Yan Pang·Xiang Wang·Zhaomiao Liu

Abstract Droplet generation in a T-shape microchannel,with a main channel width of 50 μm,side channel width of 25 μm,and height of 50 μm,is simulated to study the effects of the forced fluctuation of the bottom wall.The periodic fluctuations of the bottom wall are applied on the near junction part of the main channel in the T-shape microchannel.Effects of bottom wall’s shape,fluctuation periods,and amplitudes on the droplet generation are covered in the research of this protocol.In the simulation,the average size is affected a little by the fluctuations,but significantly by the fixed shape of the deformed bottom wall,while the droplet size range is expanded by the fluctuations under most of the conditions.Droplet sizes are distributed in a periodic pattern with small amplitude along the relative time when the fluctuation is forced on the bottom wall near the T-junction,while the droplet emerging frequency is not varied by the fluctuation.The droplet velocity is varied by the bottom wall motion,especially under the shorter period and the larger amplitude.When the fluctuation period is similar to the droplet emerging period,the droplet size is as stable as the non- fluctuation case after a development stage at the beginning of flow,while the droplet velocity is varied by the moving wall with the scope up to 80%of the average velocity under the conditions of this investigation.

Keywords Droplet length·Droplet generation·Forced fluctuation·Droplet velocity

1 Introduction

Droplet microfluidics, in which nanoliter to picoliter droplets act as individual compartments,has enabled a wide range of applications.These types of systems can save the reagent and improve the mixing or testing effect[1],prevent important fluids from pollution or chemical reaction with other materials[2],fabricate nanoparticles for photochemical synthesis due to the high surface-to-volume-ratios[3]and so on.As a result,there is a growing interest in the microfluidic techniques,especially for the formation and control of droplets or bubbles.

Droplet size and velocity are important input parameters in the microfluidic systems[4,5].Generally,the droplet size is primarily determined by the dimensions of the fluid flowing part of the chip[6].In addition, flow parameters,such as the flow rate[7,8],viscosity[5],interfacial tension[9],surface wettability[10],surface roughness[9],affect the droplet generation process.The work of Garstecki et al.[7]firstly documented the relation of the droplet length(L)and width of the main channel(w)in a T-shape microchannel as L/w=1+ αQd/Qc,where α is a constant depending on the geometry of the channel,Qdis the volumetric flow rate of the dispersed phase,and Qcis the volumetric flow rate of the continuous phase.After that,the droplet size is scaled in some other channels,such as the flow-focusing channel[11],and many modified and specific scaling laws[12–14]are reported to describe the droplet size basing on the emerging process. Many important properties of droplets,including stability,chemical reactivity,and physical characteristics are highly dependent on the parameters of the physical size and the size distribution of droplets[15,16].

Since controlling the size of droplets to raise the specific performance is in great need,there is growing interests in such investigations.In order to generate dynamically droplets with tunable size,the tunable membrane is adopted and controlled by the pressure of the chamber beside them[15,17,18].The membrane changes the width of the droplet emerging channel and finally alter the droplet size.The study by Li et al.[19]involved the perturbation into the driving of the dispersed phase in the flow-focusing channel.They found that the higher perturbation frequency reduced the droplet size,and the response to the change in flow conditions toward the hydrodynamic perturbation was significantly faster than that triggered by changes in the flow rate using typical syringe pumps.Then Huang et al.[20]also adopted this type of perturbation in the high throughput size-tunable generation system of cell containing collagen microbeads.For tunable and unequal droplets,Link et al.[21]proposed two methods,using an obstacle inside a straight channel whose drawback was that the breakup droplets flow together and an additional carefully designed separating system was needed,and a T-junction with unequal length narrow branches whose pressure drop could be increased.Similarly,Bedram and Moosavi[22]proposed a T-junction with unequal width branches.In addition,a thermally mediated breakup method in which aqueous droplets were manipulated by thermally induced surface tension gradients[23]and a valve-based method that relies on the breakup of droplets in a T-junction with a valve in one of the minor branches[24]were adopted.

All droplet microfluidic techniques rely on transport of droplets in channels[5].Typically,the hydrodynamic resistance of the flow,which is the final expression for the flow field of the continuous phase,depends on the speed of the confinement droplet in a microchannel[25].The droplet speed can also incorporate variation to the inside velocity field.For example,Harshe et al.[26]investigated the dependence on droplet mixing,which is highly controlled by the droplet velocity field,the droplet traveling velocity in a microfluidic serpentine channel,and then assessed principles of the droplet velocity influencing the mixing time.In addition,the droplet velocity was shown to be significant to the droplet fusion and breakup[21,27–29].

With the investigations of droplet size and velocity,the basic affecting factors,as well as the manipulating methods of the dropletsize and speed are achieved in the microfluidics.Based on those,the size tunable droplets emerging system,which can provide a wide droplet sizes range to match the application could be further studied.In this work,the effects of the forced deformation of the bottom wall during the droplet generation in a microchannel with amain width of 50 μm have been depicted.A new protocol to tune the droplet size and velocity is proposed,where the droplet size distribution law is scaled.Several typical shapes of the channel during the bottom wall fluctuation process are investigated to figure out the effects of the shape without the fluctuation deformation.To indicate the effect of the movable bottom wall,different fluctuation models are involved in the numerical simulation.

2 Calculation model and parameters

In this study,three dimentional(3D)geometry is used to investigate the effects of bottom wall deformation on droplet generation in the T-junction microchannel.The immiscible liquids are pumped into the microchannel through the water inlet and the oil inlet,respectively,and then merge at the right angle to generate droplets(Fig.1a),where water is set as the dispersed phase and perfluoropolyether oil as the continuous phase.The channel length is 960 μm in the x-direction,the lengths of both inlets are Lo=Lw=200 μm,and the channel height is 50μm.Widths of the side channel and main channel are ws=25 μm and w=50 μm respectively.

In order to figure out the influence of the bottom wall deformation, fluctuation is applied at a specific area on the bottom wall of main channel’s downstream.The fluctuation’s period and amplitude are varied to investigate their effects on droplet generation.The basic shape of the fluctuation can be described by Eq.(1),which is the deformation equation of the thin plate with four boundaries fixed under a uniform load.

where wzis the z-direction deformation,x,y are the x-direction and y-direction positions of the bottom wall respectively as shown in Fig.1b,D is the bending stiffness of the deformable bottom wall,l is the length of the fluctuant area on the wall(l=100 μm),q0is the uniform load which varies with time t and is controlled by the period T and load factor qfwith the following equation.

The deformation wzvaries with the location(x and y)and reaches its maximum shown as A at the center of the deformable wall in Fig.1b.Here,qf=81700 is adopted in the basic case to achieve the maximum deformation of the wall A=8 μm.It can be described in short that the deformable part of the bottom wall moves in the shape of Eq.(1)with time correlation of Eq.(2).

3 Validation of numerical simulation

Before the deformable wall channel is investigated,droplet generations in the normal solid channel with the same general geometry are analyzed and compared with the experimental and numerical methods,to validated the numerical method in this study.

Fig.1 Models of the T-junction microchannel.a Schematic of liquids flowing to generate droplets in the T-junction microchannel;b details of the bottom wall deformation are shown in dashed black ellipse of a.A is the maximum height of the dome,which is also the amplitude of the fluctuation in the deformable wall cases.When A is negative,zero and positive,the wall of the channel is accordingly expanded,straight and reduced;c schematic of oil–water contact angle θow;d details of meshes in case 4 where the boundary layer near walls of the main channel is incorporated;e details of meshes in case 6 where the local refinement of 1.25 μm in the x-direction of the junction in the main channel is incorporated with a smooth transition between two grid sizes

Table1 Properties of the immiscible liquids

3.1 Experimental and numerical methods

In this model,liquid properties are shown in Table 1.The experimental results with the polydimethylsiloxane(PDMS)T-shape channel are adopted to validate the simulation results.The general geometry,liquids and flow conditions are the same as the simulation.Liquids are pumped by syringe pumps(KD Scientific,KDS-100)at a flow rate of 0.04 m L/h for water and 0.08 m L/h for per fluoropolyether oil.The shadow processes of all the flows are recorded by a highs peed camera(Phantom v7.3,Vision Research Inc.).

The liquid flow process is simulated with the commercial software FLUENT(Ansys Inc.USA)using the volume of fluid(VOF)method to track the interface.The hydrophobic boundary condition is applied to all solid walls with an oil–water contact angle of 140°.The non-slip boundary is set for all walls,which means the velocity normal to the cross-section is zero for the continuous phase(the oil flow)at the wall.The fluctuation of the bottom wall,which is described by Eqs.(1)and(2),is applied by the dynamic mesh module with the user defined file at the designed location.The liquid flow in the microchannel is laminar with the velocity inlet condition of 0.00889 m/s(0.04 m L/h)for the water inlet,0.00889 m/s(0.08 m L/h)for the oil inlet,the pressure outlet condition of 0 Pa and the reference pressure of 101325Pa for the outlet.In the numerical simulation,the flow is modeled using the Navier–Stokes equation and the continuity equation.Simulation methods,including the pressure-implicit with splitting of operators(PISO)method,the geo-reconstruct scheme(piecewise-linear interface construction,PLIC),and the control-volume-based technique are adopted to discretize the governing equations and algebraic equations to perform the surface tension calculations,which are mentioned in the literature[30,31].

3.2 Mesh independence validation

Mesh accuracy and independence should be validated first to analyze the simulation reasonably.Results of meshes with different meshing types and grid numbers are compared in Table 2.Mesh sizes in the x-direction are documented in case 1 and case 2.Even though the size is reduced from 5 to 1.25 μm with a fourfold improvement in the accuracy and increase in the cell number,corresponding variations,which are about 7.1%in droplet length,1.6%in frequency and a slight difference of the droplet shape near the the simulations.The accuracy of y-direction and z-direction in case 3 is improved from 5 to 2.5 μm,and the cell amount is also increased to 90000 equally to case 2,but the droplet length has a 12.0%extension and the frequency has an 8.7%reduction as compared to case 1,which varies more than case 2.Therefore,the simulation is more sensitive to the grid size in the cross-section of the channel(y-direction and z-direction)since the larger variation is involved under this mesh accuracy improvement.

When the grid number remains,the mesh type is verified by adopting the boundary layer near the wall(Fig.1d)as shown in case 3 and case 4.About 6.1%variation of length in case4 is induced as compared with case3,which is derived from the mesh type(the boundary layer)because grid numbers are the same in every corresponding orientation.As the grid number of the cross-section produces a significant variation,case 5 is involved and finally yields a similar result with case 4,even though the mesh accuracy in y-direction and z-direction is improved by 50%and the cell amount is doubled.It is indicated that the meshes of case 4 and case 5 are proper in this type of simulation and the mesh independence is valid.To confirm result,case 6 is also simulated because the junction part has relation with the two liquids flow and the interface evolution.Taking computing performance into consideration,case 5 is chosen in this study.In addition,the experimental result(as shown in case 7)is also used to give some proof that case 5 is closer to real flow and simulations are in good agreement.The continuous phase film near the wall,which can be solid-like[32]and result in a longer droplet,is obviously formed in the experiment,but hard to obtain in the simulation as shown in Table 2.

4 Drop let generation under different fixed shapes of the bottom wall

For droplet generation in the T-junction microchannel,the droplet length is controlled by the liquid flow rates and the geometry parameters,especially the channel’s width[6],in the way that the droplet length grows with increasing width of the channel.Based on the fluctuation curve of Eq.(1),nine types of wall shapes,which are varied by setting the A(Fig.1b)to different values separately called Afwithout time-dependent deformation,are involved to analyze the effects of the shapes on the emerging droplet.In addition,a rationality of the 3D simulation is provided to get trends of the droplet generation with different corresponding width before investigation of the bottom wall fluctuation.

Accordingly,the shape of the channel’s bottom wall near the T-junction,which is shown in Fig.1b and supposed to be deformable in the following sections,is fixed to several shapes including that of wall expanded(Af＜0),straight(Af=0),and reduced(Af＞0).As a result,the height of the channel near the junction is changed and the average droplet length is tuned within a range about 74–82 μm orders in this analysis as shown in Fig.2.This relation shows the consistency with the result of the tunable junction width channel in Ref.[17]and also the scaling law of Garstecki’s that the droplet length is in direct proportion to the channel width[6].For this study,the droplet length L is fitted with Afin Eq.(3).

where Afis the maximum height of the dome in the fixed shape channel corresponds to A shown in the Fig.1b in the deformable wall cases and can take values in the range of?16 to 16 μm in this simulation and correlation.Although the maximum deformation near the junction wall varies from?16 to 16 μm,which is from ?32%to 32%of the main channel height,the droplet length is only changed within 7%.

Since the flow of the dispersed and continuous phases are uniform separately, the droplet emerging frequency grows to cover the reduction of the droplet size.Besides,the shape affects the droplet size chiefly,but not the monodispersity of the droplet.

5 Drop let length under different bottom wall fluctuations

To determine the influence of fluctuation,the period and amplitude,which are the key parameters of the wall shape during the droplet generation in the channel,and the droplet generation results,including the droplet length and droplet emerging frequency,are documented under these different boundary conditions.

Fig.2 The droplet length and droplet emerging frequency of various bottom wall shapes.a Droplet length;b droplet average emerging frequency

5.1 Droplet length distribution under different time periods

To document the influence of time periods on the droplet generation, fluctuations with periods in the range of 0.0096–0.0800 s are simulated.Droplet length distributions during the droplet emerging sequence are shown in Fig.3.Lengths of cases with the larger fluctuation period occupy a wider range of the droplet length,which means a worse monodispersity.The droplet sizes of the fixed shape of Af= ?8,8μm,which are simulated in last section,become the two stable limits for lengths in all fluctuating cases.A ll the droplets of different fluctuation periods in this study are seated between the size of the Af=8 μm case and that of the Af= ?8 μm case.The reason could be that for these two cases,these are the limited shapes of the bottom wall in the fluctuation,that is, the movable bottom wall is deformed between them to construct the fluctuation. Although the droplet size can fill the range between the two limited lengths, the droplet emerging frequency does not perform discriminately, but stays at 103.8 Hz,which means that droplets of different sizes emerge in a similar time interval.

The droplet sizes show some patterns along the droplet sequence as shown in Fig.3,which seems like a periodic process.Motivated by these observations,this pattern is documented to infer the scaling law of the droplet size with the deformable bottom wall.

As mentioned in Eq.(3),the droplet length has a corelationship with the height of the channel.In the varied fluctuation period cases,the amplitude A,which corresponds to Afin the fixed shape channel in the former section,varies by the following equation in the basic case:

When we suppose the Eq.(3)can also be used to predict the droplet length in the fluctuating cases,the length can be derived from Eqs.(3)–(5).

where txis the shift to achieve that the time t of the first recorded droplet emerged is set to be 0.The droplet generation time t could not be continuous but an arithmetic progression with the droplet emerging period as the common difference.So droplet lengths can be a sequence of discontinuous values seating in the predicted equation and varying with the generation time.

Droplet length sequences under various fluctuation are fitted with the form of Eq.(6),which results from the assumption of Eq.(5).

where Laveis the average length of the droplet sequence,ALis the length distribution amplitude,and TLis the length distribution period.As shown in Table 3,the fitted period completely matches the wall moving period in most cases,the average length of droplets has a narrow error with the supposed Eq.(6),which means the droplet length in the fluctuation channel is mainly controlled by the apparent heightof the channel at the emerging time.From the basic relation of the droplet length[7],it is mainly affected by the geometry and flow rates,so the fluctuation should have little influence on flow rates of liquids during the droplet emerging process,which is also proved in Fig.4d that the flow rate of the upstream(x=0 mm)is not changed by the fluctuation.

Table3 Fitted factors of the droplet length distribution

Predicted droplet lengths of a recorded droplet sequence are plotted in Fig.3(the common difference is about 0.0096 for this investigation),with which the numerical lengths are compared.When the fluctuation period is relatively long(which means the moving speed of the deformable wall is slow),such as the cases with the moving period T above 0.0192 s,the numerical droplet length meets Eq.(6)with low standard errors.For the short period fluctuation,the deformable wall affects the droplet emerging process in a way that the droplet length does not follow the channel width relation Eq.(6),as well as the long period cases.In addition,the distribution range of the numerical droplet length will be reduced as shown in Fig.3e and f where the period is under about 0.020 s.

More details of the droplet generation results are shown in Fig.4,including the average droplet length,the length difference and droplet generation time(frequency).As shown in Fig.4a,the average length is distributed around 76.1 μm within an error less than±0.3%.Equally,the droplet generation time brings out a similar trend as shown in Fig.4b since all cases are driven with the same liquid flow rate.Furthermore,for all the droplet emerging processes in this part,the droplet emerging frequencies are relatively stable at about 104 Hz, although the droplet size occupies a wide range about 73–82 μm, which indicates that the fluctuation of the bottom wall has no effect on the droplet generation time.

Fig.3 Curves of droplet length via relative flow time under different fluctuation periods.The solid circle stands for the numerical result,the circle stands for the predicted result from the fitted equation and the line is the plot of Eq.(6)under different periods.a A max = 8 μm,T = 0.060 s.b A max=8 μm,T=0.04 s.c A max=8 μm,T=0.0288 s.d A max=8 μm,T=0.020 s.e A max=8 μm,T=0.0192 s.f A max=8 μm,T=0.016 s

Fig.4 Droplet generation performance under the fluctuation wall.a Average droplet length,the solid line is the average length of all cases.b Average droplet generation frequency,the solid line is the average frequency of all cases.c Droplet length difference,which is the amplitude of the fitted relation of the droplet length under different periods and also is half of the difference between the minimum and maximum values.The dash line is the average length difference in the larger period cases,which is supposed to be the stable magnitude of length difference.The solid line is the limitation from the fixed shape cases.d The x=0 mm and x=0.1 mm lines stand for the flow rate of cross-section of the main channel at the position of x=0 mm and x=0.1 mm,which can be called Q0 and Q0.1,respectively.The net flow rate is Q0?Q0.1.The fluctuation of A is the variation of A with time during the droplet process

Fig.5 Droplet sizes under various amplitude of the bottom wall fluctuation

The documented length difference is the amplitude of the fitted plots in Fig. 3,which is half of the real length distributed range in that case. The length difference shows variation with the changing fluctuation period(Fig.4c),which is also indicated in Fig.3.That is,the length difference grows with the fluctuation period when T is below about 0.028 s,and then is stable at a value less than the half difference of droplet lengths between two limited shapes in the fixed wall cases.

5.2 Droplet length under various amplitude of bottom wall fluctuation

Three models with a droplet generation time period of 0.0096 s are simulated under the amplitudes of A=8,12,16 μm,respectively.The droplet length distribution and average droplet sizes of these models show the differences in the way that the larger fluctuation amplitude produces the larger droplet(Fig.5).For the fluctuating cases,droplets generate unstably at the beginning of the flow and become a stable generation gradually(Fig.5)with a long time average length,as shown by the dash line in Fig.5,and the emerging time is still kept at about 0.0096 s.In other words,the droplets are stable at some length after a developing process of the droplet generation when the bottom wall movement is applied on the channel. Otherwise, the droplet will keep the size since the beginning of the droplet flow as shown by the non-fluctuation case. The difference of the droplet size under various fluctuation amplitudes is slight within a relative difference of 4% of the droplet length, where the A = 8 μm case is used as the basic one mentioned in Sect. 2.

For the droplet length, the bottom wall fluctuation has limited effects and the length variation(the difference between the maximum and minimum value)is less than 7%of the average length,which is almost the variation range of the two limits cases.That is,the fixed shape cases of the same wall moving amplitude where the local height difference is 32%of the channel’s width.Fluctuations involved in this study have a gentle influence on the distribution of droplet sizes,even though the lengths fall on a predicted equation.In addition,when the fluctuation has the same frequency with the droplet generation,the monodispersity of the length is not affected,but the average length varies slightly.

6 Drop let velocity under different bottom wall fluctuation

When the bottom wall is deforming,the volume of the channel is varied,and the obvious effect is to pump the liquid flow inside the channel and finally change the liquid velocity.The droplet velocity is a typical parameter in the flow,which is indicated by the velocity of the droplet’s barycenter.In the constricted droplet flow,the droplet is considered to be symmetric,so the barycenter of the droplet should lie in the midsection of the channel.The droplet curve in the midsection of the z-direction is used to access and monitor the barycenter’s movement during the droplet flow,and finally the droplet velocity is documented.

6.1 Effects of time periods on the droplet velocity

During the droplet flow,the monitored barycenter location can be transferred to the barycenter velocity,which is also the velocity of the droplet.Under different periods of the forced bottom wall fluctuation,the droplet velocity evaluates the curve with the same trend as the movement of the wall,which can also obtain some evidence from Fig.4d that the flow rate of the downstream cross-section(x=0.1 mm)follows the movement of the wall fluctuation.As shown in Fig.6,the velocity changes in the form of sine whose period is derived from fitted velocity curves is completely consistent with the moving bottom wall.The velocity plot is relatively flat about 17000–20000 μm/s when the fluctuation period is T=0.040s as shown in Fig.6a.As the period is reduced,the droplet velocity grows gradually within a wider varying range and when T is reduced to 0.0096 s,the velocity has a lower limit about 14000 μm/s and an upper limit about 23000 μm/s.

Fig.6 Evolutions of the average droplet velocity in the main channel from x=150 μm to x=800 μm.a A max=8 μm,T=0.04 s.b A max=8 μm,T=0.0288 s.c A max=8 μm,T=0.020 s.d A max=8 μm,T=0.0192 s.e A max=8 μm,T=0.016 s.f A max=8 μm,T=0.0096 s

Fig.7 Droplet velocities under different fluctuation periods.a Average droplet length;b droplet velocity difference(half of the velocity distribution range)and relative variation

The average droplet velocity and the velocity difference for every fluctuation are also plotted as shown in Fig. 7. The average velocity is relatively uniform and all fall with in a narrow extent of 1% around 18000 μm/s, while amplitudes of the droplet velocity difference increase significantly with the reducing period of the forced movement of the bottom wall.Since the liquid driving flow rate is uniform and the average droplet size is almost stable,as mentioned in Fig.4,the average droplet velocity is similar under different fluctuations to match the uniform dispersed phase flow rate.However,the movement of the bottom wall practices the variation in the instantaneous velocity,that is,the velocity evolves with time.For Amax=8 μm cases,when the period varies from about 0.038–0.0096 s,the velocity amplitude which is half of the overall velocity difference increases from about 6%–22%.That means the velocity variation range could be doubled to 44%of the average droplet velocity.

6.2 Effects of amplitude on the droplet velocity

Besides the fluctuation period,the amplitude of the moving bottom wall is also proved to be a significant factor on the droplet velocity as shown in Fig.8.The larger droplet velocity range is obtained for the larger amplitude of the forced fluctuation.When Amax=16μm,the velocity amplitude can reach above 40%,where the velocity range during the flow is shown to cover about 10000–26000μm/s.The droplet velocity amplitude increases linearly with the Amaxof the forced bottom wall movement as shown in Table 4.

Fig.8 Droplet velocity under different fluctuation amplitudes

Table 4 Droplet velocity amplitude under different fluctuation amplitudes

The droplet flow can be affected by both the period and the amplitude obviously in the way that the higher speed of the moving wall causes the larger velocity variation during the flow.The relevant velocity variation(the difference of the maximum and minimum droplet velocity in the droplet flow)could be up to 80%of the average velocity in this study,whose wall moving amplitude is 32%(Amax=16 μm)of the channel’s width and the average droplet length is stable at 78.21μm with a standard error of0.2%.In addition,Amax=12 μm and Amax=8 μm cases can also achieve a large droplet velocity variation(about 60% and 40%, respectively)with a relatively uniform droplet size.In other words,the fluctuation of the bottom wall can force an obvious velocity variation on the droplet with little change of the generated droplet,when its period is the same as the droplet generation time.

7 Conclusions

In this article,a protocol based on bottom wall fluctuations for the droplet generation and flow control is presented.The bottom wall shapes,the period and the amplitude of wall fluctuations are investigated to document the effects of the deformation and the fluctuation in droplets generation separately.

1.The fluctuations are demonstrated to expand the droplet size range within about 7%but not change the average size and droplet emerging frequency under most of our investigating conditions,while the fixed shape of the bottom wall changes the average droplet size significantly.Droplet sizes could be predicted by the droplet-sequence length formula along the relative time when the fluctuation is forced on the bottom wall near the T-junction.

2.The wall moving fluctuation,whose frequency is the same as that of the droplet generation,can remain the droplet size stable at a specific value.A little variation(within4%)of this longtime steady magnitude is incorporated by the double-fold amplitude of the wall fluctuation after the flow development stage at the beginning.

3.The forced moving wall changes the droplet velocity in a relatively large range in the way that the velocity evolution has the same sine period with the moving wall..When the wall moving frequency is equal to the droplet emerging frequency,the droplet size has a superior stability.Therefore,the large variation range of droplet velocity(up to 80%)can be imposed by the bottom wall fluctuation without altering the droplet size and emerging interval.

AcknowledgementsThe authors are grateful for the support of the National Natural Science Foundation of China(11572013 and 11702007)and the China Postdoctoral Science Foundation(2017M 610725).