Numerical Investigation of the Thermocapillary Migration of a Water Droplet in a Microchannel by Applying Heat Source

Use your smartphone to scan this QR code and download this article ABSTRACT The migration of a small droplet has been developed during the last two decades due to its applications in industry and high technology such as MEMS and NEMS devices, Lap-On-achip, transportation of fluids and so on. There have many studies in this topic in which the energy source as driving force for moving of a droplet is quite difference like heating, magnetics, pressure, electric, laser and so on. In this study, the numerical computation is used to investigate the transient thermocapillary migration of a water droplet in a micro-channel under the effect of heating source. For tracking the evolution of the free interface between two immiscible fluids, we employed the finite element method with the two-phase level set technique to solve the Navier-Stokes equations and continuity equation coupled with the energy equation. Both the upper wall and the bottom wall of the microchannel are set to be an ambient temperature. 40mW heat source is placed at the distance of 1mm from the initial position of a water droplet. When the heat source is turned on, a pair of asymmetric thermocapillary convection vortices is formed inside the droplet and the thermocapillary on the receding side is smaller than that on the advancing side. The temperature gradient inside the droplet increases quickly at the initial times and then decreases versus time. Therefore, the actuation velocity of the water droplet first increases significantly, and then decreases continuously. Furthermore, the results also indicate that the dynamic contact angle is strongly affected by the oil flow motion and the net thermocapillary momentum inside the droplet. The advancing contact angle is always larger than the receding contact angle during actuation process.


INTRODUCTION
Recently, microfluidics technique has significantly at-2 tracted owing to its diverse applications in Lab-on-a 3 Chip devices (LOC), Micro-Electro-Mechanical Sys-4 tem (MEMS) or protein crystallization [1][2][3] . The ther-5 mocapillary migration is a great important to ma-6 nipulate the droplet behavior and optimize the per-7 formance of the behavior of the droplet-based mi-8 crofluidics 4,5 . The droplet transport behavior in a mi-9 crochannel actuated by a transient temperature gradi-10 ent has already been investigated in numerous stud-11 ies 6-14 . Brochard 6 indicated that the contact angle of 12 a liquid droplet at rest, static contact angle (SCA), is 13 altered to the dynamic contact angle (DCA) when the 14 droplet moves on a solid surface. The difference in 15 the DCA between the advancing and receding sides, 16 so-called contact angle hysteresis (CAH), is strongly 17 affected by the temperature gradient. 18 The experimental results of Chen et al. 7 , developed 19 from Ford and Nadim's work 8 , indicated that a fixed 20 CAH influences the droplet velocity and threshold 21 values much more significantly than the slip length. 22 Le et al. 9 showed the effect of upper wall condition 23 on the liquid droplet migration behavior in a mi-24 crochannel. The movement of a liquid droplet in a 25 microchannel is strengthened due to the net thermo-26 capillary momentum generated by the unequal size of 27 the two vortices inside the droplet. The results showed 28 the actuation velocity and the DCA of the droplet are 29 strongly affected by the thermal condition of upper 30 wall. In addition, the numerical results from Le et 31 al. 10 demonstrated that the silicone plug motion in-32 side capillary tube is influenced by the net thermocap-33 illary momentum generated by the temperature gra-34 dient along the gas-liquid interface and the capillary 35 force caused by the temperature difference between 36 the ends of the liquid plug. The numerical results 37 are in good agreement with the previous experimen-38 tal results 11 . Liu et al. 12 developed a lattice Boltz-39 mann phase-field model to numerically simulate the 40 thermocapillary flows in a microchannel. Their re-41 sults indicated that the contact angle strongly influ-42 ences the droplet dynamic behavior and the droplet 43 motion driven by shear flow at the inlet of a confined 44 microchannel is completely blocked by using a laser 45    Table 1.
(1) 99 Where λ denotes the amount of reinitialization pa-100 rameter, ε determines the thickness of the layer 101 around the interface and V i is the velocity vector. The 102 dense mesh must be located near the free interface 103 during migration to ensure the accuracy of the nu-104 merical simulations. The ALE technique is used to 105 ensure that the fine mesh moves simultaneously with 106 the interface. The finite element method developed by 107 Comsol Multiphysics is used to solve the governing 108 equations with the correlative boundary and initial 109 conditions, employing second-order Lagrange trian-110 gular elements. The dependency of the element num-111 ber on the simulation results has been determined to 112 ensure the accuracy of the solution.

113
The two-dimensional equations for the conservation 114 of mass, momentum, and energy for Newtonian in-115 compressible fluids are written as: Where u i and v i are the velocity components in the x-124 and z-directions, respectively; p is the pressure and 125 ρ i is the fluid density; µ i is the dynamic viscosity; C Pi 126 is the specific heat; k i is the thermal conductivity; and 127 T is the temperature. The subscripts i = "w" and i = 128 "o" represent water and hexadecane oil, respectively. 129 F x and F z are the surface tension force in the x-and z-130 directions, respectively. Q s is the heat source.

131
The dependence of fluids density on temperature can 132 be expressed as 134 Where ρ re f is the fluid density at the reference tem-135 perature, and β i is the thermal expansion coefficient 136 of the fluid.

137
The continuum surface force method developed by 138 Brackbill et al. 17 is used to deal with the existence of 139 the surface tension along the free interface. The sur-140 face tension force at the free interface can be modeled 141 by Where σ is the surface tension; δ is the Dirac delta 144 function that is a nonzero value at the droplet/air in-145 terface only; n is the unit normal vector to the inter-146 face; and κ is the local interfacial curvature. The sur-147 face tension σ can be assumed to vary linearly with 148 temperature 18 , i.e. 150 Where σ re f is the surface tension at the reference tem-

151
perature T re f and γ T = δ σ δ T is the coefficient of the 152 surface-tension.

153
The boundary conditions for the flow and tempera-154 ture field are given by : value.
247 Figure 5 shows the pressure differences (∆P = p w -248 p o ) on both side of the droplet and the variation of 249 DCA during the migration process with b s = 1 nm, 250 θ = 90 0 , W = 10 mm, and H = 1 mm. The pres-251 sure difference at the receding (∆P R ) and the advanc-252 ing side (∆P A ) of the droplet is negative and positive, 253 respectively (Figure 5a). The present results show 254 that the DCA alternates during the actuation process 255 (Figure 5b). The DCA behavior strongly depends on 256 the pressure difference acting on the droplet. The re-257 ceding contact angle (RCA, θ R ) decreases strongly 258 first and then increases significantly while the advanc-259 ing contact angle (ACA, θ A ) increases rapidly first 260 and then decrease continuously. The ACA is always 261 larger than the RCA due to the magnitude of ∆P A is 262 smaller than that of ∆P R . Since θ A > 90 > θ R and σ A > 263 σ R , σ A cosθ A -σ A cosθ A < 0. Therefore, the capillary 264 force acts against the movement of the water droplet 265 in a microchannel.

267
The transient thermocapillary migration of a water 268 droplet in a microchannel subjected to a heat source 269 has been investigating numerically. Both the upper 270 wall and the bottom wall of microchannel are set to 271 be an ambient temperature. The results indicate that 272 the actuation behavior of the droplet is strongly in-273 fluenced by a heat source. The actuation velocity 274 of the liquid droplet initially accelerates, and then 275 goes down rapidly. During the actuation process, the 276 thermocapillary vortices inside the droplet on the re-277 ceding side is always smaller than that on the ad-278 vancing side. The isotherms inside the droplet are 279 notably distorted by the thermocapillary convection. 280 The DCA of the droplet alternates versus time due 281 to the pressure difference acting on the droplet. The 282 ACA first increases rapidly and then decreases con-283 tinuously while the RCA first decreases strongly and 284 then increases significantly. The ACA is always larger 285 than the RCA during the actuation process.

301
This study is done by our self and there have not any 302 results in this paper come from other sources.