A minute magneto hydro dynamic (MHD) mixer

Transcription

1 Uniersit of Pennslania ScholarlCommons Departmental Papers (MEAM) Department of Mechanical Engineering & Applied Mechanics October 001 A minute magneto hdro dnamic (MHD) mier Haim H. Bau Uniersit of Pennslania, bau@seas.upenn.edu Jihua Zhong Uniersit of Pennslania Mingqiang Yi Uniersit of Pennslania Follow this and additional works at: Recommended Citation Bau, Haim H.; Zhong, Jihua; and Yi, Mingqiang, "A minute magneto hdro dnamic (MHD) mier" (001). Departmental Papers (MEAM) Postprint ersion. Published in Sensors and Actuators B: Chemical, Volume 79, Issues -3, October 001, pages Publisher UR: This paper is posted at ScholarlCommons. For more information, please contact librarrepositor@pobo.upenn.edu.

2 A minute magneto hdro dnamic (MHD) mier Abstract A theoretical and eperimental inestigation of a magneto hdrodnamic stirrer is presented. Such a stirrer can be used to enhance miing in micro total analsis sstems. The stirrer utilizes arras of electrodes deposited on a conduit's walls. The conduit is filled with an electrolte solution. B appling alternating potential differences across pairs of electrodes, currents are induced in arious directions in the solution. In the presence of a magnetic field, the coupling between the magnetic and electric fields induces bod (orentz) forces in the fluid. Since the electrodes can be patterned in arious was, fairl comple flow fields can be generated. In particular, in this paper, we describe the induction of cellular motion. This motion can be used to deform and stretch material interfaces and to enhance miing. The MHD stirrer does not utilize an moing parts. The eperimental obserations are in good agreement with theoretical predictions. Kewords minute stirrer, micro mier, magneto hdro dnamics, MHD Comments Postprint ersion. Published in Sensors and Actuators B: Chemical, Volume 79, Issues -3, October 001, pages Publisher UR: This journal article is aailable at ScholarlCommons:

3 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, A Minute Magneto Hdro Dnamic (MHD) Mier Haim H. Bau *, Jihua Zhong, and Mingqiang Yi Dept. Mechanical Engineering and Applied Mechanics Uniersit of Pennslania Philadelphia, PA ABSTRACT A theoretical and eperimental inestigation of a magneto hdrodnamic stirrer is presented. Such a stirrer can be used to enhance miing in micro total analsis sstems. The stirrer utilizes arras of electrodes deposited on a conduit's walls. The conduit is filled with an electrolte solution. B appling alternating potential differences across pairs of electrodes, currents are induced in arious directions in the solution. In the presence of a magnetic field, the coupling between the magnetic and electric fields induces bod (orentz) forces in the fluid. Since the electrodes can be patterned in arious was, fairl comple flow fields can be generated. In particular, in this paper, we describe the induction of cellular motion. This motion can be used to deform and stretch material interfaces and to enhance miing. The MHD stirrer does not utilize an moing parts. The eperimental obserations are in good agreement with theoretical predictions. Kewords: Minute Stirrer, Micro mier, Magneto Hdro Dnamics, MHD * All correspondence should be directed to this author. address: bau@seas.upenn.edu

4 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, INTRODUCTION In recent ears, there has been a growing interest in deeloping minute laboratories on a "chip". Often, in order to facilitate chemical and biological reactions, one needs to mi arious reagents and chemicals. Although the characteristic lengths associated with micro-deices are small, tpicall on the order of 100μm, in the case of large molecules, diffusion alone does not proide a sufficientl rapid means for miing. For eample, at room temperature, mosin's coefficient of diffusion in water is about m /s, and the time constant for diffusion along a length of 100μm is an intolerabl large 10 3 s. Commonl, one encounters onl low Renolds number flows in micro deices, and turbulence is not aailable to enhance miing. Moreoer, often it is not feasible to incorporate moing components such as stirrers into microdeices. Thus, one is forced to look for alternaties in order to make the miing process more efficient. In iew of the aboe, it is not surprising that man inestigators hae studied a ariet of schemes to enhance miing. For eample, Stremler et. al. (000), iu et. al. (000), and Yi and Bau (000) micro-fabricated two and three-dimensional "twisted" conduits in poldimethlsiloane (PMDS), silicon, and low temperature cofired ceramic tapes, respectiel. These "miers" consisted of sequences of short, straight conduits haing rectangular cross-sections. Each conduit formed a 90-degree angle with the preceding one. The bends induced two counter-rotating ortices that deformed material lines and enhanced stirring. For these deices to work effectiel, the Renolds number must be on the order of 10. Unfortunatel, man microfluidic sstems operate at much lower Renolds numbers than 10. Morone et. al. (1991), Selero and Stone (000) and Yi, Bau, and Hu (000) studied stirrers in which traelling waes were induced in a cait's wall(s). These traelling waes induced peristaltic motion in the confined liquid that facilitated fluid circulation. Another idea was put forward b ee, et. al., (000) and others who used two solenoid ales to generate pressure perturbations in the base flow at a desired frequenc and amplitude. Effectie miing has been obsered, albeit at relatiel high Renolds numbers (On the order of 10) and at fairl significant pressure perturbations. In this paper, we describe a noel, simple magneto hdrodnamic mier. Despite their unfaorable scaling (the magnitude of the magnetic forces are proportional to the fluid olume), magnetic forces offer man adantages for microdeices. In man cases, one operates with liquids that are at least slightl conductie such as biological fluids. B patterning electrodes in the flow conduits and subjecting these electrodes to potential differences, one can induce electric currents in the liquid. In the presence of a magnetic force, the currents lead to the generation of a

5 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, orentz force in a direction that is perpendicular to both the magnetic and electric fields. The magnetic field can be induced either internall b coils and soft magnetic material or a magnet embedded inside the deice or eternall with the use of electromagnets or permanent magnets. The idea of using magneto hdrodnamic forces for pumping and propulsion is not new. In preious works pertaining to microfluidic sstems, Jang and ee (000), emoff and ee (000), and Zhong, Yi, and Bau (001), among others, positioned electrodes parallel to the conduit walls. The conduits were filled with electrolte solution. When the electrodes were subjected to a potential difference in the presence of a magnetic field, the resulting orentz forces induced fluid motion along the conduit's ais. In this paper, we use a slightl different idea. Instead of positioning electrodes parallel to the conduit walls, we deposit arras of electrodes on the conduit's surface in the transerse direction. B subjecting these electrodes to aring potential differences in the presence of a magnetic field, we generate forces that drie fluid flow in arious directions in "irtual" wall-less conduits whose geometr is dictated b the positioning of the electrodes. We utilize these idea to generate secondar flows such as ma enhance miing. In the first part of the paper, we describe the theor of operation of such a stirrer. In the second part of the paper, we describe a simple prototpe of such a mier. Although the mier can be fabricated with different substrate materials ranging from silicon to ceramic tapes, we found it particularl adantageous to fabricate our prototpe with low temperature ceramic tapes (TCC). TCCs offer a rapid and inepensie means of fabricating small and moderate quantities of deices.. THEORY The stirrer consists of a liquid-filled conduit haing a rectangular cross-section. Figs. 1 and depict, respectiel, a schematic top iew and a cross-section of the conduit. The -coordinate is parallel to the conduit's walls. Uniforml spaced electrodes (denoted b the letters A, B, C, D, and E in Fig. 1) are deposited perpendicular to the conduit's walls. It is also possible to deposit electrodes on the conduit's top or in different patterns than depicted in Fig. 1. The electrodes are connected alternatel to the positie and negatie poles of a DC power suppl. As a result, electric current of densit J flows in the liquid. The direction of the electric current aries from one location to another as indicated b the hollow arrows in the figure. The liquid is subjected to a uniform magnetic field of intensit ( B ) normal to the conduit's bottom and directed out of the page. The coupling between the 3

6 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, magnetic and electric fields induces a bod (orentz) force, J B, that is perpendicular to both J and B and is directed towards the conduit's side wall. The direction of the force alternates. For eample, in the conduit segment between electrodes C and D, the force is directed in the positie -direction while in the conduit segment between the electrodes B and C, the force is directed in the opposite direction. Consequentl, in the conduit segments BC and CD, the liquid will, respectiel, moe upwards and downwards. The net effect will be the formation of conectie cells or eddies. We will show later that these eddies can deform and stretch material lines and stir the fluid. We denote the conduit's height as h, its width as W and the distance between adjacent electrodes as (see Figs. 1 and ). When the conduit is relatiel long and equipped with a large number of electrodes, we ma assume that the phenomenon is periodic in the -direction. Accordingl, we will analze a single "cell" containing a single electrode, i.e., the region enclosed between the two smmetr lines and containing electrode C in Fig. 1. The width of this cell is. The fluid motion is goerned b the Naier-Stokes equation (Batchlor, 1967) with a magnetic bod force, and the continuit equation: ρ u +ρ(u )u =- P+μ u + J B, (1) t u =0. () In the aboe, ρ is the liquid densit, u is the elocit ector, t is time, P is the thermodnamic pressure, and μ is the shear iscosit. We are neglecting graitational effects. Ohm's law states that J =σ( E +u B ), (3) where σ is the liquid's conductiit. Although this is not alwas the case in micro deices, for simplicit's sake and to allow us to illustrate the phenomenon of interest more clearl, we will assume that and W are of the same order of magnitude and that h<<. This latter assumption allows us to use "lubrication theor" or "Hele-Shaw" approimation (Batchlor, 1967, h p. ). When << 1and to the first order of approimation (i.e., neglecting terms of order h/) the pressure 4

7 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, P~P(, ) is independent of the z-coordinate and the z-component of the elocit, u z =0. Consequentl, the horizontal components of the elocit in the and directions can be epressed as ( ) ( )ˆ, ( )ˆ, ( * * z f e e u + = u ), (4) where 1 ) ( = h z z f. and are, respectiel, unit ectors in the and directions. We substitute ê ê u into equation (1) and integrate the equation oer the z coordinate. The resulting dimensionless momentum equations are: u u u p u u u h t u St Re + + = + +, (5) and ( ) ) ( Re Sign Ha p u h t St = + +. (6) In the aboe, the HS and RHS of the equations represent, respectiel, the rate of change in momentum and the arious forces (pressure, iscous, and magnetic) acting on the fluid. The Sign() function is attached to the magnetic force term to reflect the fact that this force is directed in opposite directions when <0 and >0. The equations are alid oer the domain and W W. The boundar conditions are: 0 ), ( ), ( ), ( ), ( = ± = ± = ± = ± u W W u. (7) In the aboe, we use, respectiel, h, U=σEBh /μ and μu/h as the length, elocit, and pressure scales. Re=U/ν is the Renolds number; ν ω 3 4 h St = is the Stanton number; μ σ hb Ha = is the Hartman number; ω is the frequenc of time-wise ariations in the electric field; and. (8) < = > = ) ( when when when Sign 5

8 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, Since the width of the electrode is tpicall small compared to the other characteristic dimensions, we assume in the analsis that the electrodes hae zero width. This assumption can be relaed at the epense of more cumbersome analsis. In what follows, we find it conenient to epand Sign() into the Fourier series, 4 Sign( ) ~ π n= 0 1 sin 1+ n ( n + 1) π. (9) Tpicall, for non-metallic liquids, Ha<<1. For eample, for saline solutions Ha~10-3. Moreoer, consistent with our assumption of h/<<1, we can ignore the nonlinear terms in equations (5) and (6). Finall, we will consider the time-independent case and set St=0. In terms of the stream function Ψ, ( u Ψ = and Ψ = ), the stead state ersion of equations (5) and (6) can be rewritten compactl as: 4 Ψ 3 Ψ = 6 n= 0 cos ( n + 1) π (10) with the boundar conditions, W W Ψ(, ± ) = Ψ( ±, ) = Ψ(, ± ) = Ψ( ±, ) = 0. (11) Equations (10) and (11) are conenientl soled with Mathematica (Wolfram, 1996). where Ψ = n= 0 A n ( cosh( m ) R cosh( m ) ) 1 n + π 6 ( ) ( 1+ n) 3 + π ( 1+ n) 3 ( 1 n) + cos π, (1) A n 3 6 m = π (1 + n) sinh( m W ) ( 3 + π (1 + n) ) m1 cosh( m W )sinh( m 1 W ) m cosh( m 1 W )sinh( m W ) W sinhm1 m1 R = n, m W sinhm m 1 ( 1+ n) = π, and m1 = 3 + m. The series conerges reasonabl fast at the rate of n -4. When =W=10, Fig. 3 depicts the ariations in the stream function alue, ( Ψ 0,0) Ψ e (0,0))/ Ψ (0,0) ( e,as a function of the number of terms in the series on a semi-log 6

9 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, scale. In the aboe, Ψ e =1.70 denotes the stream function alue when 100 terms are used in the summation. The figure indicates that 9 terms in the series are sufficient to obtain a precision better than 0.1%. It is an eas matter to obtain the two elocit components, u and, and the pressure field from the stream function. In the interest of conciseness, eplicit epressions are not gien here. The Mathematica notebook is aailable from the lead author upon request. When =W=10, Figs. 4, 5, 6, 7 and 8 depict contours of constant stream lines which also represent particle trajectories, the horizontal elocit u(0,) as a function of, the ertical elocit (,0) as a function of, constant pressure contour lines, and the pressure distribution p(, ) as a function of and. As is eident from Figs. 4-6, the magnetic forces induce circulator motion in the cell. Somewhat surprisingl, the ertical elocit (,0) initiall increases as increases, attains a maimum at ~1.39, and then decreases again. This decrease in the - W component of the elocit is a result of the increase of pressure at the stagnation point, (, ) = (, ), as is eident in Figs. 7 and 8. Since the absolute pressure cannot be uniquel determined and it is of no consequence in the case of incompressible flow, we normalized the pressure so that p(0,0)=0. In order to ealuate the performance of the deice as a stirrer, we compute the stretching rate of material interfaces. Consider a stirrer that is filled with two immiscible fluids; one fluid initiall occupies the region <0 and the other fluid occupies the region >0. =0 forms an interfacial line between the two fluids. We track the deformation of this interface. To this end, we track particles that are initiall located on =0. Using a 4 th order Runge-Kutta procedure, we integrate the adection equations, d u(, ) d = and = (, ), (13) dt dt with the initial conditions /<(0)</, (0)= 0 and (0)=0. Here, we use μ/(σebh) as the time scale. Although all these particles follow the streamlines, the do so at different rates. As a result, the interface is being stretched and deformed. The locations of the interface at arious times t=0, 10, 0, 30, 40, and 50 are depicted in Fig. 10. As time goes b, the interface elongates and deforms. 7

10 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, Fig. 11 quantifies the stretching of the interface. The figure depicts the normalized length ( I l( t) = ) as a function of time. In the aboe, l(t) is the length of the interface at time t. The smbols and the solid line correspond, respectiel, to computational data and a line of best fit. The data correlates well with the formula: I~ (t+1) ln(t+1). (14) In other words, the length of the interface increases slightl faster than a linear function of time. This is much better than the rate of increase due to diffusion alone (t 1/ ) but falls short of the eponential rate ehibited b chaotic adection. 3. EXPERIMENTS To illustrate that the ideas articulated here can, indeed, be put into practice, we fabricated a crude MHD stirrer. Although the stirrer can be fabricated with arious substrate materials ranging from silicon to polmers, we chose to construct our prototpe with low temperature co-fired ceramic tapes (TCC). The tapes facilitate eas, inepensie, laered manufacturing. The proide a rapid platform for the integration of passie electronics and fluidics. In their pre-fired (green) state, the ceramic tapes consist of oide particles, glass frit, and organic binder (that can be made from photo-resist). The tapes are tpicall cast with a thickness starting at 40μm. The pre-fired tapes can be machined b laser, milling, and photolithograph (when the binder is photo-resist). Metallic paths can be either printed or processed photolithographicall to form electrodes, resistors, conductors, and thermistors. Conduit sizes ma range from ~10μm to a few millimeters. Upon firing, the organic binder burns out, the oideparticles sinter, and the tapes solidif. Man tapes (>80) can be stacked together, aligned, laminated, and co-fired to form monolithic structures with comple, three-dimensional mazes of fluidic conduits, electronic circuits, and electrodes. Glass windows and other materials can be readil attached to the tapes to facilitate optical paths. Multiple laers of coils can be embedded in the tapes to generate magnetic fields. For additional details, see Bau et. al. (1998) and Kim et. al. (1998). The prototpe of the stirrer was fabricated with DuPont, TCC 951AT tapes that hae a nominal thickness of ~50μm. Rectangles of desired size were blanked from the raw material (see Fig. 1). 0μm thick and 700μm wide DuPont 5734 gold paste was hand-printed at distances 3.3mm apart to form the electrode pattern on laer (1). 8

11 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, Although it is possible to control each electrode independentl, here we connected three electrodes to a single conductor and two other electrodes to a second conductor. Fie additional laers of tpe (the laer denoted as in Fig. 1) were first laminated and then machined with a numericall controlled milling machine to accommodate an inner cait and ias (the circular holes) for electric leads. The ias were filled with DuPont 6141 paste and terminated with soldering pads (DuPont 6146). The cait can be encapsulated with another laer of tape (i.e., laer 3). In the actual eperiments, to facilitate flow isualization, we left the cait uncapped. Subsequent to the machining, the arious laers were aligned, stacked, laminated, and co-fired to form a monolithic block. The postfired cait (that was formed from 5 laers of tpe ) was 1mm deep, 4.7mm wide, and.3 mm long. The cross-section of the assembled deice is shown in Fig. 13. In our eperiments, we positioned a rectangular, permanent Neodmium magnet (Edmund Scientific, serial number CR81-37, load-carring capacit of 1-15lb) under the deice. The dimensions of the magnet were significantl larger than the cait's dimensions so that an approimatel uniform magnetic field resulted. The magnetic field can also be generated b other means such as an eternal electromagnet or coils embedded in the ceramic tapes. In the latter case, soft magnetic material such as permallo can be encapsulated in the tapes to intensif the field. In the eperiments, we filled the cait with deionized water. The resistance between the electrodes was measured in situ and the conductance was estimated to be σ= (Ω m) -1. This conductance accounts for possible dissolution of electrode materials in the water. The leads were connected to a power suppl (HP 603A) that was operated at constant oltage mode. Due to the etremel low currents, it was not possible to operate this power suppl at a constant current mode. The er low specific conductiit liquid allowed us to keep the fluid elocit er low and carr out obserations using relatiel primitie means. With a sringe, we applied a thin trace of de (Water Soluble Fluorescent iquid De, Model Red, Cole Palmer Instrument Compan, Niles, Illinois, USA) across the cait (Fig. 14) and then applied 4V across the electrodes. The current was on the order of μa, and well below the resolution leel of our power suppl (10mA). As a result of the application of a DC potential difference across the electrodes, clearl isible fluid flow was induced in the cait. The motion consisted of rotating cells with the fluid moing up in one interal between two electrodes and down in the adjacent interal. Figs. 15a, b, c, and d depict the deformation of the de induced b this fluid motion. Depending on the polarit of the electrodes, the de either moed upwards or downwards (Fig. 15a). This well organized motion could not possibl hae been caused b diffusion, and it clearl demonstrates the 9

12 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, presence of upward and downward flows in different sections of the cait. After a few seconds, the polarit of the electrodes was reersed. Since diffusion is relatiel slow and the flow is at a relatiel low Renolds number (creeping flow), the de "retracted its steps" in almost a reersible fashion (Fig. 15b) and then started deforming in the opposite direction (Fig. 15c). When the process was allowed to continue for some time, the de traced the conectie cells in good qualitatie agreement with the theoretical predictions of the preious section. In order to carr out an order of magnitude comparison with theoretical predictions, we computed the approimate time (T) that it takes a segment of de located midwa between the electrodes to moe from location = 0.4W to location =0.4W. In order to carr out this calculation, we took into consideration the fact that the ertical component of the elocit aries spatiall, i.e., T = 0.4W d 0.4W (, ). Numerical integration leads to T~9 dimensionless time units. The eperimentall measured time corresponded to about 40 time units. This relatiel large discrepanc ma hae been caused because we oerestimated the strength of the magnetic field. Moreoer, there are a number of differences between the mathematical model and the eperiment. For eample, the model assumes that the current in the liquid is uniform while in realit there was a more complicated current distribution gien the fact that the electrodes were located on the conduit's bottom. If we were to use saline solution instead of de-ionized water, the process would hae occurred at least ten times faster. At the er low currents that we used in our eperiments, we did not obsere an bubble generation. Bubble generation and electrode deterioration are likel to be a problem when higher oltages or higher conductiit solutions are used. To preent these undesirable effects, one can emplo AC currents. B appropriate snchronization of the electromagnetic field with the AC current, the direction of the orentz force would remain unaltered (emoff and ee, 000). 4. CONCUSIONS We hae demonstrated in theor and eperiment that magneto-electric forces can be used to generate comple motions in electroltic solutions. We hae applied time-independent forces to induce cellular conection in a conduit. The cellular motion stretches and deforms material lines. This process enhances miing. The rate of deformation as a function of time is slightl faster than linear. 10

13 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, Photolithograph and screen-printing allow us to form complicated electrode patterns. The interpla between the magnetic and the electric fields proides one with the means of controlling locall the direction and magnitude of the olumetric forces and, in turn, inducing desired flow patterns. In other words, MHD proides the designer of microfluidic sstems with a great amount of design fleibilit. In this paper, we hae described just one simple eample of inducing cellular conection in a conduit. The net challenge is to deise time-dependent ariations in the magnetic forces and/or electric field that will allow us to obtain a faster, perhaps eponential, rate of stretching of material lines as is common in the case of chaotic adection. ACKNOWEDGMENTS The work described in this paper was supported, in part, b DARPA through grant N to the Uniersit of Pennslania. DuPont supplied us freel with materials. REFERENCES Batchlor, G., K., 1967, An Introduction to Fluid Dnamics, Cambridge Uni. Press. Bau, H., Ananthasuresh, G., K., Santiago-Ailes, J., J., Zhong, J., Kim, M., Yi, M., Espinoza-Vallejos, P., 1998, "Ceramic Tape-Based Sstems Technolog." Micro-Electro-Mechanical Sstems (MEMS), DSC-Vol. 66, Jang J., ee, S., S., 000, Theoretical and Eperimental Stud of MHD (Magnetohdrodnamic) Micropump, Sensors and Actuators A, 80, Kim, M., Yi, M., Zhong, J., Bau, H., Hu, H., Ananthasuresh, G.K., 1998, "The Fabrication of Flow Conduits in Ceramic Tapes and the Measurement of Fluid Flow Through These Conduits." Micro-Electro-Mechanical Sstems (MEMS), DSC-Vol. 66, ee, Y-K., Tabeling, P., Shih, C., Ho, C-M, 000, Characterization of MEMS-Fabricated Miing Deice, ASME, IMECE 000, MEMS, , Orlando, Florida. emoff, A., V., ee, A., P., 000, An AC Magentohdrodnamic Micropump, Sensors and Actuators B, 63, iu, R. H., M. A. Stremler, K. V. Sharp, M. G. Olsen, J. G. Santiago, R. J. Adrian, H. Aref and D. J. Beebe, Passie Miing in a Three-Dimensional Serpentine Microchannel, Journal of Microelectromechanical Sstems, in press, 000. Morone, R. M., White, R., M., Howe, R., T., 1991, Ultrasonicall Induced Microtransport, Proceedings IEEE Workshop Micro Electro Mechanical Sstems, MEMS 95 Amsterdam, Selero, K., and Stone, H., A., 000, Peristalticall Drien Flows for Micro Miers, to be published in Phsics of Fluids. 11

14 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, Stremler, M. A., M. G. Olsen, R. J. Adrian, H. Aref, and D. J. Beebe, Chaotic Miing in Microfluidic Sstems, Solid-State Sensor and Actuator Workshop, Hilton Head, SC, June 4-8, 000. Wolfram, S., 1996, Mathematica, Cambridge. Yi, M. and Bau, H.H., 000, The Kinematics of Bend-Induced Miing in Micro-Conduits, ASME, IMECE 000, MEMS, , Orlando, Florida. Yi, M., Bau, H.H., and Hu, H., 000, A Peristaltic Meso-Scale Mier, ASME, IMECE 000, MEMS, , Orlando, Florida. Zhong, J., Yi, M., and Bau, H., H., 001, A Magnetohdrodnamic Pump Fabricated with Ceramic Tapes, submitted for publication. 1

15 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, IST OF CAPTIONS 1. A schematic (top-iew) depiction of the stirrer. The stirrer consists of a capped conduit. The electrodes, denoted with the capital letters A, B, C, D, and E; are deposited on the conduit's bottom and the are aligned perpendicular to the conduit's walls. The electrodes are connected alternatel to the positie and negatie poles of a power suppl. The -coordinate is aligned along the conduit's ais. The magnetic field (denoted b dots) is normal to and points out of the page.. A cross-section of the conduit. 3. An estimate of the relatie error resulting from truncating the infinite series after N terms. =W= Contours of constant stream function. The contours represent the particles' trajectories. =W= The horizontal component of the elocit, u(0,), is depicted as a function of. =W= The ertical component of the elocit (,0) is depicted as a function of. =W= Contours of constant pressure lines. =W= The pressure distribution p(, ) as a function of and. W== (,0) is depicted as a function of. W=. 10. The location of the interface at arious dimensionless times: t=0, 10, 0, 30, 40, and 50. W== The length of the interface as a function of the normalized time. 1. The arious laers of tapes that hae been used for the construction of the prototpe of the stirrer. aer 1 contains the electrodes. aer contains the stirrer cait and ertical ias for electrical connections. A few laers of tpe were used to obtain a cait of desired height. aer 3 sered as a coer plate. The figure is not drawn to scale. 13. A cross-section of the MHD stirrer. The laer numbers are cross-referenced with Fig. 1. Item (4) is a permanent magnet. The figure is not drawn to scale. 14. A top iew of the stirrer fabricated with ceramic tapes. The cait and the transerse electrodes are isible. At the beginning of the eperiment, a thin line of de is laid across the cait. 15. The deformation of a de line resulting from the application of the orentz forces (a). After a few seconds, the polarit of the electrodes and the direction of the orentz force are reersed, and the de returns to its initial position (b). As a result of the reersal in the direction of the orentz force, the de continues to deform in the opposite direction (c). When the process is allowed to continue for some time, the formation of eddies become apparent (d). 13

16 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, A B C D E J W V B Fig. 1: A schematic (top-iew) depiction of the stirrer. The stirrer consists of a capped conduit. The electrodes, denoted with the capital letters A, B, C, D, and E, are deposited on the conduit's bottom, and the are aligned perpendicular to the conduit's walls. The electrodes are connected alternatel to the positie and negatie poles of a power suppl. The -coordinate is aligned along the conduit's ais. The magnetic field (denoted b dots) is normal to and points out of the page. z B r Electrode h W Fig. : A cross-section of the conduit. 14

17 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, E+00 ( Ψ 0,0) Ψ e (0,0))/ Ψ (0,0) ( e 1.E-0 1.E-04 1.E N Fig. 3: An estimate of the relatie error resulting from truncating the infinite series after N terms. =W=10. Fig. 4: Contours of constant stream function. The contours represent the particles' trajectories. =W=10. 15

18 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, u(0,) Fig. 5: The horizontal component of the elocit, u(0,), is depicted as a function of. =W=10. (,0) Fig. 6: The ertical component of the elocit (,0) is depicted as a function of. =W= Fig. 7: Contours of constant pressure lines. =W=10. 16

19 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, Fig. 8: The pressure distribution p(, ) as a function of and. W== (,0) Fig. 9: (,0) is depicted as a function of. W=. 17

20 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, t=0 t=10 t=0 t=30 t=40 t=50 Fig. 10: The location of the interface at arious dimensionless times, t=0, 10, 0, 30, 40, and 50. W==10. 18

21 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, I t Fig. 11: The length of the interface as a function of the normalized time. 19

22 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, Fig. 1: The arious laers of tapes that hae been used for the construction of the prototpe of the stirrer. aer 1 contains the electrodes. aer contains the stirrer cait and ertical ias for electrical connections. A few laers of tpe were used to obtain a cait of desired height. aer 3 sered as a coer plate. The figure is not drawn to scale. 0

23 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, mm 1mm 1 4 Fig. 13: A cross-section of the MHD stirrer. The laer numbers are cross-referenced with Fig. 1. Item (4) is a permanent magnet. The figure is not drawn to scale. Fig. 14: A top iew of the stirrer fabricated with ceramic tapes. The cait and the transerse electrodes are isible. At the beginning of the eperiment, a thin line of de is laid across the cait. 1

24 Bau, H., H., Zhong, J., and Yi, M., 001, A Minute Magneto Hdro Dnamic (MHD) Mier, Sensors and Actuators B, 79/-3, a b c Fig. 15: The deformation of a de line resulting from the application of the orentz forces (a). After a few seconds, the polarit of the electrodes and the direction of the orentz force are reersed, and the de returns to its initial position (b). As a result of the reersal in the direction of the orentz force, the de continues to deform in the opposite direction (c). When the process is allowed to continue for some time, the formation of eddies becomes apparent (d). d