A finite-element algorithm for computing free-surface flows driven by arbitrary body forces is presented. The algorithm is primarily designed for the microfluidic parameter range where (i) the Reynolds number is small and (ii) force-driven pressure and flow fields compete with the surface tension for the shape of a stationary free surface. The free surface shape is represented by the boundaries of finite elements that move according to the stress applied by the adjacent fluid. Additionally, the surface tends to minimize its free energy and by that adapts its curvature to balance the normal stress at the surface. The numerical approach consists of the iteration of two alternating steps: The solution of a fluidic problem in a prescribed domain with slip boundary conditions at the free surface and a consecutive update of the domain driven by the previously determined pressure and velocity fields. For a Stokes problem the first step is linear, whereas the second step involves the nonlinear free-surface boundary condition. This algorithm is justified both by physical and mathematical arguments. It is tested in two dimensions for two cases that can be solved analytically. The magnitude of the errors is discussed in dependence on the approximation order of the finite elements and on a step-width parameter of the algorithm. Moreover, the algorithm is shown to be robust in the sense that convergence is reached also from initial forms that strongly deviate from the final shape. The presented algorithm does not require a remeshing of the used grid at the boundary. This advantage is achieved by a built-in mechanism that causes a smooth change from the behavior of a free surface to that of a rubber blanket if the boundary mesh becomes irregular. As a side effect, the element sides building up the free surface in two dimensions all approach equal lengths. The presented variational derivation of the boundary condition corroborates the numerical finding that a second-order approximation of the velocity also necessitates a second-order approximation for the free surface discretization.

1.
D.
Figeys
and
D.
Pinto
, “
Lab-on-a-chip: A revolution in biological and medical sciences
,”
Anal. Chem.
72
,
330A
(
2000
).
2.
H. A.
Stone
,
A. D.
Stroock
, and
A.
Ajdari
, “
Engineering flows in small devices: Microfluidics toward a lab-on-a-chip
,”
Annu. Rev. Fluid Mech.
36
,
381
(
2004
).
3.
N. A.
Polson
and
M. A.
Hayes
, “
Microfluidics controlling fluids in small places
,”
Anal. Chem.
73
,
312A
(
2001
).
4.
T. M.
Squires
and
S. R.
Quake
, “
Microfluidics: Fluid physics at the nanoliter scale
,”
Rev. Mod. Phys.
77
,
977
(
2005
).
5.
U.
Thiele
, “
Open questions and promising new fields in dewetting
,”
Eur. Phys. J. E
12
,
409
(
2003
).
6.
M. O.
Deville
,
P. F.
Fischer
, and
E. H.
Mund
,
High-Order Methods for Incompressible Fluid Flow
(
Cambridge University Press
,
Cambridge
,
2002
).
7.
O. C.
Zienkiewicz
and
R. L.
Taylor
,
The Finite Element Method. Volume 3: Fluid Dynamics
(
Butterworth-Heinemann
,
Oxford
2000
).
8.
A.
Ramos
,
H.
Morgan
,
N. G.
Green
, and
A.
Castellanos
, “
AC electrokinetics: A review of forces in microelectrode structures
,”
J. Phys. D
31
,
2338
(
1998
).
9.
Z.
Guttenberg
,
A.
Rathgeber
,
S.
Keller
,
J. O.
Rädler
,
A.
Wixforth
,
M.
Kostur
,
M.
Schindler
, and
P.
Talkner
, “
Flow profiling of a surface-acoustic-wave nanopump
,”
Phys. Rev. E
70
,
056311
(
2004
).
10.
A.
Wixforth
,
Ch.
Strobl
,
Ch.
Gauer
,
A.
Toegl
,
J.
Scriba
, and
Z.
Guttenberg
, “
Acoustic manipulation of small droplets
,”
Anal. Bioanal. Chem.
379
,
982
(
2004
).
11.
Z.
Guttenberg
,
H.
Müller
,
H.
Habermüller
,
A.
Geisbauer
,
J.
Pipper
,
J.
Felbel
,
M.
Kielpinski
,
J.
Scriba
, and
A.
Wixforth
, “
Planar chip device for PCR and hybridization with surface acoustic wave pump
,”
Lab Chip
5
,
308
(
2005
).
12.

In an experimental realization, the body force may be caused by a surface-acoustic wave (SAW) that travels over the surface of a substrate and rushes into the droplet.10 The fast motion of the SAW is damped by the fluid, giving rise to a body force via the acoustic streaming effect.46

13.
K.
Sritharan
,
C. J.
Strobl
,
M. F.
Schneider
,
Z.
Guttenberg
, and
A.
Wixforth
, “
Acoustic mixing at low Reynolds numbers
,”
Appl. Phys. Lett.
88
,
054102
(
2006
).
14.
V.
Cristini
and
Y.-C.
Tan
, “
Theory and numerical simulation of droplet dynamics in complex flows—A review
,”
Lab Chip
4
,
257
(
2004
).
15.
J. U.
Brackbill
,
D. B.
Kothe
, and
C.
Zemach
, “
A continuum method for modeling surface tension
,”
J. Comput. Phys.
100
,
335
(
1992
).
16.
Y.
Renardy
and
M.
Renardy
, “
PROST: A parabolic reconstruction of surface tension for the volume-of-fluid method
,”
J. Comput. Phys.
183
,
400
(
2002
).
17.
S.
Popinet
and
S.
Zaleski
, “
A front-tracking algorithm for accurate representation of surface tension
,”
Int. J. Numer. Methods Fluids
30
,
775
(
1999
).
18.
B.
Lafaurie
,
C.
Nardone
,
R.
Scardovelli
,
S.
Zaleski
, and
G.
Zanetti
, “
Modelling merging and fragmentation in multiphase flows with SURFER
,”
J. Comput. Phys.
113
,
134
(
1994
).
19.
A.
Smolianski
, “
Finite-element/level-set/operator-splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfaces
,”
Int. J. Numer. Methods Fluids
48
,
231
(
2005
).
20.
C.
Pozrikidis
,
Boundary Integral and Singularity Methods for Linearized Viscous Flow
(
Cambridge University Press
,
Cambridge
,
1992
).
21.
A. Z.
Zinchenko
,
M. A.
Rother
, and
R. H.
Davis
A novel boundary-integral algorithm for viscous interaction of deformable drops
,”
Phys. Fluids
9
,
1070
(
1997
).
22.
H.
Saito
and
L. E.
Scriven
, “
Study of coating flow by the finite element method
,”
J. Comput. Phys.
42
,
53
(
1981
).
23.
S. F.
Kistler
and
L. E.
Scriven
, “
Coating flows
,” in
Computational Analysis of Polymer Processing
, edited by
J. R. A.
Pearson
(
Applied Science
,
Barking, Essex
,
1983
) Chap. 8.
24.
C.
Cuvelier
and
R. M. S.
Schulkes
, “
Some numerical methods for the computation of capillary free boundaries governed by the Navier-Stokes equations
,”
SIAM Rev.
32
,
355
(
1990
).
25.
C.
Cuvelier
,
A.
Segal
, and
A. A.
van Steenhoven
,
Finite Element Methods and Navier-Stokes Equations
(
Reidel
,
Dordrecht
,
1986
).
26.
R.
Aris
,
Vectors, Tensors, and the Basic Equations of Fluid Mechanics
(
Dover
,
New York
,
1989
).
27.
L. E.
Scriven
, “
Dynamics of a fluid interface. Equation of motion for Newtonian surface fluids
,”
Chem. Eng. Sci.
12
,
98
(
1960
).
28.
E.
Bänsch
, Habilitation thesis,
Albert-Ludwigs-Universität Freiburg
, Freiburg,
1998
.
29.
R. A.
Cairncross
,
P. R.
Schunk
,
T. A.
Baer
,
R. R.
Rekha
, and
P. A.
Sackinger
, “
A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion
,”
Int. J. Numer. Methods Fluids
33
,
375
(
2000
).
30.
M. A.
Walkley
,
P. H.
,
P. K.
Jimack
,
M. A.
Kelmanson
, and
J. L.
Summers
, “
Finite element simulation of three-dimensional free-surface flow problems
,”
J. Sci. Comput.
24
,
147
2005
.
31.
M.
Renardy
, “
Imposing ‘no’ boundary condition at outflow: Why does it work?
Int. J. Numer. Methods Fluids
24
,
413
(
1997
).
32.
K. A.
Brakke
, “
The surface evolver
,”
Exp. Math.
1
,
141
(
1992
).
33.
M.
Brinkmann
, Ph.D. thesis,
Universität Potsdam
, Potsdam,
2002
.
34.
L. D.
Landau
and
E. M.
Lifshitz
,
Fluid Mechanics
(
Pergamon
,
Oxford
,
1963
).
35.
G.
Dziuk
, “
An algorithm for evolutionary surfaces
,”
Numer. Math.
58
,
603
(
1991
).
36.
K.
Deckelnick
and
K. G.
Siebert
, “
$W1,∞$-convergence of the discrete free boundary for obstacle problems
,”
IMA J. Numer. Anal.
20
,
481
(
2000
).
37.
H.
Lamb
,
Hydrodynamics
(
Dover
,
New York
,
1932
).
38.
H.
von Helmholtz
, “
Zur Theorie der stationären Ströme in reibenden Flüssigkeiten
,”
Nat. Med.
V
,
1
(
1869
). Reprinted in Ref. 39, pp. 223–230.
39.
H.
von Helmholtz
,
Wissenschaftliche Abhandlungen
(
Barth
,
Leipzig
,
1882
), Vol.
1
.
40.
B. A.
Finlayson
,
The Method of Weighted Residuals and Variational Principles
(
,
New York
,
1972
).
41.

As a simple demonstration, we consider a problem that is similar to the well-known catenary setup. A curve, representing the one-dimensional surface, in two-dimensional space is assumed to be fixed at its end points. A homogeneous force should act in the direction normal to the connecting line between the end points. If the curve is physically realized by a chain of rigid or elastic elements, its shape is given by a catenary or a parabola, respectively. But if the curve has to be realized by the free surface of a fluid, this problem is ill-posed and does not possess a stationary solution. The free surface then can only compensate normal forces and its shape would be a straight line. On the other hand, its curvature would then be zero and therefore could not compensate the external force. Hence, no solution exists.

42.
M.
Behr
, “
On the application of slip boundary condition on curved boundaries
,”
Int. J. Numer. Methods Fluids
45
,
43
(
2004
).
43.
M. A.
Walkley
,
P. H.
,
P. K.
Jimack
,
M. A.
Kelmanson
, and
J. L.
Summers
, “
On the calculation of normals in free-surface problems
,”
Commun. Numer. Methods Eng.
20
,
343
(
2004
).
44.
I. N.
Bronshtein
and
K. A.
Semendyayew
,
Handbook of Mathematics
(
Harri Deutsch
,
Thun, Frankfurt a. M.
,
1985
).
45.
The C++ Finite Element Library libMesh is written by
B. S.
Kirk
,
J. W.
Peterson
,
R.
Stogner
, and
S.
Petersen
. Its source-code is available from http://libmesh.sourceforge.net.
46.
W. M.
Nyborg
, “
Acoustic streaming
,”
Phys. Acoust.
2B
,
265
(
1965
).
47.
A.
Wixforth
and
Z.
Guttenberg
(private communication).
48.
J.
Guven
, “
Membrane geometry with auxiliary variables and quadratic constraints
,”
J. Phys. A
37
,
L313
(
2004
).
49.
R.
Capovilla
,
J.
Guven
, and
J. A.
Santiago
, “
Deformations of the geometry of lipid vesicles
,”
J. Phys. A
36
,
6281
(
2003
).
50.
U.
Seifert
, “
Configurations of fluid membranes and vesicles
,”
46
,
13
(
1997
).
51.
C. J.
Strobl
,
C.
Schäflein
,
U.
Beierlein
,
J.
Ebbecke
, and
A.
Wixforth
, “
Carbon nanotube alignment by surface acoustic waves
,”
Appl. Phys. Lett.
85
,
1427
(
2004
).
52.
M.
Kostur
,
M.
Schindler
,
P.
Talkner
, and
P.
Hänggi
, “
Chiral separation in microflows
,”
Phys. Rev. Lett.
96
,
014502
(
2006
).