SESSION 11

STABILITY AND TRANSITIONS

Chairmen: Y. Hagiwara, P. Angeli

ON THE GENERATION OF VORTICITY AT LIQUID-LIQUID INTERFACES

Marcus Gruen, Manfred J. Hampe
Fachgebiet Thermische Verfahrenstechnik
Technische Hochschule Darmstadt
Petersenstr. 30, D-64387 Darmstadt, Germany

The onset of Marangoni convection is a phenomenon frequently encountered in liquid-liquid extraction systems. The common explanation of Marangoni effects is that these are due to a gradient of the interfacial tension in the two dimensional phase boundary. A more subtle theoretical approach is to discard the concept of a two dimensional interface and to model an interface as a three-dimensional interfacial region. If this is done, the interface is seen to become 'structured', the structure being related to the concentration gradients across and along the interface. In phase equilibrium, the three dimensional interface needs additional variables in order to specify the thermodynamic state. These are the vectorial 'structure' to be defined for every component and the vectorial 'interfacial potential' which is also defined for every component. In non-equilibrium situations, gradients of the interfacial potential are driving forces for the change of the interfacial structure. The interfacial structure may be transported, created or annihilated. Hence, a balance equation for a tensorial 'structure transport' can be formulated. If the rules of irreversible thermodynamics are followed, the two tensorial driving forces for momentum transport and structure transport, the gradient of the velocity and the gradient of the interfacial potential resp., interact and give rise to the phenomena known as the Marangoni effect and the inverse Marangoni effect.

If a two phase system with a three dimensional interface is modelled in terms of continuum mechanics, a slightly modified Navier-Stokes equation for momentum transport and momentum transfer than for single phase systems or for a system with a two dimensional boundary is obtained. The modified Navier-Stokes equation does contain terms with driving forces due to gradients of the interfacial potential. Classical rheology has some difficulty with explaining the onset of vorticity in incompressible media. Helmholtz in 1858 stated that 'particles formerly free of vorticity will stay so forever'. Marangoni effects in liquid-liquid systems - and the onset of turbulence generally - are the experimental disproof of his statement.

The question discussed in this paper is, how vorticity is generated under certain conditions at a liquid- liquid interface within the framework of a theory assuming the interface to be three dimensional. It will be shown that at a three dimensional interface, vorticity is easily generated even for the incompressible fluids of a liquid-liquid system. There is a fundamental difference between a planar, horizontal liquid-liquid interface and the curved interface of a drop rising or falling in the gravitational field. The formation of phase boundaries, e. g. the formation of drops at nozzles, will almost always induce vorticity. The onset of vorticity depends crucially on the purity of the interface and on the effect of impurities on the interfacial rheological and related phenomenological parameters.

ANALYSIS OF NONPARALLEL FLOW EFFECTS ON THE INSTABILITY OF A CAPILLARY JET IN ANOTHER IMMISCIBLE FLUID

L. Tadrist, St. Radev
Institut Universitaire des Systemes Thermiques Industriels - CNRS -UMR 139
Universite de Provence, Centre de Saint-Jerome
Av. Esc. Normandie-Niemen,13397 Marseille Cedex 20 - France;
Institute of Mechanics, Bulgarian Academy of Sciences
4,acad. G. Bonchev str.,1113 Sofia, Bulgaria

In most of the experiments on the instability of a liquid capillary jet the axis of the latter is directed vertically. In these conditions the jet flow is subjected to the action of the gravity forces which results into jet contraction and acceleration. When the experimental results are to be compared to some theoretical model the preferences are given to the analvtical models (e.g. Rayleigh (1878), Tomotika (1935), Yuen (1968)) which are derived for fully cylindrical jets i.e. under the assumption that the gravity effects are negligible. This assumption appears to work well when some integral parameters of the instability such as jet length and drop sizes are compared. However it could be expected that locally in the vicinity of a Diven jet cross section the disturbances behaviour differs from that predicted for fully cylindrical jet.

In our presentation we will report some measurements of jet diameter oscillatinns in time in a cross section fixed along the jet axis. The oscillations of the diameter are made visible by a laser sheet oriented perpendicularly to jet axis. (For more details concerning the measuring technique see Tadrist et al. ( 1991 )). Jet contraction is controled by varying the density of the surrounding fluid (continuous phase). One example of these oscillations is shown in the figure below corresponding to the case of a water jet flowing in dodecane. Increasing continuous phase density leads to a more regular signal the frequency of which could be related to the fastest growing one. This conclusion is based as well on the FFT of the signal.

1. Rayleigh L., Proc. Lond. Math. Soc. 10 ( 1878).4- l 3.
2. Tadrist L., E. Alaoui, R. Occelli, J. Pantaloni, Exper. in Fluids 12 ( 1991 ) 67-75.
3. Tomotika S., Proc. R. Soc. Lond. A150 ( 1935) 327-337.
4. Yuen M.-C. J., Fluid Mech. 33 (1968) 151-163

Temporal Signal of the Jet Diameter Obtained by Photoelectrical Cell
(Along an ordinate axis the photoelectrical cell tension is plotted)