Magnetic manipulation on the coalescence of a ferrofluid drop at its bulk surface

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Droplet (bubble) breakup at the microscale

The droplet (bubble) breakup into two or more daughter droplets (bubbles) proves a valid way to improve the production efficiency. It requires systematic research on the complex mechanisms encountered in the droplet (bubble) breakup. Common geometries for the droplet (bubble) breakup include the T-junction, Y-junction and flow-focusing junction, etc.
The droplet breakup was firstly investigated by Link et al. [72] in a microfluidic T-junction and an obstruction-mediated geometry at large capillary numbers. Based on the Rayleigh-Plateau instability, a model for the critical capillary number was proposed to characterize the transition between the breakup and non-breakup regimes in the T-junction, as shown in Fig. 1-5(a-j). For the breakup design with obstruction, the sizes of the resulting daughter drops were sensitive to the precise placement of the obstacle, as displayed in Fig. 1-5(k-m). Each breakup technique mentioned above showed advantages and disadvantages, where the precise control on the droplet size distribution could be realized in a T-junction, while the production efficiency could be obtained in the space-saving design with obstructions.
Extended to moderate values of the capillary number, the theoretical prediction for the critical condition determining the droplet breakup in a symmetric T-junction was improved by Leshansky and Pismen [73]. The surface tension competed with the viscous force in the narrow gap between the droplet and the channel wall. It should be noted that this theory could not be directly applied for the droplet breakup in an asymmetric T-junction, where the breakup process producing daughter drops with unequal sizes was intrinsically dynamical and dominated by the flow direction.
Jullien et al. [74] also performed the droplet breakup in a symmetric T-junction where the capillary number varied in three orders of magnitude. Three regimes were observed included the non-breakup, breakup with gaps and breakup with permanent obstruction. A critical droplet length for breakup, independent of the capillary number, was proposed by the physical arguments.
An improved asymmetric T-junction with branch arms of identical lengths and distinct cross sections was employed to form unequal-sized daughter droplets [75]. The power-law correlation in function of the capillary number showing the transition line between the breakup and non-breakup regimes was given. Among the range of the capillary number, the volume of the droplets with unequal sizes was merely affected by the branch width ratios and independent of the capillary numbers. The device could be optimized by using higher branch width ratios and lower capillary numbers to further decrease the droplet size and improve the production efficiency.
Apart from the breakup regimes and size distribution of the daughter droplets in a T-junction, the emergence of satellite droplets during the droplet breakup was paid attention and explored as well [76]. The thread underwent the stretching, fluid drainage and bounce back processes prior to the formation of satellite droplets. It was revealed that both the superficial velocity of the mother droplet and the droplet viscosity exhibited a positive effect on the size of the satellite droplets. A critical capillary number of 0.03 was observed, where the size of the main satellite droplet kept constant below this value and increased with the capillary number above this value.
The deformation and breakup of droplets were also investigated in a symmetric focusing junction. For droplets of low viscosities [77, 78], the phase diagrams were plotted to characterize the critical transition conditions between various regimes including the non-breakup, single breakup and double breakup. The number and size of the daughter droplets were jointly determined by the flow rate and the length of the mother droplet. The critical lengths of the mother droplet and the first daughter droplet showed excellent agreements between the theoretical and experimental results. In the highly viscous systems [79, 80], stable viscous stratifications and a series of daughter droplets could be observed when the liquid of high viscosity served as the second continuous phase. The size difference between the first two daughter droplets was dominated by the capillary number and the thickness of the liquid film between the droplet and channel wall. Under the circumstance where the highly viscous liquid was employed as the dispersed phase, the non-breakup together with the convective and absolute breakup were observed. The scaling law of the maximum droplet elongation as a function of the initial droplet size and the capillary number was established.
Besides the Newtonian and non-Newtonian aqueous liquids referred above, the ferrofluid was also employed to investigate the droplet breakup at both the T-junction and Y-junction, as exhibited in Fig. 1-6. In a symmetric T-junction [81, 82], three breakup types with permanent, partly obstruction and without obstruction were observed. Both regimes with obstruction could be divided into the squeezing, transition and pinch-off stages. In the regimes with obstruction, the power-law exponents of the minimum neck width with time increased with the magnetic flux density in the squeezing and transition stages and kept constant in the pinch-off stage. The sizes of the daughter droplets and breakup dynamics could be actively controlled by the non-uniform magnetic field. For the breakup of ferrofluid droplets in a Y-junction [83], the size distribution could be manipulated by varying the flow rates and the strength of the magnetic field. A fitted correlation incorporating the magnetic Bond number was attained to predict the volume ratio between two daughter ferrofluid droplets.

PhD dissertation for University of Lorraine and Tianjin University

In addition to the above-mentioned droplet breakup under various circumstances, the bubble breakup also attracted extensive attention to figure out the similarities and differences in comparison with the droplet breakup. Common regimes of a bubble moving through a T-junction involves the breakup with permanent and partly obstruction, breakup with gaps, non-breakup with partly obstruction and gaps [84-86]. For the breakup regimes with partly and permanent obstructions, the critical length of the mother bubble was characterized by the head and rear velocities of the bubble to describe the transition between the two regimes. The breakup dynamics of a bubble corresponding to the regimes with partly and permanent obstructions were also investigated systematically, including the evolution of the minimum neck width, superficial velocity and pressure difference in each sub-stages. The breakup time decreased with the superficial velocity of the mother bubble while nearly independent of the liquid viscosity and the length of the mother bubble.
The T-junction divergences within a symmetric and asymmetric loop were also employed to induce the bubble breakup. In the symmetric loop with two branches of the same length [87, 88], the bubbles could break or do not break at the first T-junction. The breakup regimes were the same as those in the simple T-junction without a loop. The transition model between the breakup and non-breakup regimes was described by a power-law relationship, showing that the extension of the bubble was related to the capillary number and the viscosity ratio between the gas and liquid phases. Furthermore, the critical dimensionless width of the bubble neck was determined as 0.5-0.6: the bubble breakup was irreversible and fast below the critical value while reversible and slow above the value. In the asymmetric loop where two branches with different lengths were divided at the first T-junction, the bubble breakup was affected by the hydrodynamic feedback from the daughter bubbles in the downstream [89]. The size ratio between two bubbles in the branches exhibited a nonmonotonic relationship with the two-phase flow rate ratio and liquid viscosity. The resistance effect of the downstream bubble was related to the capillary number of the longer bubble. While the representative resistance induced by the shorter bubble required further experimental and theoretical studies.
As supplementary to the experimental results, the numerical simulations were also conducted to carry out thorough studies on the droplet (bubble) breakup. The asymmetric breakup of droplet in a novel T-junction with a valve in one of the branches was investigated by the VOF based numerical algorithm [90]: In the side branch with a valve, daughter droplets with smaller sizes could be acquired by decreasing the capillary number. It was found that the breakup time of droplet was independent of the valve ratio. The pressure drop decreased linearly with the valve ratio in the absence of the tunnel, while irrelevant to the valve ratio with the formation of a tunnel.
With the multiphase Lattice Boltzmann method, the droplet breakup in microfluidic T-junctions was simulated, where the channel walls in one branch were smooth and inhomogeneous in another [91]: Under the effect of contact angle hysteresis, the daughter droplets with smaller sizes always appeared in the nonideal branch. Moreover, the daughter droplet in the nonideal branch moved towards the outlet in a relatively low velocity or maintained blocked near the junction.
The color-gradient model of Lattice Boltzmann method was performed to investigate the asymmetric breakup of droplet in a T-junction [92]: Under a pressure difference at two branches which was expressed by the Darcy-Weisbach equation, the breakup into unequally sized daughter droplets followed two steps of the filling stage and the breakup stage. The volume ratio of droplet in each branch could be adjusted by the asymmetric condition of the T-junction.
The breakup of ferrofluid droplets in a T-junction was also numerically investigated [93]. In the presence of an asymmetric magnetic field, five regimes including the non-breakup and a new splitting in hybrid mode (SHM) were observed. The mother droplet with larger size suppressed the sensibility of the droplet volume ratio to the strength of the magnetic field. A correlation model in function of the capillary number and the magnetic Bond number was built to determine the transition line between the breakup and non-breakup regimes.
The hydrodynamic behaviors of the bubble breakup and deformation were numerically investigated as well. Based on the boundary element model, the interface evolution and flow field of a semi-infinite bubble moving through a Y-junction bifurcation were calculated by Calderon et al. [94]. The splitting ratio was found to increase with the driving bubble pressure and decrease with the angle between the branch arms. The bubble velocity was significantly affected by the viscous loss at the bifurcation. The recirculation zone near the contact line was observed at a higher pressure and wall shear stress, which laid a foundation for the design of microfluidic devices where fluid mixing was desired.
Chen et al. [95] made use of the interface tracking and finite element methods to investigate the bubble deformation in a symmetric bifurcation. The incompressible Navier-Stokes equation was employed to characterize the effects of the gravity, inertia, and surface tension on the flow pattern and the shear stress gradient. The gravity showed noteworthy influence and led to the changes in the bubble profile and higher pressure and shear stress gradients in the upper branch. The increased pressure gradient near the tip of the bifurcation corner could be observed only in the case of asymmetric bubble breakup.$

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Drop coalescence and spreading

Expect for the multiphase flows in confined microchannels at microscales, the free-surface drop impact is another vital research field. Classified by the surface condition below the falling drop, the drop impact comprises two distinct behaviors on a liquid surface and a solid surface.

Drop impact and coalescence on a liquid surface

The splashing phenomenon is defined as the ejection of numerous small-sized droplets as a result of the severe deformation of a liquid interface following the drop impact [98]. It’s widely encountered in the problems of the industrial and environmental applications [99]. According to the shape, dynamical behavior of the liquid corolla, the splashing could be classified into two categories of the “crown splashing” and the “prompt splashing” [100]. The disintegration of the liquid corolla leads to the ejection of large quantities of droplets. Both the Froude number and the Weber number were employed to determine the splashing behavior, where the deep liquid splashing was related to both dimensionless numbers while the shallow liquid splashing was only relevant to the Weber number [101-103].
Coalescence is a widely encountered physical phenomenon involving various length scales, such as the formation of solar storms in the universe, raindrop formation, ink-jet printing, oil recovery, emulsification and cell-cell coalescence in micro-biological systems [104-109]. When the gap between two leading edges becomes infinitesimal, an initial liquid bridge forms under van der Waals force [110]. Once the drops come into contact with each other, the neck will rapidly expand due to its lower pressure than the center of the drops. As indicated in Fig. 1-8, the drop coalescence involves the following four types: (1) Drop coalescence at a liquid-liquid interface. (2) Drop coalescence at an air-liquid interface. (3) Coalescence of two spreading drops on a solid substrate. (4) Coalescence of two pendant drops. Each type of drop coalescence will be introduced in detail in the following paragraphs.
Drop coalescence could be observed at a planar liquid-liquid interface in a 2D Hele-Shaw cell, as shown in Fig. 1-8(a). The coalescence of a single glycerol drop at the interface between the glycerol aqueous solution and the silicone oil was jointly investigated by the high-speed camera and the PIV technique [111]. The drop of the water/glycerol mixture was generated through a glass tube and controlled by an electronic valve. As soon as the drop was released from the end of the glass tube, it descended through the superior silicone oil. Prior to the interfacial rupture, the drop was basically motionless on the thin liquid film above the underlying liquid-liquid interface. After the rupture, the receding free edge of the liquid film was pulled by the capillary force, forming a large opening where the drop sank beneath the bulk liquid. The increasing viscosity of the ambient silicone oil could result in a smaller velocity of the receding free edge and lower vorticity value in the shrinking capillary ring. Then the study on the drop coalescence continued at the glycerol aqueous solution/silicone oil interface [112-114]: The dimensionless Ohnesorge number could distinguish the coalescing regimes. The partial coalescence and a secondary drop were observed in the inertia regime where Oh < 1. For the viscous regime where Oh > 1, the fully coalescence was observed without any secondary drop. Similarly, the drop coalescence at the liquid-liquid interface between the glycerol aqueous solution and Exxsol D80 oil was also investigated [115]. The 2D PIV system with the maximum frequency of 1000 fps was employed to track the velocity fields inside the coalescing drop. The high concentration of the surfactant Span 80 in the oil phase facilitated the interface deformation before the film ruptured and resulted in a longer time for the film rupture. Immediately after the film rupture, two counter-rotating vortices emerged inside the drop on either side of the rupture point. The intensities of the two vortices significantly decreased with the surfactant concentration. The kinetic energy per unit mass corresponding to various surfactant concentrations was also calculated, indicating that the kinetic energy per unit mass was distributed at the bottom of the drop at the early coalescing stages while distributed near the upper part of the drop at later stages.

Table of contents :

Chapter 1 Literature review
1.1 Multiphase flows at the microscale
1.1.1 Droplet (bubble) formation at the microscale
1.1.2 Droplet (bubble) breakup at the microscale
1.2 Drop coalescence and spreading
1.2.1 Drop impact and coalescence on a liquid surface
1.2.2 Drop impact and spreading on a solid surface
1.3 Passive and active manipulation on multiphase flows
1.3.1 Passive methods
1.3.2 Active methods
1.4 Outline
Chapter 2 Experimental section
2.1 Experimental devices
2.1.1 High-speed camera
2.1.2 Ultra-high-speed DC electrical device
2.1.3 High-speed micro-PIV
2.1.4 Syringe pump
2.1.5 Microfluidic devices
2.2 Experimental liquids
2.3 Analytical methods
2.3.1 High-speed images
2.3.2 Electrical signals
2.3.3 Velocity fields
2.3.4 Liquid properties
Chapter 3 Droplet formation in T-junction and flow-focusing devices
3.1 Introduction
3.2 Experimental section
3.2.1 Liquid properties for the droplet formation in the T-junction device
3.2.2 Liquid properties for the droplet formation in the flow-focusing device
3.3 Results and discussion
3.3.1 Formation of elastic droplets in a microfluidic T-junction
3.3.2 Droplet formation in elastic fluid at a flow-focusing microchannel
3.4 Summary
Chapter 4 Stretching and breakup of elastic droplets in consecutive flow-focusing device
4.1 Introduction
4.2 Experimental section
4.3 Results and discussion
4.3.1 Breakup regimes of the droplets in the flow-focusing junction
4.3.2 Dynamics of the droplet stretch-rebound
4.3.3 Dynamics of the droplet stretch-breakup
4.4 Summary
Chapter 5 Initial coalescence of an aqueous drop at a planar liquid surface
5.1 Introduction
5.2 Experimental section
5.3 Results and discussion
5.4 Summary
Chapter 6 Drop impact, spreading and breakup on a solid surface
6.1 Introduction
6.2 Experimental section
6.3 Results and discussion
6.3.1 Initial contact and spreading of a pendant drop at solid surface
6.3.2 Filament thinning of the liquid neck
6.4 Summary
Chapter 7 Magnetic manipulation on the coalescence of a ferrofluid drop at its bulk surface
7.1 Introduction
7.2 Experimental section
7.3 Results and discussion
7.3.1 Initial coalescence of a ferrofluid drop without a magnetic field
7.3.2 Initial coalescence of a ferrofluid drop under a magnetic field
7.4 Summary
Chapter 8 Conclusions, innovations and perspectives
8.1 General conclusions
8.2 Innovations
8.3 Perspectives


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