We analyze the migration characteristics of a droplet in an oscillatory flow field in a parallel plate microconfinement. Using phase field formalism, we capture the dynamical evolution of the droplet over a wide range of the frequency of the imposed oscillation in the flow field, drop size relative to the channel gap, and the capillary number. The latter two factors imply the contribution of droplet deformability, commonly considered in the study of droplet migration under steady shear flow conditions. We show that the imposed oscillation brings an additional time complexity in the droplet movement, realized through temporally varying drop shape, flow direction, and the inertial response of the droplet. As a consequence, we observe a spatially complicated pathway of the droplet along the transverse direction, in sharp contrast to the smooth migration under a similar yet steady shear flow condition. Intuitively, the longitudinal component of the droplet movement is in tandem with the flow continuity and evolves with time at the same frequency as that of the imposed oscillation, although with an amplitude decreasing with the frequency. The time complexity of the transverse component of the movement pattern, however, cannot be rationalized through such intuitive arguments. Towards bringing out the underlying physics, we further endeavor in a reciprocal identity based analysis. Following this approach, we unveil the time complexities of the droplet movement, which appear to be sufficient to rationalize the complex movement patterns observed through the comprehensive simulation studies. These results can be of profound importance in designing droplet based microfluidic systems in an oscillatory flow environment.

In the present paper we focus on deriving the modified Smoluchowski slip velocity of second-order fluids, for electroosmotic flows over plane surfaces with arbitrary non-uniform surface potential in the presence of thin electric double layers (EDLs). We employ matched asymptotic expansion to stretch the electric double layer and subsequently apply regular asymptotic expansions taking the Deborah number ($De$) as the gauge function. Modified slip velocities correct up to $O(De^{2})$ are presented. Two sample cases are considered to demonstrate the effects of viscoelasticity on slip velocity: (i) an axially periodic patterned potential and (ii) a step-change-like variation in the surface potential. The central result of our analysis is that, unlike Newtonian fluids, the electroosmotic slip velocity for second-order fluids does not, in general, align with the direction of the applied external electric field. Proceeding further forward, we show that the slip velocity in a given direction may, in fact, depend on the applied electric field strength in a mutually orthogonal direction, considering three dimensionality of the flow structure. In addition, we demonstrate that the modified slip velocity is not proportional to the zeta potential, as in the cases of Newtonian fluids; rather it depends strongly on the gradients of the interfacial potential as well. Our results are likely to have potential implications so far as the design of charge modulated microfluidic devices transporting rheologically complex fluids is concerned, such as for mixing and bio-reactive system analysis in lab-on-a-chip-based micro-total-analysis systems handling bio-fluids.

Performance of a hydrocyclone as a size separation unit is popularly judged using empirical or phenomenological models, with a predefined design under consideration. In contrast, here we develop a fundamental basis for analyzing the classification behavior of any hydrocyclone using experiment, simulation, and concurrent theory. Considering the G force (defined as the ratio of the centrifugal force to the weight of the suspended particle under consideration) distribution that has implications in creating separation of the suspended particles, here we bring out its consistent dependence on the essential governing parameters. The present estimation seems to agree well with the existing notions and available literature data, irrespective of the hydrocyclone size. Thus, our analysis is expected to provide a consistent basis for design of any tailormade hydrocyclone.

We envisage the spreading behavior of a two-dimensional droplet under a thin-film-based paradigm, under a perfect wetting condition, while the droplet is placed over a nonisothermal substrate. Starting from the onset of thin-film behavior (or equivalently beyond the inertia-dominated initial stage), we identify the existence of mutually contrasting multiple scaling regimes defining the spreading behavior at different time scales. This is attributable to the time-stage-wise upsurge of capillarity or thermocapillarity over the other. In particular, the spreading behavior is characterized by the foot-width (w) evolution with time (t) in a power-law fashion w ∼ t(α), with α being the spreading exponent, defining the rate of spreading. Following pertinent thin-film and subsequent similarity analysis, we identify different asymptotes of α over disparate temporal scales, leading to the characterization of different scaling regimes over the entire spreading event starting from the inception of thin-film behavior. Reported literature data are found to correspond well to the present interpretations and estimations.

We study the influence of disjoining pressure for moving long bubbles inside cylindrical capillaries. Towards that end, consistent thin-film equations, for the annular region separating the bubble from the channel surface, are presented with specific emphasis on three different attributes: (a) the van der Waals interaction, formalized by the classical Lifshitz form of disjoining pressure; (b) the nonuniformity in film thickness, accommodated by the necessary corrections in the disjoining pressure; and (c) the electrostatic component of disjoining pressure, reminiscent of the electrostatic interactions in the presence of surface charges. The present thin-film analysis appositely uncovers the existence and the breakdown of the two-thirds power law for minimum film thickness behavior. This is attributed to the alteration in the characteristic length scales governing the underlying physics, as quantitatively established by our consistent scaling analysis. In the breakdown regimes, the characteristic length scales are found to be composed of the suitable combinations of the capillary number and the physics driven intrinsic length scales. The characteristics of the breakdown regime reported by us appear to match excellently with reported experimental data in the low capillary number regime. Towards unveiling the possible implications of slope and curvature dependence of disjoining pressure, our analysis reveals that the consequent correction term endorses an order two-thirds power of the capillary number contribution without alerting the governing length scales. The subsequent asymptotic analysis reveals that this correction may be neglected to the leading order approximation. Finally, we consider the electrostatic component of the disjoining pressure which may be realized in the presence of surface charges. Our analysis reveals that the significance of the electrostatic interaction is realized over a very small capillary number regime, leading towards the departure from the two-thirds power law type behavior. Reasonably good agreement is obtained with reported experimental data over this regime.