Effect of surfactant on motion and deformation of compound droplets in arbitrary unbounded Stokes flows

Abstract

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.

Formation of compound droplets during fragmentation of turbulent buoyant oil jet in water

“…There is considerable literature about the behaviour of an isolated compound droplet subjected to external shear. Reduced deformation of the internal droplet with increasing has been shown for of the same range by Stone & Leal (1990), Mandal, Ghosh & Chakraborty (2016) and Kim & Dabiri (2017). In contrast, substantial deformations are reported for higher and low interfacial tension of the interior droplet by Smith, Ottino & de la Cruz (2004).…”

Electrically modulated dynamics of a compound droplet in a confined microfluidic environment

“…Based on small-deformation perturbation analysis, the inner and outer droplet shapes can be represented as (Mandal et al. 2016 b ) Here and denote the corrections in shape of the interface at different orders of perturbation, namely and , respectively. The corrections can be expressed in the following form as where and can again be expressed as The constant coefficients in the above equations are provided in § 2.2 of the supplementary material.…”

“…After applying the non-dimensionalization scheme, the governing electric potential distribution ( ) for the th ( ) fluid reads The essential boundary conditions take the form where the normal unit vector in the outward direction at the interface is denoted by (Mandal, Ghosh & Chakraborty 2016 b ). We have employed the general notation to denote any quantity at the interface .…”

“…where the normal unit vector in the outward direction at the interface Π ij is denoted by n ij (Mandal, Ghosh & Chakraborty 2016b). We have employed the general notation [ ] Π ij to denote any quantity at the interface Π ij .…”

Electro-Hydrodynamics of Emulsion Droplets: Physical Insights to Applications

“…The high voltage is supplied across the electrodes and dynamics are captured by using cameras. In these studies, some authors have also introduced surfactants at the liquid-liquid interface [ 27 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 , 82 , 83 , 84 , 85 , 86 , 87 , 88 , 89 ]. A summary of the most prominent works related to droplet EHD are summarized in Table 1 .…”