摘要
本研究以數值方法探討不同平面蜿蜒微流道在各種雷諾數下的流動及混合情形。由於當雷諾數相當大時，使用網格法模擬高Schmidt數流體內之對流-擴散傳輸，數值擴散的情形無法有效避免，因此在本研究中使用粒子反向追跡結合擴散模式的方法與蒙地卡羅法模擬流體的混合行為，此兩種方法的模擬結果具有一致性。由本研究的結果中發現，在低雷諾數時，流體的流動情形與混合行為在不同形狀的蜿蜒微流道中大致相同；然而隨著雷諾數上升，流體流動情形及混合效率會因流道外形的差異而有所不同，而且以網格法求得的混合度則明顯高估。本研究比較了具有S型與反S型模組之蜿蜒微流道、具有直角Z型與直角反Z型模組之蜿蜒微流道、以及具有Z型與反Z型模組之蜿蜒微流道，由本研究的模擬結果得知:具有Z型與反Z型之蜿蜒微流道有最佳的混合；其次為具有直角Z型與直角反Z型之蜿蜒微流道。流體在蜿蜒微流道中經過Z型與反Z型模組構成之流道時，銳角轉彎所產生的渦流範圍及強度皆較大，渦流可拉伸及扭曲流體的交界面，使流體的接觸面積增加，而提高混合效率，故在相同的雷諾數下，具有Z型與反Z型模組之蜿蜒微流道有較佳的混合表現。

Extended Abstract

Numerical simulation for mixing of fluids in a serpentine microchannel with Z-type and reversed Z-type modules

Author : Kai-Cheng Hsu
Advisor : Chih-Yang Wu
Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan
SUMMARY

In this work, we compare the flow and mixing in a serpentine microchannel with multiple circular arcs, a serpentine microchannel with multiple right-angled bends and a serpentine microchannel with multiple acute-angled bends. The particle-tracking simulation with an approximation diffusion model and the Monte Carlo simulation are employed to study the mixing behaviors of fluids in serpentine microchannels. These non-grid methods yield consistent results, while the grid method obviously overestimates the mixing performance. The simulation results of the non-grid methods show that the mixing performance of the serpentine microchannel with multiple acute-angled bends is the best among these mixers, and that of the serpentine microchannel with multiple right-angled bends is the second best one. When fluids go through the acute-angled bends, vortices can be developed stronger and bigger at high Reynolds numbers. Multi-directional vortices are able to stretch and distort the interface between different fluids and improve the mixing performance effectively.
Key words : serpentine microchannels, microfluidics mechanics, fluids mixing, numerical diffusion, particle tracking

INTRODUCTION

Micromixers have been paid attention in medical and biological fields over last few decades. Unfortunately, due to the Reynolds numbers in microchannels are low and the flows are laminar, the mixing process is extremely slow and mainly relies on molecular diffusion. Thus, many micromixers have been designed in recent years, and can be mainly categorized as active and passive types. Passive micromixers purely depend on the characteristic of geometry to enhance the mixing efficiency. Active ones, however, need external force to perturb the flow and it is difficult to be fabricated. Most passive micromixers resort to generate vortices or lateral advection to fold and stretch the flow, so the contact area is able to increase and mixing efficiency can enhance greatly. Particularly, fluids in a serpentine microchannel can continuously change the flow direction and develop multi-vortices because of centrifugal force. This assists fluids to achieve better mixing efficiency in microchannels.

NUMERICAL METHOD

This work uses numerical methods to investigate the mixing performance and flow characteristics in micromixers based on a serpentine microchannel with multiple circular arcs, right-angled bends or acute-angled bends. To illustrate the geometry of the micromixers, the micromixer including two inlet channels, a L-shaped microchannel from inlets to the first Z-shaped microchannel, a serpentine channel based on successive Z-shaped and reversed Z-shaped units, and an outlet channel, as shown in Fig. 1, is taken as an example. The Z-shaped microchannel has bends. Rhodamine B in DI water is injected to the inlet channel A and DI water is injected to the inlet channel B. These two fluids are assumed to have the same Schmidt number defined as:
， (1)
where is the kinematic viscosity, is the diffusion coefficient. The other important parameter of the mixing flow is Reynolds number defined as:
， (2)
where is the mean velocity, is the hydraulic diameter and is the kinematic viscosity of fluids. The method proposed by Matsunaga et al. (Lab on a chip, 2013, V. 13, pp, 1515-1521) is used to simulate the mixing phenomena in micromixers. The method includes the fluid particle-tracking (illustrated in Fig. 2) based on the velocity field preliminarily solved by grid-based simulation and the numerical solution of a modeling equation taking molecular diffusion into account.
The other method, the Monte Carlo simulation, proposed by Hathhorn et al. (Hydraulic Engineering Journal, vol. 123, no. 12, pp. 1157-1160, 1997) is also adopted to study the fluid mixing in micromixers. The displacement of particles is defined by advection displacement and random walk, which can be expressed as:
， (3)
where is advection velocity field obtained by the gird method, is a time step,
is a random walk representing particle diffusion and is defined as:
， (4)
where is the random value with Gaussian distribution, is a unit vector with random direction, and is the diffusion coefficient.
To compare the mixing performance, the degree of mixing, M, at the target plane defined as
， (5)
where donates the standard deviation of the concentration at the target plane and donates the standard deviation of the concentration when fluids are separated.

RESULT AND DISCUSSION

The concentration distribution at the exit in the serpentine microchannel with multiple acute-angled bends at by using three methods are shown in Fig. 4. The result of the grid method shows the excessive diffusion between interfaces because of numerical diffusion at high number. The Monte Carlo method and the particle tracking with ADM method can generate the reliable result and they are in good consistence with each other. Fig. 5 is the degree of mixing, M, in different target plane solved by the three methods. It’s observed that the degree of mixing solved by grid method is overestimated.
Fig. 6 shows the degree of mixing, M, at the exit at four Re numbers in the serpentine micromixer with multiple right-angled bends and that with multiple acute-angled bends. The degrees of mixing at low Reynolds numbers ( ) are almost the same for these two micromixers because they have equal diffusion time. However, the degree of mixing is much higher in the serpentine micromixer with multiple acute-angled bends at . Fig. 7 shows the concentration distributions at outlet in two micromixers at . It’s also evident that the contact area at the exit is more in a serpentine micromixers with multiple acute-angled bends because of significantly stronger vortices. The strong vortices can stretch and fold the interfaces of fluids, and so enhance the degree of mixing.

CONCLUSION

The strength of vortices and lateral advection is different in the three micromixers considered at high Reynolds numbers ( ). Since the mixing mechanism is dominated by convection at , the difference of separation vortices is significant when flows go through the right-angled bends or the acute-angled bends. The flow structure of the latter is much more complicated and so enhances fluid mixing more effectively.

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