## 1.Microfluidics and electro-hydrodynamics

We investigate micro scale flows in the presence of electric field using both experiments and modeling. Electro-hydrodynamic instabilities in microchannels are employed for efficient mixing of miscible liquids or microdroplet formation for immiscible liquids. The mathematical methods used include linear stability analysis, long wave analysis and solving eigenvalue problems.

### 1.1. Model and Physical System

The physical system consists of two immiscible liquids with different physical properties flowing in a microchannel. The liquids are Newtonian or non-Newtonian. The interface is allowed to deflect.

Figure 1. Physical system

In a microchannel Reynolds number is small as the channel is small. Therefore, the liquids flow side by side sharing a common flat interface. To obtain the droplet of one phase into another, the interface needs to be deflected. External electric field is applied either parallel (Fig. 1a) or normal (Fig. 1b) to the flat interface. The voltage at which the interface loses its flatness is recorded as the critical voltage, which is determined via linear stability analysis. For that purpose momentum equation for velocity and Laplace equation for voltage are solved for both phases. Surface coupled model is assumed as there is a jump in the physical properties of the liquids at the interface. The liquids may be assumed perfect dielectric, leaky dielectric or conducting.

Figure 2. Effect of the electric field

For fast electric charge relaxation times, i.e. when the charge relaxation time is much smaller compared to the fluid time scale, without solving the pertubaton equations, conditions for which the electric field has a stabilizing or destabilizing effect derived analytically and presented on a conductivity ratio versus permittivity ratio graph. As seen in Fig. 2, there are six regions. Parallel electric field has a destabilizing effect in regions 1 and 4, whereas normal field has a destabilizing effect in regions 1, 2, 4, and 5. However, the magnitude of the electric field to destabilize the interface is not known. In regions 3 and 6 the electric field has a stabilizng effect, i.e. the interface cannot be deflected by applying an electric field.

### 1.2.Experiments

Two different types of micro channels are machined out of Plexiglas to apply an electric field either parallel (Fig. 3a) or normal (Fig. 3b) to the flat interface.

Figure 3. Micro channels to apply an electric field parallel (a) or normal (b) electric field.

Various liquid pairs are used and the critical voltage, i.e. the voltage at which interface becomes unstable is recorded using a setup as in the figure.

Figure 4. Experimental setup

A typical data includes set of pictures (Fig. 5) and the aim is to determine the critical voltage.

Figure 5. Set of pictures from an experiment showing the evolution of the interfacial instability between ethylene glycol and 50 cSt silicone oil flowing at 10 μL/min in a micro channel of 1.0mm width. The critical voltage of 880V is reached in (b). The rupture occurred in (h). The electric field is turned off in (o) and the flat interface is recovered in (t).

### References

- Eribol, P. and Uguz, A.K 2015 Experimental investigation of electrohydrodynamic instabilities in micro channels European Physical Journal - Special Topics 224, 423-432 (DOI: 10.1140/epjst/e2015-02371-5)
- Eribol, P. 2014 An experimental study on the stability of the interface between immiscible liquids in a micro channel subject to an electric field M.S. Thesis, Dept. of Chemical Eng., Boğaziçi Univ. CHE 2014 E75).
- Nurocak, A. 2012 Effect of the direction of the electric field on the interfacial instability between a newtonian fluid and a viscoelastic polymer M.S. Thesis, Dept. of Chemical Eng., Boğaziçi Univ. (CHE 2012 N87)
- Ersoy, G. 2011 Investigation of the interfacial instability between a Newtonian fluid and a polymeric fluid under the influence of an electric field for microfluidics applications M.S. Thesis, Dept. of Chemical Eng., Boğaziçi Univ. (CHE 2011 E78)
- Nurocak, A. and Uguz, A.K 2013 Effect of the Direction of the Electric Field on the Interfacial Instability between a Passive Fluid and a Viscoelastic Polymer European Physical Journal - Special Topics 219, 99-110 (DOI: 10.1140/epjst/e2013-01785-3)
- Ersoy, G. and Uguz, A.K. 2012 Electro-hydrodynamic instability in a microchannel between a Newtonian and a non-Newtonian liquid Fluid Dynamics Research 44, 031406.1- 031406.12 (DOI: 10.1088/0169-5983/44/3/031406)
- Uguz, A.K. and Aubry, N. 2008 Quantifying the linear stability of a flowing electrified two-fluid layer in a channel for fast electric times Physics of Fluids 20, 092103.1-092103.10 (DOI: 10.1063/1.2976137)
- Uguz, A.K., Ozen, O., and Aubry, N. 2008 Electric field effect on a two-fluid interface instability in channel flow for fast electric times Physics of Fluids 20, 031702.1-031702.4 (DOI: 10.1063/1.2897313)
- Uguz, A.K. and Aubry, N. 2008 Analytical study of the instability between two liquids flowing in a channel and subjected to parallel or normal electric field Proceedings of IMECE 2008, IMECE2008-67971

### General references

- L. Johns and R. Narayanan, Interfacial Instability (Springer-Verlag, New York, 2002).
- S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981)
- W. Guo, G. Labrosse, and R. Narayanan, The Application of the Chebyshev-Spectral Method in Transport Phenomena: Lecture Notes in Applied and Computational Mechanics (Springer-Verlag, Berlin/Heidelberg, 2013), Vol. 68.
- G. Labrosse, Méthodes Spectrales (Ellipses, Paris, 2011).
- ImageJ : http://imagej.nih.gov/ij/