#### Date of Award

Fall 2009

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

#### Department

Mathematical Sciences

#### First Advisor

Demetrius T. Papageorgiou

#### Second Advisor

Michael Siegel

#### Third Advisor

Michael R. Booty

#### Fourth Advisor

Peter G. Petropoulos

#### Fifth Advisor

Pushpendra Singh

#### Abstract

The nonlinear dynamics of viscous perfectly conducting liquid jets or threads under the action of a radial electric field are studied theoretically and numerically here. The field is generated by a potential difference between the jet surface and a concentrically placed electrode of given radius. A long-wave nonlinear model that is used to predict the dynamics of the system and in particular to address the effect of the radial electric field on jet breakup is developed, Two canonical regimes are identified that depend on the size of the gap between the outer electrode and the unperturbed jet surface. For relatively large gap sizes, long waves are stabilized for sufficiently strong electric fields but remain unstable as in the non-electrified case for electric field strengths below a critical value, For relatively small gaps, an electric field of any strength enhances the instability of long waves as compared to the non-electrified case. Accurate numerical simulations are carried out based on our nonlinear models to describe the nonlinear evolution and terminal states in these two regimes. It is found that jet pinching does not occur irrespective of the parameters, Regimes are identified where capillary instability leads to the formation of stable quasi-static microthreads (connected to large main drops) whose radius decreases with the strength of the electric field. The generic ultimate singular event described by our models is the attraction of the jet surface towards the enclosing electrode and its contact with the electrode in finite time. A self-similar closed form solution is found that describes this event with the interface near touchdown having locally a cusp geometry. The theory is compared with the time-dependent simulations with excellent agreement.

In addition a core-annular flow problem is considered to include the external viscous fluid. A full problem simulation, based on a boundary integral technique is carried out to capture the full dynamics of the electrified viscous jet in the zero Reynolds number limit. Pinching solutions of either electrified or non-electrified viscous jets are obtained and the instantaneous velocity field and flow patterns are studied numerically near breakup, As the electric field strength increases, the size and shape of the drops are changed dramatically compared with the non-electrified problem. However, the local dynamics remain the same as shown in the non-electrified capillary breakup problem, since the main and satellite liquid masses joined by a collapsing neck have the same potential and would not feel the strong influence of the external field. The pinching is suppressed if the field strength is sufficiently large and another type of breakup behavior appears. Briefly speaking, the interface is attracted and touches the outer electrode in the radial direction in a similar phenomenon found for a single jet problem, This type of terminal state is also described by a lubrication model in the thin annulus limit. A comparison between the boundary-integral simulations and the asymptotic results is also carried out.

#### Recommended Citation

Wang, Qiming, "Nonlinear evolution of annular layers and liquid threads in electric fields" (2009). *Dissertations*. 199.

https://digitalcommons.njit.edu/dissertations/199