This study deals with the motion and deformation of a compound drop system, subject to arbitrary but Stokesian far-field flow conditions, in the presence of bulk-insoluble surfactants. We derive solutions for fluid velocities and the resulting surfactant concentrations, assuming the capillary number and surface Péclet number to be small, as compared with unity. We first focus on a concentric drop configuration and apply Lamb’s general solution, assuming the far-field flow to be arbitrary in nature. As representative case studies, we consider two cases: (i) flow dynamics in linear flows and (ii) flow dynamics in a Poiseuille flow, although for the latter case, the concentric configuration does not remain valid in general. We further look into the effective viscosity of a dilute suspension of compound drops, subject to linear ambient flow, and compare our predictions with previously reported experiments. Subsequently, the eccentric drop configuration is addressed by using a bipolar coordinate system where the far-field flow is assumed to be axisymmetric but otherwise arbitrary in nature. As a specific example for eccentric drop dynamics, we focus on Poiseuille flow and study the drop migration velocities. Our analysis shows that the presence of surfactant generally opposes the imposed flows, thereby acting like an effective augmented viscosity. Our analysis reveals that maximizing the effects of surfactant makes the drops behave like solid particles suspended in a medium. However, in uniaxial extensional flow, the presence of surfactants on the inner drop, in conjunction with the drop radius ratio, leads to a host of interesting and non-monotonic behaviours for the interface deformation. For eccentric drops, the effect of eccentricity only becomes noticeable after it surpasses a certain critical value, and becomes most prominent when the two interfaces approach each other. We further depict that surfactant and eccentricity generally tend to suppress each other’s effects on the droplet migration velocities.
We pinpoint the limitations in traditional electroviscous analysis for narrow fluidic confinements. We show that because of neglecting the convective transport of ions originated out of the established streaming field itself, the traditional approach may result in physically inconsistent flow rate predictions. We show that the larger the value of an ionic Peclet number and narrower the confinement, the more conspicuous are these erroneous predictions, within threshold limits of the nondimensional zeta potential. We come up with an improved mathematical model to overcome such discrepancies.
An obliquely inclined circular water jet, impinging on a flat horizontal surface, confers a series of hydraulic jump profiles, pertaining to different jet inclinations and jet velocities. These jump profiles are non-circular, and can be broadly grouped into two categories, based on the angle of jet inclination, φ, made with horizontal. Jumps corrosponding to the range (25° < φ≤ 90°) are observed to be bounded by smooth curves, whereas those corresponding to φ≤ 25° are characterized by distinct corners. The present work attempts to find a geometric and hydrodynamic characterization of the spatial patterns formed as a consequence of such non-circular hydraulic jump profiles. Flow-visualization experiments are conducted to depict the shape of demarcating boundaries between supercritical and subcritical flows, and the corresponding radial jump locations are obtained. Theoretical calculations are also executed to obtain the radial locations of the jumps with geometrically smooth profiles. Comparisons are subsequently made between the theoretical predictions and the experimental observations, and a good agreement between these two can be observed. Jumps with corners, however, turn out to be comprised of strikingly contrasting profiles, which can be attributed to the ‘jump–jet’ interaction and the ‘jump-jump’ interaction mechanisms. A phenomenological explanation is also provided, by drawing an analogy from the theory of shock-wave interactions.
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