Mathematical modelling of electrohydrodynamic phenomena in liquid bridges

In this thesis, we study fluid flow in a horizontal liquid bridge, considering gravity and electrohydrodynamic effects. We study two geometries based on the boundary types: liquid bridges between parallel vertical walls and between identical horizontal cylinders. Liquid bridges are ubiquitous in nature and application, arising e.g. in foams, machine lubrication, and porous materials. Our analysis of the model systems paves the way for understanding and, therefore, controlling liquid bridges in systems where there is an interplay of geometrically structured surfaces and electric fields.

For the frst geometry, we formulate full governing equations including appropriate boundary conditions. A simplifed two-dimensional, no-electric-field problem is presented, and equilibrium bridge shapes are computed using MATLAB. In addition, several direct numerical simulations are carried out using COMSOL Multiphysics software to investigate the stability of the liquid bridges and electrohydrodynamic effects.

It is found that to ensure accuracy of the simulations signifcant mesh refinements are needed leading to signifcant increase in computational time.

For the second geometry, liquid bridges suspended between two horizontal cylindrical electrodes are studied using simplified model equations, experiments, and direct numerical simulations. In the experiments, oil droplets are suspended between two parallel electrodes and observed as the electric field is varied. The dynamics of both electrified and non-electrified liquid bridges are recorded using high-speed cameras.

We find that the mathematical model based on the Stokes equations with the Maxwell stress tensor incorporating the effect of the electric field aligns well with experimental observations over a wide range of parameter values. We also develop a reduced-order model based on the Onsager variational principle, in which the dynamics is obtained via a minimisation principle on a Raleighian functional consisting of a dissipation functional and a term derived from the free energy of the system. The reduced-order model is solved both in the presence and absence of gravity. Using a boundaryelement method to solve the electric-field equations, we analyse the influence of the electric field resulting from an imposed potential difference between the electrodes on the dynamics of liquid bridges. We observe that electrified liquid bridges move up and develop stable, flatter interfaces, and can capture more liquid. Direct numerical simulations with COMSOL using the full governing equations with appropriate boundary conditions also validate the mathematical models.

In conclusion, our study demonstrates that such complex phenomenon as electrohydrodynamics in liquid bridges can be well described using two mathematical models: the Stokes equations incorporating the Maxwell stress sensor and the reduced-order model derived using the Onsager variational principle. Each model has its advantages and limitations, providing valuable insights into the behaviour of electrified liquid bridges.