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Uniaxial, extensional flows in liquid bridges

Collaborator: E. Bänsch  

Cooperation with: C. Berg (Universität Bremen, ZARM)

Description:

Owing to their constant deformation rates linear flow fields are commonly used for rheological tests, e.g., to measure the fluid viscosity or the deformation behavior of the whole sample or components in it.

Experimentally, these flow fields may be realized as extensional flows in a stretched bridge of viscous liquid under microgravity conditions, where the bridge is held exclusively between two circular concentric support membranes. To derive an almost linear flow field, the stretching velocity has to increase exponentially in time. Furthermore, in order to achieve a deformation rate as constant as possible, both in space and time, in our case the radius of the membranes is decreased accordingly to maintain a (nearly) cylindrical shape during stretching.

A mathematical description of the system leads to a free boundary problem for the Navier-Stokes equations: Let $\Omega=\Omega(t)$ denote the--a priori unknown--domain occupied by the fluid. Find a vector-valued velocity field ${\bf v }={\bf v }(t,{\bf x})$ and a pressure field $p=p(t,{\bf x})$ such that

   

with V_f the normal velocity of the free surface $\Gamma_f$, holds.

Since all data are axisymmetric and the flow is laminar we may assume a two-dimensional axisymmetric configuration.

For the numerical simulation of the problem we use a finite element method with the following key ingredients: a variational formulation of the curvature of the free boundary, yielding an accurate, dimension-independent and simple-to-implement approximation for the curvature; a stable time discretization, semi-implicit w.r.t. the treatment of the curvature terms, which on the one hand avoids a CFL-like restriction of the time step as in common ``explicit'' treatments of the curvature terms and on the other hand decouples the computation of the geometry from that of the flow field. This approach has proven to be both efficient and robust (see [2]).

 

Fig. 1: Experiment, stretched bridge of castor oil at different time instants

 

Fig. 2: Simulation, strain rate distribution of stretched bridge at different time instants

References:

1.  E. BÄNSCH, C. BERG, Uniaxial, extensional flows in liquid bridges, in preparation.
2.  E. BÄNSCH, Finite element discretization of the Navier-Stokes equations with a free capillary surface, Numer. Math., 88 (2001), pp. 203-235.
3.  S. BERG, R.  KRÖGER, H.J. RATH, Measurement of extensional viscosity by stretching large liquid bridges in microgravity, J. Non-Newtonian Fluid Mech., 55 (1994), pp. 307-319.
4.  C. BERG, M. DREYER,H.J. RATH, A large fluid-bridge device to measure the deformation of drops in uniaxial extensional flow fields, Meas. Sci. Technol., 10 (1999), pp. 956-964.

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