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Chaotic mixing on a micromixer with barriers embedded
Chaos, Solitons and Fractals 33 (2007) 1362–1366 www.elsevier.com/locate/chaos
Chaotic mixing on a micromixer with barriers embedded Ruijin Wang a
a,b
, Jianzhong Lin
a,c,* ,
Huijun Li
a
Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China b Mechanical School, Zhejiang University of Science and Technology, Hangzhou 310012, China c China Jiliang University, Hangzhou 310018, China Accepted 23 January 2006
Abstract Rapid progresses in technology of the biochip or electrochemical engineering require high-performance micromixer. Research on the flow in the microchannel is the foundation of the design for passive micromixer. In this paper, a micromixer embedded with barriers was studied numerically. It is found that the barriers periodically embedded in the microchannel are beneficial to the chaotic mixing in that the barriers can form a group of linked twist maps that possess the Bernoulli property and the chaotic advection in these regions. The mixing efficiency in the diffusion channel with barriers is much higher than that without the barriers. 2006 Elsevier Ltd. All rights reserved.
1. Introduction The integrated micro-fluidic devices known as the micro-total analysis systems (l-TAS) have many advantages such as extremely low volume consumption for both detector and sample, inexpensive and small, extremely short sample-toresult time, compared with the traditional analytical devices. Diffusion of species is often critical to the operation, accuracy and efficiency of micro-fluidic devices for DNA analysis, mass spectrometry, biosensors, surface patterning and other applications [1–5]. However, the velocity distribution is the most important factor to impact the species diffusion. The specifically arranged barriers in the microchannel can change the flow pattern, as we know, by making up two or more linked twist maps (LTM) and consequential chaos [6–8]. These barriers can shift the center of the closed streamlines periodically. There are some investigations concerned with the LTM in literature. Qian and Bau [9] showed that periodical variety of the flow patterns results in chaotic flow. Stroock et al. [10] reported that when the fluid is driven axially by a pressure gradient, the ridges on the floor of the channel give rise to a transverse flow. Another variation [11] on the cavity flow is the exploitation of time-dependent changes in geometry by adding a secondary baffle, in this case the portrait changes from one figure of eight to another, but the location of the central hyperbolic point has been shifted. Chang and Cho
* Corresponding author. Address: Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China. Tel.: +865 71879 52882; fax: +865 71879 51464. E-mail addresses: [email protected] (R. Wang), [email protected] (J. Lin).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.099
R. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 1362–1366
1363
[12] developed an alternating whirl-lamination (AWL-type) mixer, composed of rotationally arranged microblocks and dividing microchannels, the micromixer can reduce effectively the mixing length over wide ranges of flow rate. Kim et al. [13] presented a new chaotic passive micromixer, in which the chaotic flow is induced by periodic perturbation of the velocity field due to periodically embedded barriers along the top surface of the channel, while a helical type of flow is obtained by slanted grooves on the bottom surface in the pressure driven flow. The aim of present work is to investigate how the barriers change the flow patterns in the microchannel by numerical simulation, and provide a powerful auspice for the design of the micromixer.
2. Linked twist maps (LTM) 2.1. Maping and mixing The motion of fluid particles is described mathematically with a map or mapping. Let R denote the region occupied by the fluid. We refer to points in R as fluid particles. The flow of fluid particles is mathematically described by a smooth, invertible transformation, or map, of R into R, denoted by S, also having a smooth inverse. The application of S to the domain R, denoted S(R), is referred to as one advection cycle. Similarly, n advection cycles are obtained by n repeated applications of S, denoted Sn(R). For a specific fluid particle p, the trajectory of p is the sequence of points {. . ., Sn(p), . . . , S1(p), S(p), S2(p), . . . , Sn(p), . . . }. Let A denote any subdomain of R. Then l(A) denotes the volume of A. Thus l is a function that assigns the volume to any subdomain of R. Incompressibility of the fluid is expressed by stating that, as any subdomain of R is stirred, its volume remains unchanged, i.e., l(A) = l(S(A)). Mixing is a critical concept. Within the domain R, let B denote a region of black fluid and let W denote any other region. Mathematically, we denote the amount of black fluid that is contained in W after n applications of the mixing process by l(Sn(B) \ W), that is the volume of Sn(B) that ends up in W after n advection cycles. Then the fraction of black fluid contained in W is given by l(Sn(B) \ W)/l(W). 2.2. Chaos and Bernoulli property It was said that there was still no universally accepted definition for chaos [13]. Nevertheless, in practice, a map may be called chaotic if the orbits have some positive Lyapunov exponents. Lyapunov exponent is a number associated with an orbit, describing its stability in the linear approximation. One must note here that a Lyapunov exponent is an infinite-time averagePvalue. Consider the set of bi-infinite sequences, where each element in the sequence P is either ‘0’ or ‘1’. We call this set 2 , where the superscript two denotes the two ‘symbols’, 0 or 1. So an element of 2 has the form: { . . . Sn, Sn+1 . . . , S1, S0, S1, . . . , Sn, Sn+1 . . . }, where Si is either 0 or 1, forPall i. The two infinite sequences in our 2 bi-infinite sequence are separated by the period. We define a map from into itself, called the Bernoulli shift. The shift map, denoted by r, acts on a bi-infinite sequence by shifting the period one place to the right, i.e., r = {. . .Sn, Sn+1 . . . , S1, S0, S1, . . . , Sn, Sn+1 . . . }. It is obvious that the Bernoulli shift has an infinite number of periodic orbits of all periods. This proved that the Bernoulli property embodies a deterministic chaotic system to behave. 2.3. Linked twist maps We will describe a type of map or advection cycle-LTM, which will be rigorously shown to possess the Bernoulli property. Consider a region of fluid, possibly a cross-flow in a 3D flow in a channel with an axial flow in the direction normal to the page, containing a stagnation point surrounded by closed streamlines. The stagnation point is on the horizontal axis, offset to the left of the centre. Imagine that the flow pattern is altered at some later time and by some mechanism. The alteration involves moving the stagnation point to the right of the centre. The motion of a fluid particle from the beginning of a half-cycle to the end of the same half-cycle is described with a twist map. For the closed streamlines in each half-cycle, let (r, h) denote streamline coordinates-plane, that is, on a streamline r is constant and h is an angular variable that increases monotonically in time. The map of particles from the beginning to the end of a half cycle is given by S(r, h) = (r, h + g(r)). We will see that the function g(r) is the key here. We can now define LTM over the entire advection cycle. Let A1 denote an annulus whose inner (denoted r1i) and outer (denoted r1o) boundaries are streamlines centred at (c, 0) in the first half of the advection cycle. Let A2 denote another annulus constructed in the same way and centred at (c, 0) in the second half of the advection cycle. Let S1(r, h) = (r, h + g1(r)) be a twist map defined on A1 with dg1/dr 5 0 and gðr1i Þ ¼ 2pn, for some integer n. Let S2(r, h) = (r, h + g2(r)) be a twist map defined on A2 with dg2/dr 5 0 and gðr2i Þ ¼ 2pm, for some integer m, the map defined by S2 \ S1 on A1 [ A2 is referred to as a LTM.
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R. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 1362–1366
Fig. 1. Schematic of the BEM (the depth is 100 lm).
2.4. Barriers embedded micromixer On the basis of the foundations of chaotic mixing, a BEM shown in Fig. 1, was designed for mixing or dispensing, and then numerical research was carried out to see whether the velocity profile is beneficial to the mixing or not. In rough analogy to the structures described in Ref. [14], the BEM studied numerically here placed six barriers purposely in diffusion channel so that periodic advection cycle could be performed.
3. Mathematic model and numerical scheme The equations governing the impressible flow in a micro-channel are the continuity equation and N–S equation. The no-slip boundary condition is valid because here the Knudsen number is about 5–400 · 106, much less than 103. With the consideration of the diffusion between two species, the non-dimensional equation for the diffusions of multi-species is: oC 1 þ ðV rÞC ¼ r2 C; ot ScRe
ð1Þ
where C is the species concentration, V is the flow velocity vector, Sc = m/D is the Schmidt number with D and m being the diffusion coefficient and the kinematic viscosity of the species, respectively, and Re = Uh/m is the Reynolds number with U and h being the convective velocity and the channel width, respectively. In this paper, D is of the order of 109–1011 m2/s, and m is of the order of 106 m2/s, thus Sc is of the order of 103–105 and Re is of the order of 101–103. Therefore, the diffusion term is comparable to the convective term and is not negligible. Eq. (1) together with the continuity equation and N–S equation are solved using finite volume algorithms. The simulation domain under consideration is covered by hexahedral grid. For each cell, discrete versions of the differential equations are derived after the domain being discretized into a finite set of control volumes or cells.
4. Results and discussion The properties of the liquids injected in inlet 1 and inlet 2 are the same as that listed in Ref. [4] in numerical simulation. The numerical results show that the shifts of the velocity profiles are very strong. Fig. 2 depicts the flow
R. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 1362–1366
(a)
(b)
(c)
(d)
1365
(e)
Fig. 2. The flow patterns in the region of barriers (a) line 5, (b) line 6, (c) line 7, (d) line 8 and (e) line 9.
1.0
1
Variance
0.8
0.6
2 1 without barrier 2 with barriers
0.4
0.2
0
2
4
6 8 10 12 Downstream distance / mm
14
Fig. 3. The variance versus the downstream distance.
patterns near the barriers. The shift of the stagnation point of the subdomains can be seen evidently, and the shift can construct LTMs and form blinking flow. Thereby, this microstructure possesses Bernoulli property and benefits to the rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i. PN 2 mixing efficiency. Using the mixing index, the variance of the volume fraction is defined as r ¼ N i¼1 ðC i CÞ PN with Ci being the volume fraction, C ¼ N the average volume fraction value over all of the cells, N the quani¼1 C i tity of the cells in the calculated region. It can be seen in Fig. 3 that the mixing index downstream decreases along with downstream distance from the joint. Especially, it decreases more rapidly in the region with barriers than without barriers, because there are LMTs in that region.
5. Conclusion Here we present a quantitative investigation of the spatial distribution of diffusing species and mixing efficiency in the pressure-driven microfluidic mixer using numerical simulation. Following consequences are drawn: (1) The specially embedded periodic barriers are beneficial to the chaotic mixing because they can shift the flow pattern and produced blinking flow, that is, they can form a group of linked twist maps (LTMs) that possess Bernoulli property and chaotic advection in this regions. (2) The mixing efficiency in diffusion channel with barriers is much better than that without barrier.
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R. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 1362–1366
Acknowledgements We acknowledge the financial supports of National Natural Science Foundation of China (No. 20299030), Key Project of Zhejiang Provincial Education Department (No. 20041174) and Key Project of the Science and Technology Department of Zhejiang Province (No. 2005C21102)
References [1] Beebe DJ, Mensing GA, Walker GM. Physics and applications of microfluidics in biology. Ann Rev Biomed Eng 2002;4:261–86. [2] Wang RJ, Lin JZ, Li ZH. Analysis of electro-osmotic flow characteristics at joint of capillaries with saltation f-potential and dimension. Biomed Microdev 2005;7:131–5. [3] Wong PK, Lee YK, Ho CM. Deformation of DNA molecules by hydrodynamic focusing. J Fluid Mech 2003;497:55–65. [4] Wang RJ, Lin JZ, Li ZH. Study on the impacting factors of transverse diffusion in the micro-channel of T-sensor. J Nanosci Nanotechnol 2005;5:1281–6. [5] Yoomin Ahn, Tkahito Ono, Masayoshi Esashi. Si multiprobes integrated with lateral actuators for independent scanning probe applications. J Micromech Microeng 2005;15:1224–9. [6] Froncioni AM, Muzzio FJ, Peskin RL. Chaotic motion of fluid and solid particles in plane wakes. Chaos, Solitons & Fractals 1997;8:109–30. [7] Ottino JM, Souvaliotis A, Metcalef G. Chaotic processes—new problems and computational issues. Chaos, Solitons & Fractals 1995;6:425–31. [8] Liu M, Muzzio FJ, Peskin RL. Quantification of mixing in a periodic chaotic flows. Chaos, Solitons & Fractals 1994;4:869–93. [9] Qian S, Bau HH. A chaotic electroosmotic stirrer. Anal Chem 2002;74:3616–25. [10] Stroock AD, Dertinger SKW, Ajdari A, Mezic I, Stone HA, Whitesides GM. Chaotic mixer for microchannels. Science 2002;295:647–51. [11] Jana SC, Tjahjadi M, Ottino JM. Chaotic mixing of viscous fluids by periodic changes of geometry: the baffle-cavity system. AIChE J 1994;40:1769–81. [12] Chang SH, Cho YH. Static micromixers using alternating whirls and lamination. J Micromech Microeng 2005;15:1397–405. [13] Kim DS, Lee SW, Kwon TH, Lee SS. A barrier embedded chaotic micromixer. J Micromech Microeng 2004;14:798–805. [14] Horner M, Metcalfe G, Wiggins S, Ottino JM. Transport mechanisms in open cavities: effects of transient and periodic boundary flows. J Fluid Mech 2002;452:199–229.
Chaotic mixing on a micromixer with barriers embedded Ruijin Wang a
a,b
, Jianzhong Lin
a,c,* ,
Huijun Li
a
Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China b Mechanical School, Zhejiang University of Science and Technology, Hangzhou 310012, China c China Jiliang University, Hangzhou 310018, China Accepted 23 January 2006
Abstract Rapid progresses in technology of the biochip or electrochemical engineering require high-performance micromixer. Research on the flow in the microchannel is the foundation of the design for passive micromixer. In this paper, a micromixer embedded with barriers was studied numerically. It is found that the barriers periodically embedded in the microchannel are beneficial to the chaotic mixing in that the barriers can form a group of linked twist maps that possess the Bernoulli property and the chaotic advection in these regions. The mixing efficiency in the diffusion channel with barriers is much higher than that without the barriers. 2006 Elsevier Ltd. All rights reserved.
1. Introduction The integrated micro-fluidic devices known as the micro-total analysis systems (l-TAS) have many advantages such as extremely low volume consumption for both detector and sample, inexpensive and small, extremely short sample-toresult time, compared with the traditional analytical devices. Diffusion of species is often critical to the operation, accuracy and efficiency of micro-fluidic devices for DNA analysis, mass spectrometry, biosensors, surface patterning and other applications [1–5]. However, the velocity distribution is the most important factor to impact the species diffusion. The specifically arranged barriers in the microchannel can change the flow pattern, as we know, by making up two or more linked twist maps (LTM) and consequential chaos [6–8]. These barriers can shift the center of the closed streamlines periodically. There are some investigations concerned with the LTM in literature. Qian and Bau [9] showed that periodical variety of the flow patterns results in chaotic flow. Stroock et al. [10] reported that when the fluid is driven axially by a pressure gradient, the ridges on the floor of the channel give rise to a transverse flow. Another variation [11] on the cavity flow is the exploitation of time-dependent changes in geometry by adding a secondary baffle, in this case the portrait changes from one figure of eight to another, but the location of the central hyperbolic point has been shifted. Chang and Cho
* Corresponding author. Address: Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China. Tel.: +865 71879 52882; fax: +865 71879 51464. E-mail addresses: [email protected] (R. Wang), [email protected] (J. Lin).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.099
R. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 1362–1366
1363
[12] developed an alternating whirl-lamination (AWL-type) mixer, composed of rotationally arranged microblocks and dividing microchannels, the micromixer can reduce effectively the mixing length over wide ranges of flow rate. Kim et al. [13] presented a new chaotic passive micromixer, in which the chaotic flow is induced by periodic perturbation of the velocity field due to periodically embedded barriers along the top surface of the channel, while a helical type of flow is obtained by slanted grooves on the bottom surface in the pressure driven flow. The aim of present work is to investigate how the barriers change the flow patterns in the microchannel by numerical simulation, and provide a powerful auspice for the design of the micromixer.
2. Linked twist maps (LTM) 2.1. Maping and mixing The motion of fluid particles is described mathematically with a map or mapping. Let R denote the region occupied by the fluid. We refer to points in R as fluid particles. The flow of fluid particles is mathematically described by a smooth, invertible transformation, or map, of R into R, denoted by S, also having a smooth inverse. The application of S to the domain R, denoted S(R), is referred to as one advection cycle. Similarly, n advection cycles are obtained by n repeated applications of S, denoted Sn(R). For a specific fluid particle p, the trajectory of p is the sequence of points {. . ., Sn(p), . . . , S1(p), S(p), S2(p), . . . , Sn(p), . . . }. Let A denote any subdomain of R. Then l(A) denotes the volume of A. Thus l is a function that assigns the volume to any subdomain of R. Incompressibility of the fluid is expressed by stating that, as any subdomain of R is stirred, its volume remains unchanged, i.e., l(A) = l(S(A)). Mixing is a critical concept. Within the domain R, let B denote a region of black fluid and let W denote any other region. Mathematically, we denote the amount of black fluid that is contained in W after n applications of the mixing process by l(Sn(B) \ W), that is the volume of Sn(B) that ends up in W after n advection cycles. Then the fraction of black fluid contained in W is given by l(Sn(B) \ W)/l(W). 2.2. Chaos and Bernoulli property It was said that there was still no universally accepted definition for chaos [13]. Nevertheless, in practice, a map may be called chaotic if the orbits have some positive Lyapunov exponents. Lyapunov exponent is a number associated with an orbit, describing its stability in the linear approximation. One must note here that a Lyapunov exponent is an infinite-time averagePvalue. Consider the set of bi-infinite sequences, where each element in the sequence P is either ‘0’ or ‘1’. We call this set 2 , where the superscript two denotes the two ‘symbols’, 0 or 1. So an element of 2 has the form: { . . . Sn, Sn+1 . . . , S1, S0, S1, . . . , Sn, Sn+1 . . . }, where Si is either 0 or 1, forPall i. The two infinite sequences in our 2 bi-infinite sequence are separated by the period. We define a map from into itself, called the Bernoulli shift. The shift map, denoted by r, acts on a bi-infinite sequence by shifting the period one place to the right, i.e., r = {. . .Sn, Sn+1 . . . , S1, S0, S1, . . . , Sn, Sn+1 . . . }. It is obvious that the Bernoulli shift has an infinite number of periodic orbits of all periods. This proved that the Bernoulli property embodies a deterministic chaotic system to behave. 2.3. Linked twist maps We will describe a type of map or advection cycle-LTM, which will be rigorously shown to possess the Bernoulli property. Consider a region of fluid, possibly a cross-flow in a 3D flow in a channel with an axial flow in the direction normal to the page, containing a stagnation point surrounded by closed streamlines. The stagnation point is on the horizontal axis, offset to the left of the centre. Imagine that the flow pattern is altered at some later time and by some mechanism. The alteration involves moving the stagnation point to the right of the centre. The motion of a fluid particle from the beginning of a half-cycle to the end of the same half-cycle is described with a twist map. For the closed streamlines in each half-cycle, let (r, h) denote streamline coordinates-plane, that is, on a streamline r is constant and h is an angular variable that increases monotonically in time. The map of particles from the beginning to the end of a half cycle is given by S(r, h) = (r, h + g(r)). We will see that the function g(r) is the key here. We can now define LTM over the entire advection cycle. Let A1 denote an annulus whose inner (denoted r1i) and outer (denoted r1o) boundaries are streamlines centred at (c, 0) in the first half of the advection cycle. Let A2 denote another annulus constructed in the same way and centred at (c, 0) in the second half of the advection cycle. Let S1(r, h) = (r, h + g1(r)) be a twist map defined on A1 with dg1/dr 5 0 and gðr1i Þ ¼ 2pn, for some integer n. Let S2(r, h) = (r, h + g2(r)) be a twist map defined on A2 with dg2/dr 5 0 and gðr2i Þ ¼ 2pm, for some integer m, the map defined by S2 \ S1 on A1 [ A2 is referred to as a LTM.
1364
R. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 1362–1366
Fig. 1. Schematic of the BEM (the depth is 100 lm).
2.4. Barriers embedded micromixer On the basis of the foundations of chaotic mixing, a BEM shown in Fig. 1, was designed for mixing or dispensing, and then numerical research was carried out to see whether the velocity profile is beneficial to the mixing or not. In rough analogy to the structures described in Ref. [14], the BEM studied numerically here placed six barriers purposely in diffusion channel so that periodic advection cycle could be performed.
3. Mathematic model and numerical scheme The equations governing the impressible flow in a micro-channel are the continuity equation and N–S equation. The no-slip boundary condition is valid because here the Knudsen number is about 5–400 · 106, much less than 103. With the consideration of the diffusion between two species, the non-dimensional equation for the diffusions of multi-species is: oC 1 þ ðV rÞC ¼ r2 C; ot ScRe
ð1Þ
where C is the species concentration, V is the flow velocity vector, Sc = m/D is the Schmidt number with D and m being the diffusion coefficient and the kinematic viscosity of the species, respectively, and Re = Uh/m is the Reynolds number with U and h being the convective velocity and the channel width, respectively. In this paper, D is of the order of 109–1011 m2/s, and m is of the order of 106 m2/s, thus Sc is of the order of 103–105 and Re is of the order of 101–103. Therefore, the diffusion term is comparable to the convective term and is not negligible. Eq. (1) together with the continuity equation and N–S equation are solved using finite volume algorithms. The simulation domain under consideration is covered by hexahedral grid. For each cell, discrete versions of the differential equations are derived after the domain being discretized into a finite set of control volumes or cells.
4. Results and discussion The properties of the liquids injected in inlet 1 and inlet 2 are the same as that listed in Ref. [4] in numerical simulation. The numerical results show that the shifts of the velocity profiles are very strong. Fig. 2 depicts the flow
R. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 1362–1366
(a)
(b)
(c)
(d)
1365
(e)
Fig. 2. The flow patterns in the region of barriers (a) line 5, (b) line 6, (c) line 7, (d) line 8 and (e) line 9.
1.0
1
Variance
0.8
0.6
2 1 without barrier 2 with barriers
0.4
0.2
0
2
4
6 8 10 12 Downstream distance / mm
14
Fig. 3. The variance versus the downstream distance.
patterns near the barriers. The shift of the stagnation point of the subdomains can be seen evidently, and the shift can construct LTMs and form blinking flow. Thereby, this microstructure possesses Bernoulli property and benefits to the rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i. PN 2 mixing efficiency. Using the mixing index, the variance of the volume fraction is defined as r ¼ N i¼1 ðC i CÞ PN with Ci being the volume fraction, C ¼ N the average volume fraction value over all of the cells, N the quani¼1 C i tity of the cells in the calculated region. It can be seen in Fig. 3 that the mixing index downstream decreases along with downstream distance from the joint. Especially, it decreases more rapidly in the region with barriers than without barriers, because there are LMTs in that region.
5. Conclusion Here we present a quantitative investigation of the spatial distribution of diffusing species and mixing efficiency in the pressure-driven microfluidic mixer using numerical simulation. Following consequences are drawn: (1) The specially embedded periodic barriers are beneficial to the chaotic mixing because they can shift the flow pattern and produced blinking flow, that is, they can form a group of linked twist maps (LTMs) that possess Bernoulli property and chaotic advection in this regions. (2) The mixing efficiency in diffusion channel with barriers is much better than that without barrier.
1366
R. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 1362–1366
Acknowledgements We acknowledge the financial supports of National Natural Science Foundation of China (No. 20299030), Key Project of Zhejiang Provincial Education Department (No. 20041174) and Key Project of the Science and Technology Department of Zhejiang Province (No. 2005C21102)
References [1] Beebe DJ, Mensing GA, Walker GM. Physics and applications of microfluidics in biology. Ann Rev Biomed Eng 2002;4:261–86. [2] Wang RJ, Lin JZ, Li ZH. Analysis of electro-osmotic flow characteristics at joint of capillaries with saltation f-potential and dimension. Biomed Microdev 2005;7:131–5. [3] Wong PK, Lee YK, Ho CM. Deformation of DNA molecules by hydrodynamic focusing. J Fluid Mech 2003;497:55–65. [4] Wang RJ, Lin JZ, Li ZH. Study on the impacting factors of transverse diffusion in the micro-channel of T-sensor. J Nanosci Nanotechnol 2005;5:1281–6. [5] Yoomin Ahn, Tkahito Ono, Masayoshi Esashi. Si multiprobes integrated with lateral actuators for independent scanning probe applications. J Micromech Microeng 2005;15:1224–9. [6] Froncioni AM, Muzzio FJ, Peskin RL. Chaotic motion of fluid and solid particles in plane wakes. Chaos, Solitons & Fractals 1997;8:109–30. [7] Ottino JM, Souvaliotis A, Metcalef G. Chaotic processes—new problems and computational issues. Chaos, Solitons & Fractals 1995;6:425–31. [8] Liu M, Muzzio FJ, Peskin RL. Quantification of mixing in a periodic chaotic flows. Chaos, Solitons & Fractals 1994;4:869–93. [9] Qian S, Bau HH. A chaotic electroosmotic stirrer. Anal Chem 2002;74:3616–25. [10] Stroock AD, Dertinger SKW, Ajdari A, Mezic I, Stone HA, Whitesides GM. Chaotic mixer for microchannels. Science 2002;295:647–51. [11] Jana SC, Tjahjadi M, Ottino JM. Chaotic mixing of viscous fluids by periodic changes of geometry: the baffle-cavity system. AIChE J 1994;40:1769–81. [12] Chang SH, Cho YH. Static micromixers using alternating whirls and lamination. J Micromech Microeng 2005;15:1397–405. [13] Kim DS, Lee SW, Kwon TH, Lee SS. A barrier embedded chaotic micromixer. J Micromech Microeng 2004;14:798–805. [14] Horner M, Metcalfe G, Wiggins S, Ottino JM. Transport mechanisms in open cavities: effects of transient and periodic boundary flows. J Fluid Mech 2002;452:199–229.