Numerical Investigation of Electroosmotic Flow. in Convergent/Divergent Micronozzle

Transcription

1 Appled Mathematcal Scences, Vol. 5, 2011, no. 27, Numercal Investgaton of Electroosmotc Flow n Convergent/Dvergent Mcronozzle V. Gnanaraj, V. Mohan, B. Vellakannan Thagarajar College of Engneerng Madura, Tamlnadu, Inda vgnanaraj@yahoo.com Abstract A fundamental understandng of the transport phenomena n mcrofludc channels s crtcal for systematc desgn and precse control of such mnaturzed devces towards the ntegraton and automaton of Lab-on- a-chp devces. Electroosmotc flow s wdely used to transport and mx fluds n mcrofludc systems. Electroosmotc transport n convergent dvergent mcronozzle s sgnfcant n many applcatons, such as bomolecular transport, DNA transport n cell patch clamps. The goal of ths paper s to develop a theoretcal model of electroosmotc flow n mcronozzle to gan a better understandng of transport phenomena n mcrofludcs channels. Numercal study of electroosmotc flow through convergent dvergent nozzle mcrochannels has been developed n ths paper. The governng equatons consst of a 2D Posson-Boltzman equaton and a 2D Naver-Stoke's equaton wth Electrc Double Layer(EDL) feld. The potental dstrbuton of the EDL n the channel s obtaned by solvng the non-lnear 2D Posson Blotzmann equaton usng fnte element technque. Then the soluton for the 2D Naver-Stoke s equaton for the velocty dstrbuton s also obtaned. The velocty dstrbuton profles are obtaned usng COMSOL Multphyscs software. Keywords: mcronozzle, eleectroosmotc flow, Posson- Boltzmann equaton, Naver-Stokes equaton. 1. Introducton When the flud s flowng n channels the frcton of the wall wll resst flowng. In partcular f the channels are n mcro scale or nano scale, the frcton of the channels wall s very strong. In order to make the flud flow n mcrochannel, t s necessary to apply external forces. At present electrcal feld s wdely used.

2 1318 V. Gnanaraj, V. Mohan, B. Vellakannan Dutta and Leghton [1] presented a theoretcal study on dsperson n a rectangular mcrochannel bounded by two quarter-crcular dsks. Kesuke Horuch et. al[2] analysed Electroosmotc flow n trapezodal mcrochannel. Electroosmotc flow velocty measurements n a square mcrochannel were studed by Shou-Shng Hseh et. al. Electroosmotc flow n cerpentne channel numecally smulated by A. S. Rawool[4] and found that electroosmoss nduces secondary flow patterns n the straght porton of the channel n addton to secondary vortces at bends. The effects of channel heght, electrolyte concentraton, surface potental, EDL thckness and externally appled elctrc feld on the velocty profle of traangular mcrochannels are numercally studed by Vatheeswaran and Mohan,V[5]. The meshless collocaton method was appled by V. Gnanaraj V. Mohan [6] to solve the Posson-Boltzmann equaton and the Naver-Stokes equatons governng the flow through crcular and trangular mcrochannel. The volumetrc flow rates wth dfferent ratos for a fxed hydraulc dameter were compared and they found that there was a sgnfcance nfluence of the cross secton geometry on the flow feld. In ths paper we analyze the Electroosmotc flow through a convergent/ dvergent mcronozzle 1.2 Electroosmotc flow and ts applcatons Electroosmoss[3] s the bulk movement of lqud relatve to a surface due to an externally appled feld. Ths phenomena was observed by Rauss nearly two centures ago. Electroosmotc flow provde an alternatve to pump wth better control and no movng parts. Ths mechansm s used to delver and control lqud sample of nanovolumes n mcrochannels of mnaturzed devces used for bo analytcal systems. 2. Problem Descrpton In the present work we consder a convergent dvergent mcronozzle as shown n the fg. 1. The cross secton of convergent/dvergent mcronozzle has length L and nlet heght 2h and outlet heght 2h 0.The channel contans electrolyte soluton. An external force s appled along the axs of the channel. Due to the nteracton between the charged on n the electrolyte soluton and the statc charges on the delectrc channel walls. Electrc Double Layer [EDL] s formed. We assume that the flow s ncompressble and the flud s ndependent of temperature. The velocty s steady and fully developed and there s no externally mposed pressure gradent.

3 Numercal nvestgaton of electroosmotc flow 1319 Fg Posson-Boltzmann equatons and Boundary Condtons Electroosmotc flow n convergng/dvergng mcronozzle plays vtal role n DNA transport n cell patch clamps. To analyze the characterstcs of eelectroosmotc n convergent/dvergent mcronozzle, a 2D Posson-Boltzmann equaton and a 2D Naver-Stoke s equaton were used to model the Electrc Double Layer feld and flow feld n the cross secton of convergent/dvergent mcronozzles. We consder a convergent/ dvergent mcro nozzle of Length L and and nlet heght 2h and outlet heght 2h 0. We assume that the flow s ncompressble and the flud s ndependent of temperature. The velocty s steady and fully developed and there s no externally mposed pressure gradent. The theory of electrostatcs gves the relatonshp between the electrcal potental,ψ, and the net-charge densty per unt volume ρ e, at any pont n the lqud s descrbed by the two-dmensonal Posson equaton: 2 ρ e ψ = ( 1 ) εε 0 where ε s the delectrc constant of the soluton and ε 0 s the permttvty of vacuum. Assumng that the equlbrum Boltzmann dstrbuton s applcable, the on number concentraton per unt volume n an electrolyte soluton s of the form: ε 0 = x c 2 J -1 m -1 s the permttvty of the vacuum. Assumng, the Boltzmann dstrbuton s applcable, the on number concentraton per unt volme n an electrolyte soluton s of the form α eψ n = = n exp ( 2 ) kbt where n & α are the bulk onc concentraton and the valence of type on respectvely. e = x c s the charge the proton, k b = x J. k -1

4 1320 V. Gnanaraj, V. Mohan, B. Vellakannan s Boltzmann s constant and T s absolute temperature. The net volumetrc charge densty ρe s proportonal to the concentraton dfference between catons and anons and s gven by ρ = e n n α e ( 3 ) α eψ = α n exp n kbt For symmetrc electrolyte soluton KCl (z: z =1:1) soluton t takes the form α eψ ρ = e e α n exp n kbt eψ = e n snh k bt 2 ( 4 ) Substtutng ths n to the Posson s equaton, the two dmensonal non-lnear Posson-Boltzmann equaton s 2 ψ = 2en eψ snh ε 0ε kbt ( 5 ) eψ Let Ψ =. kbt Then ( 5 ) becomes 2 2 Ψ = k snh ( Ψ) ( 6 ) e n where k = ε 0ε kbt Equaton (6) s subject to the followng boundary condtons. Ψ Inlet : x = 0, = 0 Ψ Out let : x = L, = 0 Wall : Ψ= ς

5 Numercal nvestgaton of electroosmotc flow Velocty flow equatons and Boundary condtons Under our assumptons and usng non dmenson varables the velocty equaton 2 Eρe becomes u = ( 7 ) μ usng ( 4 ) n ( 7 ) we have 2 1 eψ u= 2 en E sn h ( 8 ) μ kt b 2en Let K = E μ Now equaton ( 8 ) becomes 2 2 u u + = K snh( Ψ) 2 2 x Equaton (9) s subject to the followng boundary condtons. u Inlet : x = 0, = 0 u Out let : x = L, = 0 Wall : Ψ= ς ( 9) 3. Use of COMSOL Multphyscs We apply COMSOL Multphyscs classcal PDE Modes to solve the coupled equatons equaton ( 6) and ( 9). Equaton ( 6 ) together wth ts boundary condtons are solved usng COMSOL Multphyscs Helmholtz equaton PDE whle (9 ) s solved usng PDE Mode Posson equaton. Fg. 2 Velocty dstrbuton at the centre Fg 3. Velocty dstrbuton for Varous values of K

6 1322 V. Gnanaraj, V. Mohan, B. Vellakannan 4. RESULTS AND DISCUSSIONS In ths secton we llustrate the electroknetc effect n twodmensonal electroosmotc flow n the absence of an appled pressure drop. The Debye layer thckness λ D, whch depends on the bulk onc concentraton of the lqud nsde the channel and plays a key role n producng electro-osmotc flow, s typcally small compared wth the wdth h(x) of the channel. A decrease of bulk concentraton wll ncrease the Debye thckness and decrease the dmensonless parameter K= h(x)/λd. Snce the Debye layer establshes the charge separaton that leads to electro-osmotc flow, a change n K wll affect the electro-osmotc velocty profle. For large values K we see that the velocty has a flat profle across the channel, except n a narrow regon near the wall. Ths s the expected electro-osmotc velocty profle. As K becomes smaller, departure from electro-osmotc flow becomes notceable. When the Debye thckness s comparable wth the characterstc length scale h, the nternal potental s non-zero far away from the walls. 5. CONCLUSIONS Numercal smulaton of the mcro flow n convergent mcrochannel s gven n ths paper based on EOF mechansm. The dmensonless equatons of zeta potental and electrc feld potental are gven by analyss. The control equaton s smplfed by use of fluxon functon. Smulaton conclusons show: 1. The gradent of zeta potental s 0 along x drecton, that s the drecton along the mcrochannel wall. 2. The dstrbuton of electrc feld s unform. 3. The EOF volecty away from the wall s equal, the fgure of EOF volecty lke as plug. 4. All above conclusons are the theory base for mcrofludc control n desgnng mcropump. References [1] Dutta D, Leghton DT (2001) Dsperson reducton n pressure drven flow through mcroetched channels. Anal Chem.73: [2] Kesuke Horuch Prashanta Dutta, Cecla D., Experment and smulaton of mxed flows n a trapezodal mcrochannel, Mcroflud Nanoflud, [3] Probsten, R.F.(1994) Physcochemcal Hydronomcs, An Introducton, Second Edton, John Wley and Sons,Newyork.

7 Numercal nvestgaton of electroosmotc flow 1323 [4] A.S.Rawool,Sushantra K.Mtra,Numercal smulaton of electroosmotc flow n serpentne channels, Mcroflud-Nanoflud,2(2006): [5] Vatheeswaran Gnanaraj, V. Mohan, Numercal Smulaton of Electroosmotc Flowthrough Trangular Mcrochannel, Bulletn of the Amercan Physcal Socety, 60th Annual Meetng of the Dvson of Flud Dynamcs, Volume 52, No. 12,2007 [6] V. Gnanaraj V. Mohan, Numercal Smulaton of Electroosmotc Flow through Crcular and Trangular Mcrochannel wth Dfferent Aspect Ratos, Computatonal Mechancs 2009, Part 3, Part 2, 249, Sprnger. Receved: December, 2010