12 results
Coupled bulk and interfacial transport of surfactants governs the settling of a drop towards a wall
- Sayali N. Jadhav, Shubhadeep Mandal, Uddipta Ghosh
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- Journal:
- Journal of Fluid Mechanics / Volume 987 / 25 May 2024
- Published online by Cambridge University Press:
- 24 May 2024, A41
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Surfactant-like impurities are omnipresent in multiphase emulsions and may substantially affect the motion of small droplets by altering their interfacial properties. Usually these surfactants are soluble in the bulk and undergo adsorption–desorption onto the interface which modifies their surface concentration and hence their overall influence on droplet motion. Yet, the impact of the bulk solubility and transport of surfactants on droplet dynamics, especially in the presence of bounding walls, remains poorly understood. As such, in this article, we assess the impact of bulk soluble surfactants on the settling of a spherical drop towards a plane wall. We consider coupled bulk and interfacial transport of surfactants, mediated by adsorption–desorption processes and construct a semi-analytical framework for arbitrary values of ‘bulk interaction parameter’, which dictates the strength of adsorption–desorption kinetics compared with bulk diffusion. Our results indicate that while mass exchange between the bulk and the interface can remobilize the drop, a finite bulk diffusion rate restricts this process and therefore slows down the drop. This also results in bulk concentration depletion near the south pole and accumulation near the north pole, the extent of which becomes strongly asymmetric with an enhanced intensity of depletion, as the drop approaches the wall. Presence of the wall and bulk solubility are found to aid each other towards remobilizing the drop by aptly modifying the interfacial concentration. Our results may provide fundamental insights into the kinetics of surfactant-laden drops, with potential applications in food and pharmaceutical industries, separation processes, etc.
Electrohydrodynamic settling of drop in uniform electric field: beyond Stokes flow regime
- Nalinikanta Behera, Shubhadeep Mandal, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 881 / 25 December 2019
- Published online by Cambridge University Press:
- 24 October 2019, pp. 498-523
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The electrohydrodynamics of a weakly conducting buoyant drop under the combined influence of gravity and a uniform electric field is studied computationally, focusing on the inertia-dominated regime. Numerical simulations are performed for both perfectly dielectric and leaky dielectric drops over a wide range of dimensionless parameters to explore the interplay of fluid inertia and electrical stress to govern the drop shape and charge convection. For perfectly dielectric drops, the fluid inertia alters the drop shape and the deformation behaviour of the drop follows a non-monotonic path. The drop shape at steady state exhibits the transition from oblate to prolate shape on increasing the electric field strength, in sharp contrast to the cases concerning the Stokes flow regime. Similar behaviour is also obtained for leaky dielectric drops for certain fluid properties. For leaky dielectric drops, the fluid inertia also affects the convective transport of charges at the drop surface and thereby alters the drop dynamics. Unlike the Stokes flow regime, where surface charge convection has little effect on the settling speed, the same modifies the drop settling speed quite significantly in the finite inertial regime depending on the combination of electrical conductivity ratio and permittivity ratio. For oblate drops at low capillary number, charge convection alters drop shape, while keeping the nature of deformation unaltered. However, for relatively large capillary number, the oblate drop transforms into a dimpled shape due to charge convection. For all cases, an interesting fact is noticed that under the combined action of electric and inertial forces, the resultant deformation is less than the summation of the deformations caused by individual effects, when inertial effects are strong. These results are likely to provide deep insights into the interplay of various nonlinearities towards altering electrohydrodynamic settling of drops and bubbles.
Electrorheology of a dilute emulsion of surfactant-covered drops
- Antarip Poddar, Shubhadeep Mandal, Aditya Bandopadhyay, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 881 / 25 December 2019
- Published online by Cambridge University Press:
- 24 October 2019, pp. 524-550
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We investigate the effects of surfactant coating on a deformable viscous drop under the combined action of shear flow and a uniform electric field. Employing a comprehensive three-dimensional approach, we analyse the non-Newtonian shearing response of the bulk emulsion in the dilute suspension regime. Our results reveal that the location of the peak surfactant accumulation on the drop surface may get shifted from the plane of shear to a plane orthogonal to it, depending on the tilt angle of the applied electric field and strength of the electrical stresses relative to their hydrodynamic counterparts. The surfactant non-uniformity creates significant alterations in the flow perturbation around the drop, triggering modulations in the bulk shear viscosity. Overall, the shear-thinning or shear-thickening behaviour of the emulsion appears to be greatly influenced by the interplay of surface charge convection and Marangoni stresses. We show that the balance between electrical and hydrodynamic stresses renders a vanishing surface tension gradient on the drop surface for some specific shear rates, rendering negligible alterations in the bulk viscosity. This critical condition largely depends on the electrical permittivity and conductivity ratios of the two fluids and orientation of the applied electric field. Also, the physical mechanisms of charge convection and surface deformation play their roles in determining this critical shear rate. As a consequence, we obtain new discriminating factors, involving electrical property ratios and the electric field configuration, which govern the same. Consequently, the surfactant-induced enhancement or attenuation of the bulk emulsion viscosity depends on the electrical conductivity and permittivity ratios. The concerned description of the drop-level flow physics and its connection to the bulk rheology of a dilute emulsion may provide a fundamental understanding of a more complex emulsion system encountered in industrial practice.
Electrical switching of a surfactant coated drop in Poiseuille flow
- Antarip Poddar, Shubhadeep Mandal, Aditya Bandopadhyay, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 870 / 10 July 2019
- Published online by Cambridge University Press:
- 07 May 2019, pp. 27-66
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Electrical effects can impart a cross-stream component to drop motion in a pressure-driven flow, due to either an asymmetric charge distribution or shape deformation. However, surfactant-mediated alterations in such migration characteristics remain unexplored. By accounting for three-dimensionality in the drop motion, we analytically demonstrate here a non-trivial switching of drop migration with the aid of a surfactant coating on its surface. We establish this phenomenon as controllable by exploiting an interconnected interplay between the hydrodynamic stress, electrical stress and Marangoni stress, manifested so as to achieve a net interfacial force balance. Our results reveal that under different combinations of electrical conductivity and permittivity ratios, the relative strength of the electric stress with respect to the hydrodynamic stress, the applied electric field direction and the surfactants alter the longitudinal and cross-stream velocity components of the droplets differently. The effect of drop deformation on its speed is found to be altered with the increased sensitivity of the surface tension to the surfactant concentration, depending on the competing effects of the electrohydrodynamic flow modification and the tip stretching phenomenon. Further, with a suitable choice of electrical property ratios, the Marangoni effects can be exploited to direct the drop in reaching a final transverse position towards or away from the channel centreline. These results may turn out to be of immense consequence in providing an insight to the underlying complex physical mechanisms dictating an intricate control on the drop motion in different directions.
Effect of transverse temperature gradient on the migration of a deformable droplet in a Poiseuille flow
- Sayan Das, Shubhadeep Mandal, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 850 / 10 September 2018
- Published online by Cambridge University Press:
- 12 July 2018, pp. 1142-1171
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Intricate manipulation of droplets in fluidic confinements may turn out to be critically important for achieving their controlled transverse distributions. Here, we study the migration characteristics of a suspended deformable droplet in a parallel plate channel under the combined influence of a constant temperature gradient in the transverse direction and an imposed pressure driven flow. An outstanding question concerning the resultant non-trivial dynamical features that we address here pertains to the nonlinearity that results as a consequence of the shape deformation, which does not permit us to analyse the combined transport as a mere linear superposition of the results for the thermocapillary and imposed flow driven droplet migration in an effort to obtain the final solution. For the analytical solution, an asymptotic approach is used, where we neglect any effect of inertia or thermal convection of the fluid in either of the phases. To obtain a numerical solution, we use the conservative level set method. We perform numerical simulations over a wide range of governing parameters and obtain the dependence of the transverse steady position of the droplet on different parameters. In order to address practical microfluidic set-ups, the influence of a bounding wall as well as the effect of thermal convection and finite shape deformation on the cross-stream migration of the droplet is investigated through numerical simulations. Increase in the thermal Marangoni stress shifts the steady-state transverse position of the droplet further away from the channel centreline, for any particular value of the capillary number (which signifies the ratio of the viscous force to the surface tension force). The confinement ratio, which is the ratio of the droplet radius to the channel height, plays an important role in predicting the transverse position of the droplet and thus has immense consequences for the design of droplet-based microfluidic devices with enhanced functionalities. A large confinement ratio drives the droplet towards the channel centre, whereas a smaller confinement ratio causes the droplet to move towards the wall. Moreover, for a fixed droplet radius and constant imposed temperature gradient, an increase in the channel height results in an increase in the time required for the droplet to reach the steady-state position. However, the final steady-state position of the droplet is independent of its initial position but at the same time dependent on the droplet phase thermal conductivity. A larger droplet thermal conductivity compared with the carrier phase results in a steady-state droplet position closer to the channel centreline. A higher fluid inertia, on the other hand, shifts the steady-state position towards the channel wall.
Sedimentation of a surfactant-laden drop under the influence of an electric field
- Antarip Poddar, Shubhadeep Mandal, Aditya Bandopadhyay, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 849 / 25 August 2018
- Published online by Cambridge University Press:
- 18 June 2018, pp. 277-311
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The sedimentation of a surfactant-laden deformable viscous drop acted upon by an electric field is considered theoretically. The convection of surfactants in conjunction with the combined effect of electrohydrodynamic flow and sedimentation leads to a locally varying surface tension, which subsequently alters the drop dynamics via the interplay of Marangoni, Maxwell and hydrodynamic stresses. Assuming small capillary number and small electric Reynolds number, we employ a regular perturbation technique to solve the coupled system of governing equations. It is shown that when a leaky dielectric drop is sedimenting in another leaky dielectric fluid, the Marangoni stress can oppose the electrohydrodynamic motion severely, thereby causing corresponding changes in the internal flow pattern. Such effects further result in retardation of the drop settling velocity, which would have otherwise increased due to the influence of charge convection. For non-spherical drop shapes, the effect of Marangoni stress is overcome by the ‘tip-stretching’ effect on the flow field. As a result, the drop deformation gets intensified with an increase in sensitivity of the surface tension to the local surfactant concentration. Consequently, for an oblate type of deformation the elevated drag force causes a further reduction in velocity. For similar reasons, prolate drops experience less drag and settle faster than the surfactant-free case. In addition to this, with increased sensitivity of the interfacial tension to the surfactant concentration, the asymmetric deformation about the equator gets suppressed. These findings may turn out to be of fundamental significance towards designing electrohydrodynamically actuated droplet-based microfluidic systems that are intrinsically tunable by varying the surfactant concentration.
Drop deformation and emulsion rheology under the combined influence of uniform electric field and linear flow
- Shubhadeep Mandal, Sudipta Sinha, Aditya Bandopadhyay, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 841 / 25 April 2018
- Published online by Cambridge University Press:
- 23 February 2018, pp. 408-433
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Electrohydrodynamics of a leaky dielectric suspended drop subjected to the combined influence of a uniform electric field and linear velocity field is analysed analytically and numerically. In the limit of small charge convection and small shape deformation, an analytical solution is obtained for the deformed drop shape when the imposed linear flow is of uniaxial extensional type with the extensional component aligned in the direction of the electric field. This perturbation approach is then applied towards obtaining the effect of a uniform electric field on the effective extensional rheology of a dilute emulsion. Key results indicate that the magnitude and sense of drop deformation not only depends on the material properties of the drop and medium but is also governed by the strength of the applied electric field relative to the applied flow field. The interfacial charge convection is found to increase or decrease the drop deformation depending on the direction of electrohydrodynamic flow and relative strength of electric field. The electrohydrodynamic flow and drop deformation modulates the effective extensional viscosity of the emulsion. Importantly, the presence of the electric field leads to strain-rate-dependent effective extensional viscosity of the emulsion. The emulsion is found to exhibit strain-rate thinning/thickening behaviour depending on the drop to medium charge relaxation time scale. The analytically obtained drop shape and deformation are in excellent agreement with numerical simulations for the small deformation ranges.
Effect of temperature gradient on the cross-stream migration of a surfactant-laden droplet in Poiseuille flow
- Sayan Das, Shubhadeep Mandal, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 835 / 25 January 2018
- Published online by Cambridge University Press:
- 27 November 2017, pp. 170-216
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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 Péclet number. The present study is focused on the following two limiting situations of surfactant transport: (i) surface-diffusion-dominated surfactant transport considering small surface Péclet number, and (ii) surface-convection-dominated surfactant transport considering high surface Péclet 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.
The effect of uniform electric field on the cross-stream migration of a drop in plane Poiseuille flow
- Shubhadeep Mandal, Aditya Bandopadhyay, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 809 / 25 December 2016
- Published online by Cambridge University Press:
- 15 November 2016, pp. 726-774
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The effect of a uniform electric field on the motion of a drop in an unbounded plane Poiseuille flow is studied analytically. The drop and suspending media are considered to be Newtonian and leaky dielectric. We solve for the two-way coupled electric and flow fields analytically by using a double asymptotic expansion for small charge convection and small shape deformation. We obtain two important mechanisms of cross-stream migration of the drop: (i) shape deformation and (ii) charge convection. The second one is a new source of cross-stream migration of the drop in plane Poiseuille flow which is due to an asymmetric charge distribution on the drop surface. Our study reveals that charge convection can cause a spherical non-deformable drop to migrate in the cross-stream direction. The combined effect of charge convection and shape deformation significantly alters the drop velocity, drop trajectory and steady state transverse position of the drop. We predict that, depending on the orientation of the applied uniform electric field and the relevant drop/medium electrohydrodynamic parameters, the drop may migrate either towards the centreline of the flow or away from it. We obtain that the final steady state transverse position of the drop is independent of its initial transverse position in the flow field. Most interestingly, we show that the drop can settle in an off-centreline steady state transverse position. Two-dimensional numerical simulations are also performed to study the drop motion in the combined presence of plane Poiseuille flow and a tilted electric field. The drop trajectory and steady state transverse position of the drop obtained from numerical simulations are in qualitative agreement with the analytical results.
Effect of surfactant on motion and deformation of compound droplets in arbitrary unbounded Stokes flows
- Shubhadeep Mandal, Uddipta Ghosh, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 803 / 25 September 2016
- Published online by Cambridge University Press:
- 19 August 2016, pp. 200-249
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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.
Uniform electric-field-induced lateral migration of a sedimenting drop
- Aditya Bandopadhyay, Shubhadeep Mandal, N. K. Kishore, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 792 / 10 April 2016
- Published online by Cambridge University Press:
- 03 March 2016, pp. 553-589
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We investigate the motion of a sedimenting drop in the presence of an electric field in an arbitrary direction, otherwise uniform, in the limit of small interface deformation and low-surface-charge convection. We analytically solve the electric potential in and around the leaky dielectric drop, and solve for the Stokesian velocity and pressure fields. We obtain the correction in drop velocity due to shape deformation and surface-charge convection considering small capillary number and small electric Reynolds number which signifies the importance of charge convection at the drop surface. We show that tilt angle, which quantifies the angle of inclination of the applied electric field with respect to the direction of gravity, has a significant effect on the magnitude and direction of the drop velocity. When the electric field is tilted with respect to the direction of gravity, we obtain a non-intuitive lateral motion of the drop in addition to the buoyancy-driven sedimentation. Both the charge convection and shape deformation yield this lateral migration of the drop. Our analysis indicates that depending on the magnitude of the tilt angle, conductivity and permittivity ratios, the direction of the sedimenting drop can be controlled effectively. Our experimental investigation further confirms the presence of lateral migration of the drop in the presence of a tilted electric field, which is in support of the essential findings from the analytical formalism.
Electro-osmosis of superimposed fluids in the presence of modulated charged surfaces in narrow confinements
- Shubhadeep Mandal, Uddipta Ghosh, Aditya Bandopadhyay, Suman Chakraborty
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- Journal:
- Journal of Fluid Mechanics / Volume 776 / 10 August 2015
- Published online by Cambridge University Press:
- 10 July 2015, pp. 390-429
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In the present study, we attempt to analyse the electro-osmotic flow of two superimposed fluids through narrow confinements in the presence of axially modulated surface charges. We attempt to solve for the flow structure as well as the interface deformation by both analytical and numerical techniques. Approximate analytical solutions are obtained through asymptotic analysis for low deformations, whereas numerical solutions are obtained by applying the phase field formalism; the numerical solutions are obtained for small as well as large interfacial deformations. The analytical solutions are derived only for the transient deformation of the interface, neglecting the transience in the flow, i.e. the flow is assumed to be quasisteady. The numerical solutions, however, are derived including the effects of inertia and transients in the flow. We attempt to compare our analytical and numerical results and explore the effects of several physico-chemical parameters on the deformation of the interface as well as the nature of the flow. Our analysis reveals that parameters such as the modulation wavelength, surface tension (described through the capillary number), viscosity ratio, permittivity ratio and extent of asymmetry in the potential on the two walls are the major contributors to the deformation and the resulting flow features.