Controlled stabilization of rotating toroidal drops in viscous linear flow

Abstract

Toroidal drops, embedded in viscous flow, have a large range of stationary shapes that are challenging to compute numerically due to their inherent instability. When both the drop and the outer fluid are Newtonian liquids, the only reported cases of such stable configurations are of highly expanded drops rotating in an axisymmetrical extensional flow. In this study, we propose a method for stabilizing the stationary shapes of inherently unstable rotating toroidal drops, embedded in extensional or biextensional flow, by subjecting the system to feedback control stabilization. The proposed controller is designed using a two-state dynamic model of the system and is tested on a high-order nonlinear dynamic model of the drop deformation. It is demonstrated that, through this simplified feedback-control-centred approach, an extended collection of stabilized stationary solutions is generated, which spans the range from highly expanded drops to almost collapsed ones. In the latter region, that was never obtained in previous studies, multiplicity of solutions is identified. Furthermore, our method is more accurate and more efficient compared with the previously used ones.

Stationary dimpled drops under linear flow

“…In the dynamic computations (Zabarankin et al 2013;Malik et al 2020), the solution was assumed to be stationary if (2.11) was established to a desired degree of accuracy. We note that, following Malik et al (2020Malik et al ( , 2022, with a mesh of 100 surface elements, the criterion for stationarity is set at max(|u n |) ≤ O(10 −3 ). The obtained singly connected stationary shapes do not change for an indefinitely long time, and are addressed as stable with respect to axisymmetric perturbations.…”

“…For a control signal, we used the capillary number Ca that is proportional to the intensity of the outer flow. A similar approach was used in Malik et al (2022) to stabilize toroidal drop shapes. Detailed descriptions of the algorithm and obtained results are presented in the forthcoming sections.…”

“…That report was short of displaying the entire span of the capillary numbers for these solutions due to the increasing instability level that was encountered with shapes in the region near the inner collapse of the tori. The latter obstacle was overcome by Malik et al (2022), where the range of Ca was extended by employing a control algorithm to stabilize those branches of deformed tori and complete the missing regions of Ca for the various Bond numbers. It is evident and quite remarkable that the points of near collapsed tori reported in Malik et al (2022) practically coincide with those obtain as Ca collapse in this work and reported in figure 8.…”

“…Malik, Lavrenteva & Nir (2021) reported on the branch of toroidal stationary shapes existing under the same conditions as simply connected ones. Malik et al (2022) used a feedback-control approach to extend the branch of toroidal drops up to near collapse. Yet the branches of singly connected and toroidal shapes remained unconnected in cases with and without rotation.…”

“…The main goal of this study is to construct the connecting branch in the case when the outer linear flow consists of rotational and axisymmetric (extensional or compressional) components. The method that was employed is based on the use of classical control theory similar to that employed in Malik et al (2022). Methods of control theory are employed extensively and successfully to stabilize various types of flow, e.g.…”