Sayan Das

Migration of a surfactant-laden droplet in non-isothermal Poiseuille flow

The motion of a surfactant-laden viscous droplet in the presence of non-isothermal Poiseuille flow is studied analytically and numerically. Specifically, the focus of the present study is on the role of interfacial Marangoni stress generated due to imposed temperature gradient and non-uniform distribution of bulk-insoluble surfactants towards dictating the velocity and direction of motion of the droplet when the background flow is Poiseuille. Assuming the thermal convection and fluid inertia to be negligible, we obtain the explicit expression for steady velocity of a non-deformable spherical droplet when the droplet is located at the centerline of the imposed unbounded Poiseuille flow and encountering a linearly varying temperature field. Under these assumptions, the interfacial transport of surfactants is governed by the surface Peclet number which represents the relative strength of the advective transport of surfactant molecules over the diffusive transport. We obtain analytical solution for small and ...

Cross-stream migration of a surfactant-laden deformable droplet in a Poiseuille flow

The motion of a viscous deformable droplet suspended in an unbounded Poiseuille flow in the presence of bulk-insoluble surfactants is studied analytically. Assuming the convective transport of fluid and heat to be negligible, we perform a small-deformation perturbation analysis to obtain the droplet migration velocity. The droplet dynamics strongly depends on the distribution of surfactants along the droplet interface, which is governed by the relative strength of convective transport of surfactants as compared with the diffusive transport of surfactants. The present study is focused on the following two limits: (i) when the surfactant transport is dominated by surface diffusion, and (ii) when the surfactant transport is dominated by surface convection. In the first limiting case, it is seen that the axial velocity of the droplet decreases with increase in the advection of the surfactants along the surface. The variation of cross-stream migration velocity, on the other hand, is analyzed over three different regimes based on the ratio of the viscosity of the droplet phase to that of the carrier phase. In the first regime the migration velocity decreases with increase in surface advection of the surfactants although there is no change in direction of droplet migration. For the second regime, the direction of the cross-stream migration of the droplet changes depending on different parameters. In the third regime, the migration velocity is merely affected by any change in the surfactant distribution. For the other limit of higher surface advection in comparison to surface diffusion of the surfactants, the axial velocity of the droplet is found to be independent of the surfactant distribution. However, the cross-stream velocity is found to decrease with increase in non-uniformity in surfactant distribution.

Effect of temperature gradient on the cross-stream migration of a surfactant-laden droplet in Poiseuille flow

The motion of a viscous droplet in unbounded Poiseuille flow under the combined influence of bulk-insoluble surfactant and linearly varying temperature field aligned in the direction of imposed flow is studied analytically. Neglecting fluid inertia, thermal convection and shape deformation, asymptotic analysis is performed to obtain the velocity of a force-free surfactant-laden droplet. The droplet speed and direction of motion are strongly influenced by the interfacial transport of surfactant, which is governed by surface Peclet number. The present study is focused on the following two limiting situations of surfactant transport: (i) surface-diffusion-dominated surfactant transport considering small surface Peclet number, and (ii) surface-convection-dominated surfactant transport considering high surface Peclet number. Thermocapillary-induced Marangoni stress, the strength of which relative to viscous stress is represented by the thermal Marangoni number, has a strong influence on the distribution of surfactant on the droplet surface. The present study shows that the motion of a surfactant-laden droplet in the combined presence of temperature and imposed Poiseuille flow cannot be obtained by a simple superposition of the following two independent results: migration of a surfactant-free droplet in a temperature gradient, and the motion of a surfactant-laden droplet in a Poiseuille flow. The temperature field not only affects the axial velocity of the droplet, but also has a non-trivial effect on the cross-stream velocity of the droplet in spite of the fact that the temperature gradient is aligned with the Poiseuille flow direction. When the imposed temperature increases in the direction of the Poiseuille flow, the droplet migrates towards the flow centreline. The magnitude of both axial and cross-stream velocity components increases with the thermal Marangoni number. However, when the imposed temperature decreases in the direction of the Poiseuille flow, the magnitude of both axial and cross-stream velocity components may increase or decrease with the thermal Marangoni number. Most interestingly, the droplet moves either towards the flow centreline or away from it. The present study shows a critical value of the thermal Marangoni number beyond which the droplet moves away from the flow centreline which is in sharp contrast to the motion of a surfactant-laden droplet in isothermal flow, for which the droplet always moves towards the flow centreline. Interestingly, we show that the above picture may become significantly altered in the case where the droplet is not a neutrally buoyant one. When the droplet is less dense than the suspending medium, the presence of gravity in the direction of the Poiseuille flow can lead to cross-stream motion of the droplet away from the flow centreline even when the temperature increases in the direction of the Poiseuille flow. These results may bear far-reaching consequences in various emulsification techniques in microfluidic devices, as well as in biomolecule synthesis, vesicle dynamics, single-cell analysis and nanoparticle synthesis.

Influence of interfacial slip on the suspension rheology of a dilute emulsion of surfactant-laden deformable drops in linear flows

The present study deals with the effect of interfacial slip on the deformation and emulsion rheology of a dilute suspension of droplets in a linear flow. The droplets are laden with surfactants that are bulk-insoluble and get transported only along the interface. An asymptotic approach is adopted for the present analysis in order to tackle the nonlinearity present due to deformation of droplets. The analysis is carried out under two different limiting scenarios namely: surface diffusion-dominated-surfactant transport and surface convection-dominated surfactant transport. For either of the limiting cases we look into the droplet dynamics for two commonly encountered bulk flows - uniaxial extensional and simple shear flow. Under the assumption of negligible fluid inertia in either phase, it is shown that slip at the droplet interface significantly affects the surfactant-induced Marangoni stress and hence droplet deformation and emulsion rheology. Presence of interfacial slip not only brings about a decrease in the droplet deformation but also reduces the effective viscosity of the emulsion. The fall in both droplet deformation and effective viscosity is found to be more severe for the limiting case of surface convection-dominated surfactant transport. For the case of an imposed simple shear flow, the normal stress differences generated due to droplet deformation are affected as well due to the presence of interfacial slip.

Influence of complex interfacial rheology on the thermocapillary migration of a surfactant-laden droplet in Poiseuille flow

The effect of surface viscosity on the motion of a surfactant-laden droplet in the presence of a non-isothermal Poiseuille flow is studied, both analytically and numerically. The presence of bulk-insoluble surfactants along the droplet surface results in interfacial shear and dilatational viscosities. This, in turn, is responsible for the generation of surface-excess viscous stresses that obey the Boussinesq-Scriven constitutive law for constant values of surface shear and dilatational viscosities. The present study is primarily focused on finding out how this confluence can be used to modulate droplet dynamics in the presence of Marangoni stress induced by nonuniform distribution of surfactants and temperature along the droplet surface, by exploiting an intricate interplay of the respective forcing parameters influencing the interfacial stresses. Under the assumption of negligible fluid inertia and thermal convection, the steady-state migration velocity of a non-deformable spherical droplet, placed at the centerline of an imposed unbounded Poiseuille flow, is obtained for the limiting case when the surfactant transport along the interface is dominated by surface diffusion. Our analysis proves that the droplet migration velocity is unaffected by the shear viscosity whereas the dilatational viscosity has a significant effect on the same. The surface viscous effects always retard the migration of a surfactant-laden droplet when the temperature in the far-field increases in the direction of the imposed flow although the droplet always migrates towards the hotter region. On the contrary, if a large temperature gradient is applied in a direction opposite to that of the imposed flow, the direction of droplet migration gets reversed. However, for a sufficiently high value of dilatational surface viscosity, the direction of droplet migration reverses. For the limiting case in which the surfactant transport along the droplet surface is dominated by surface convection, on the other hand, surface viscosities do not have any effect on the motion of the droplet. These results are likely to have far-reaching consequences in designing an optimal migration path in droplet-based microfluidic technology.

Effect of interfacial slip on the deformation of a viscoelastic drop in uniaxial extensional flow field

The effect of interfacial slip on the deformation of a viscoelastic droplet, suspended in another viscoelastic medium, in the presence of a uniaxial extensional flow, is studied analytically. Using the Oldroyd-B constitutive relation, the Stokes flow problem is solved in the limit of a small capillary number and small Deborah number. Experimentally observed interfacial velocity slip is incorporated using a Navier slip boundary condition. The interfacial slip significantly reduces the magnitude of droplet deformation when the droplet has larger viscosity as compared with the suspending phase. The droplet shape becomes less ellipsoidal in the presence of slip. The effect of slip diminishes for low viscosity droplets. Slip effectively weakens the dependence of the droplet deformation on the droplet to medium viscosity ratio. The viscoelasticity of the suspending phase plays a dominant role on the droplet deformation as compared with the viscoelasticity of the droplet phase when there is velocity slip at the ...

Surfactant-induced retardation in lateral migration of droplets in a microfluidic confinement

The present study deals with the effect of surfactants on the cross-stream migration of droplets in a confined fluidic environment, both experimentally and theoretically. Presence of an imposed flow induces droplet deformation and disturbs the equilibrium that results in subsequent surfactant redistribution along the interface. This further creates a gradient in surface tension, thus generating a Marangoni stress that significantly alters the droplet dynamics. On subsequent experimental investigation, it is found that presence of surfactants reduces the cross-stream migration velocity of the droplet. High-speed photography is utilized to visualize the transport of droplets in a microfluidic channel. It is shown that the channel confinement significantly enhances the surfactant-induced retardation of the droplet. In addition, a larger surfactant concentration is found to induce a greater reduction in cross-stream migration velocity of the droplet, the effect of which is reduced when the initial transverse position of the droplet is shifted closer to the channel centerline. To support our experimental results, an asymptotic approach is adopted to solve the flow field in the presence of bulk-insoluble surfactants and under the assumption of small shape deformation. A good match between our theoretical prediction and the experimental results is obtained. The present analysis provides us with a wide scope of application towards various droplet-based microfluidic as well as medical diagnostic devices where manipulation of droplet trajectory is a major issue.