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Computing stationary free-surface shapes in microfluidics

### Abstract

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.

© 2006 American Institute of Physics

Received 04 February 2006
Accepted 08 September 2006
Published online 16 October 2006

Acknowledgments:
We gratefully acknowledge our experimental partners in the group of Professor Achim Wixforth, University of Augsburg, and the developers of the *libmesh* project. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via Grant No. 1517/25-1, SFB 486 (Project B13), and the Graduiertenkolleg: Nichtlineare Probleme in Analysis, Geometrie und Physik.

Article outline:

I. INTRODUCTION
II. STATEMENT OF THE PROBLEM
A. The basic equations
B. The parameter regime
III. CONTINUOUS DESCRIPTION OF THE PROBLEM
A. Physical aspects of the free-surface boundary condition: First variations
B. Splitting the problem into two numerical systems
C. Second variation with respect to the surface parametrization
IV. DISCRETIZATION OF THE PROBLEM
A. The fluidic system
B. The geometric system
C. Controlling the tangential displacements of boundary nodes
V. SUMMARY OF THE ALGORITHM
VI. NUMERICAL EXPERIMENTS
A. The instability of a “direct explicit update” algorithm
B. Testing the accuracy of the modified Newton algorithm
C. A deformed microdroplet
VII. SUMMARY AND OUTLOOK

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/content/aip/journal/pof2/18/10/10.1063/1.2361291

2006-10-16

2016-02-12

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