and singular menisci

Outline

1 2

21 | | , ,1H H

HX Y

N

1 2

2

0

1 | | 0

gH

H H gH

N

Capillary rise. Laplace equation in Cartesian coordinates

Normal vector to the

free liquid surface

Laplace equation of capillarity

with the body force = gravity

( , )Z H X Y Profile of the free surface

Boundary conditions

Introducing the unit normal vector n

pointing to the fiber exterior , the

Young equation reads

( , , 0)x yn nn

cos . N n

1 2

21 | | cos .x y

H HH n n

X Y

At infinity where the meniscus levels off,

we have

2 2 : 0r X Y H

:

is the fiber profile

in the (X,Y) - section

y

x

CB

n

A A

Math model in dimensionless variables

( , ) , =

/ /

fiber fiberx y L L area

h H g

r R

,

,

,

1 2

2: 1 | | 0h h h

1 2

2: 1 | | cosh

hn

2 2 : 0r x y h

y

x

CB

n

A A

Numerics: Orr FM, Brown RA, Scriven LE. 1977 J. Colloid Interface Sci. 60, 137–147.

Pozrikidis C. 2010 Computation of three-dimensional hydrostatic menisci. IMA J. Appl.

Math.75, 418–438.

Surface evolver by K.Brakke, http://www.susqu.edu/brakke/evolver/evolver.html

16

,0

, (1) :

( , ) ( , ) ( ),i i

x y O

h x y h x y O

1 2

,0 2 ,0: 1 | | 0i ih h

,0

1 2,0 2: 1 | | cos .

ii h

hn

Method of matched asymptotic expansions, <<1

Lo LL. 1973, J. Fluid Mech. 132, 65–78

Meniscus on an elliptical wire as seen from different angles

Inner region

Inner region

Outer region

Outer region

Outer region

Outer region

Working hypothesis: Two regions, two different asymptotics: In the inner region, the shape of meniscus is significantly affected by the fiber shape

In the outer region, , the shape of meniscus is universal and is described by a linear equation

0h h 2 2 : 0r x y h

, (1 / )x y O

C. Zhao and V.Vekselman

17

Outer region,

Inner region

Inner region

Outer region

Outer region

Outer region

Outer region

In the leading order, the shape of meniscus is described by the boxed term.

0h h

2 2 : 0r x y h

, (1 / )x y O

"2006-01-14 Surface

waves" by Roger

McLassus -

http://commons.wikime

dia.org/wiki/File:2006-

0114_Surface_waves.j

pg#mediaviewer/File:2

006-01-

14_Surface_waves.jpg

0

1

cos sinn n n

n

C K r C n D n

o,0

0h (r,φ) = K r

General solution:

n K r

Royal Society Proceedings A, 2014, vol. 470 no. 2168, 20140113

Observe that all ripples become circular at infinity; same phenomenon is shaping meniscus…

are the modified Bessel functions, constants Cn ???

Linking two expansions by the first integral

Inner expansion of the outer expansion specifies C0 and provides the boundary condition for the inner problem

cosl hd

Integrate over

0h N

where l is the dimensionless fiber perimeter.

\ \M M M M

i o

C C C C

hd hd hd h d h d

,0

0

cos cosln ln

2 2 2[ ]

Eo i

r

l l eh K r r

0

, 0.5772E

Meniscus on an arbitrarily shaped fiber as a problem of minimal surfaces

,

,

,

1 2

2: 1 | | 0h h

1 2

2: 1 | | cosh

hn

cos: ( , ) ln ln

2 2

El er h r r

y

x

CB

n

A A

0

cos( , ) ( , ) ln ln 2

2

meniscus Elh x y h x y K r r e

When h is obtained, meniscus shape is calculated using the van Dyke construction:

20

• Intro to wetting and capillarity as related to fibers

• Wetting of shaped fibers. Formulation through matched asymptotic expansion

• Hodograph method and singular menisci

Outline

Analogy: 2D flow of non-Newtonian fluids through porous media

Entov VM, Goldstein RV (1994) Qualitative Methods in Continuum Mechanics.

,

,

,

: cos J n

cos: ( , ) ln ln

2 2

El er h r r

: 0, ( )h J J J J

2| | ( ), ( ) ,

1

( ) 0, ( ) 0

Jh J J

J

J J

Continuity equation Non-linear Darcy’s equation with flux J

y

x

CB

z

J

JJ

J

A A

1 1

At the fiber surface the normal component

of the flux is constant

The stream function

,

,

,

/ , /x yJ y J x

.

,x yJ J

Two equivalent pairs

( , )J

| | 0, cos , sinx yJ J J J J J

( , )J ( , )h J

Assumption on the functional behavior

0, J

Hodograph transformation for the stream function and meniscus height

,

,

,

Sokolovsky transformation

1 2 1 2

2 21 , 1h h

J J J JJ J

( , )h J ( , )J

2 1ln 1 1 ln arccosht J J J

,h h

t t

Leading to the Cauchy-Riemann equations

,W h i t i And complex potentials

cos sinh sin cosh ,

sin sinh cos cosh .

dx t dh t d

dy t dh t d

Chaplygin relations with the (x,y) coordinates

.

Example: meniscus on a circular cylinder ,

,

,

Using the (,t) plane, one infers: implying

y

x

CB

z

J

JJ

J

A A

1 1

tC

i

0

i B

A

A

t

W

( )hC

i

0

2iQB

A

A

h

2 2 0h t

1 2h C t C

2 2cos ln cos ln 4Eh r r e

With the aim of the boundary conditions one, reproduces Lo’s solution

Physical plane Hodograph plane Plane of complex potential

Meniscus on a oval fiber with = 0 ,

,

Using `inverse' method of truncated Fourier series :

And follow the grid

Hodograph plane Plane of complex potential

B

W

,0ihC

i

0

2i l

A

A

Ch

tC

i

0

iB

A

A

j

jt

a b

2( ) 12

O

lW h a e

: , [0, ]j jt i where

.

26

Singularities of contact line on smooth fibers

Royal Society Proceedings A, 2014, vol. 470 no. 2168, 20140113

27

x

y

z

c

O

O

x

Singularities of contact line on fibers with sharp edges

When the model of a locally flat liquid surface becomes incompatible with the contact angle condition?

Shaded region corresponds to the solid fiber

Cross-section

Shaping menisci at the chemically uniform edges

O

X

1 10

H G HGr

r r r r r

Gravity is less

important

1 222

2

1( , ) 1

H HG r

r r

( , ) ( )H r rF const

Laplace equation

in cylindrical

coordinates

Similarity

solution

Looking at the

meniscus from

the top

Fiber

2

3 22 2

( )(1 )0

1

F F F

F F

Solution 1 2( ) cos sinF A A

Domains of analyticity

.

2 2sin cos cos( )cos( ) 0

0 2

2

3 2

3 2

Square of admissible

angles where continuous

solution exists.

internal

external P. Concus and R.

Finn, PNAS, 63 (2),

292 (1969)

Generalization of the Concus-Finn condition

Mech. Res.Comm.2014, 62, 62–167

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.

NEW SYNGULARITY

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.