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

Abstract

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 surfactantladen 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 governed surfactant transport considering high surface Péclet number. Thermocapillary-induced Marangoni stress, strength of which relative to viscous stress is represented by thermal Marangoni number, has strong influence on the distribution of surfactant on the droplet surface. Temperature field not only affects the axial velocity of the droplet but also has significant 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 Poiseuille flow, the droplet migrates towards the flow centerline. The magnitude of both axial and crossstream velocity components increases with the thermal Marangoni number. However, when the imposed temperature decreases in the direction of 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 centerline or away from it. Present study shows a critical value of the thermal Marangoni number beyond which the droplet moves away from the flow centerline which is in sharp contrast to the motion of a surfactant-laden droplet in isothermal flow for which droplet always moves towards the flow centerline.

Effect of surfactant on the settling of a drop towards a wall

“…These studies indicate that surface impurities might have multifold effects on the motion and deformation of drops and bubbles (Stone & Leal 1990;Tsemakh et al 2004;Vlahovska et al 2005). On one hand, presence of surfactants may augment interfacial deformation (Stone & Leal 1990;Vlahovska et al 2005;Mandal et al 2017b), while at the same time, it may also lead to cross-stream migration of drops (Hanna & Vlahovska 2010;Schwalbe et al 2011;Pak et al 2014;Das et al 2018b) and resist motion at the interface by altering the surface tension (Stone & Leal 1990;Hanna & Vlahovska 2010). The aforementioned studies outline a wide variety of methods to probe into the flow dynamics in the presence of surface impurities; these include analytical investigations (Haber & Hetsroni 1972;Sadhal & Johnson 1986;Stone & Leal 1990;Pawar & Stebe 1996;Vlahovska, Bławzdziewicz & Loewenberg 2009;Pak et al 2014;Mandal et al 2016), assuming small deformation (small capillary number (Ca) limit) as well as numerical studies for finite interfacial deformation (Stone & Leal 1990;Milliken, Stone & Leal 1993;Pozrikidis 2001;James & Lowengrub 2004;Vlahovska et al 2005Vlahovska et al , 2009, which unveil the rich physics associated with multiphase flows.…”

“…It may, therefore, be inferred that presence of impurities in the form of surfactants, can potentially alter the dynamics of drops and bubbles and hence should be accounted for, when modelling the motion of such entities. As such, a large number of investigations on the effect of surface impurities on droplet motion have been carried out (Levich & Krylov 1969;Sadhal & Johnson 1983;Oguz & Sadhal 1988;Stone & Leal 1990;Cuenot, Magnaudet & Spennato 1997;Wang, Papageorgiou & Maldarelli 1999;Hanna & Vlahovska 2010), which includes, for instance, motion of compound drops with surfactants (Tsemakh, Lavrenteva & Nir 2004;Mandal et al 2016), settling of surfactant-laden drops and bubbles (Chen & Stebe 1996;Poddar et al 2018), motion of surfactant-laden drops in Poiseuille flow (Hanna & Vlahovska 2010;Pak, Feng & Stone 2014;Mandal, Bandopadhyay & Chakraborty 2015;Das et al 2017;Das, Mandal & Chakraborty 2018b), to name just a few. A number of studies have also been carried out (Vlahovska, Bławzdziewicz & Loewenberg 2002;Vlahovska, Loewenberg & Blawzdziewicz 2005;Pal 2011;Danov et al 2012;Mandal et al 2017b;Das, Bhattacharjee & Chakraborty 2018a;Sengupta, Walker & Khair 2018), to probe into the influence of surfactants on the bulk rheology of emulsions.…”

“…2018), motion of surfactant-laden drops in Poiseuille flow (Hanna & Vlahovska 2010; Pak, Feng & Stone 2014; Mandal, Bandopadhyay & Chakraborty 2015; Das et al. 2017; Das, Mandal & Chakraborty 2018 b ), to name just a few. A number of studies have also been carried out (Vlahovska, Bławzdziewicz & Loewenberg 2002; Vlahovska, Loewenberg & Blawzdziewicz 2005; Pal 2011; Danov et al.…”

“…2014; Das et al. 2018 b ) and resist motion at the interface by altering the surface tension (Stone & Leal 1990; Hanna & Vlahovska 2010). The aforementioned studies outline a wide variety of methods to probe into the flow dynamics in the presence of surface impurities; these include analytical investigations (Haber & Hetsroni 1972; Sadhal & Johnson 1986; Stone & Leal 1990; Pawar & Stebe 1996; Vlahovska, Bławzdziewicz & Loewenberg 2009; Pak et al.…”

Shape stability and flow behaviour of a phase change material based slurry in coupled fluid-thermo-electrical fields for electronic device cooling

“…Additionally, the electrokinetics and thermal transport of electrolyte has been studied by Das et al [21][22][23] in a wide range, showing that the overall electric double layer (EDL) electrostatic potential distribution results in a significant reduction of the effective EDL thickness, thus be of overwhelming significance in nanofluidic transport. However, the crossstream migration driven by electric field [24] or temperature gradient [25] may accelerates or slows down the PCM energy charging rate but still unclarified in the field of energy storage.…”