Keywords:ductivity uids and describe various applications of MHD such as uid pumping, ow con-trol in uidic networks, uid stirring and mixing, circular liquid chromatography, thermalreactors, and microcoolers.

2008 Elsevier Ltd. All rights reserved.

growal, ph

and Lee, 2000; Lemoff and Lee, 2000; Huang et al., 2000; Bau, 2001; Sadler et al., 2001; Zhong et al., 2002; Bau et al.,2002, 2003; Sawaya et al., 2002; West et al., 2002, 2003; Ghaddar and Sawaya, 2003; Bao and Harrison, 2003a,b; Eijkelet al., 2004; Wang et al., 2004; Arumugam et al., 2005, 2006; Qian and Bau, 2005b; Homsy et al., 2005, 2007; Affanni andChiorboli, 2006; Aguilar et al., 2006; Kabbani et al., 2007; Patel and Kassegne, 2007; Duwairi and Abdullah, 2007; Ho,2007), stirrers (Bau et al., 2001; Yi et al., 2002; Qian et al., 2002; Gleeson and West, 2002; Xiang and Bau, 2003; Gleeson

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* Corresponding author. Tel.: +1 215 898 8363; fax: +1 215 573 6334.E-mail address: bau@seas.upenn.edu (H.H. Bau).

Mechanics Research Communications 36 (2009) 1021

Contents lists available at ScienceDirect

Mechanics Research Communicationsdoi:10.1016/j.mechrescom.2008.06.013plant that integrates common laboratory procedures ranging from ltration and mixing to separation and detection. The var-ious operations are done automatically within a single platform. To achieve these tasks, it is necessary to propel, stir, andcontrol uids. Since in many applications, one uses buffers and solutions that are electrically conductive, one can transmitelectric currents through the solutions. In the presence of an external magnetic eld, the interaction between the electriccurrents and magnetic elds results in Lorentz body forces, which, in turn, can be used to propel and manipulate uids. Thisis the domain of magneto-hydrodynamics (MHD).

The application of MHD to pump, conne, and control liquid metals and ionized gases is well-known (Woodson and Mel-cher, 1969; Davidson, 2001). The application of MHD to weakly conductive electrolyte solutions is somewhat more compli-cated due to electrodes electrochemistry. Recently, various MHD-based microuidic devices including micro-pumps (JangMicrouidicsLab-on-a-chipMagneto-hydrodynamicsMHDLorentz forceMicro-pumpChaotic StirrerFluid ManipulationMixingChaos

1. Introduction

In recent years, there has been anology, chemical reactors, and medicing interest in developing lab-on-a-chip (LOC) systems for bio-detection, biotech-armaceutical, and environmental monitoring. LOC is a minute chemical processingMagneto-hydrodynamics based microuidics

Shizhi Qian a, Haim H. Bau b,*aDepartment of Aerospace Engineering, Old Dominion University Norfolk, VA 23529-0247, USAbDepartment of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, 220 South 33rd Street, Philadelphia, PA 19104-6315, USA

a r t i c l e i n f o

Article history:Received 24 March 2008Received in revised form 19 June 2008Available online 4 July 2008

a b s t r a c t

In microuidic devices, it is necessary to propel samples and reagents from one part of thedevice to another, stir uids, and detect the presence of chemical and biological targets.Given the small size of these devices, the above tasks are far from trivial. Magnetohydro-dynamics (MHD) offers an elegant means to control uid ow in microdevices without aneed for mechanical components. In this paper, we review the theory of MHD for low con-

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tion. Mspecie2005b

volumfor mo

We ad

tions

where

S. Qian, H.H. Bau /Mechanics Research Communications 36 (2009) 1021 11constant (F = 96484.6 C/mol), R is the universal gas constant, T is the absolute temperature of the electrolyte solution, N is thetotal number of species present in the electrolyte solution, and u B is the induction term.

Under steady state conditions,

r Nk 0; k 1; . . . ;N 5and the current ux

J FXNk1

zkNk: 6

The potential in the electrolyte solution is governed by the local electroneutrality condition:

XNk1

zkck 0: 7

Electroneutrality holds everywhere except in the thin Debye screening layer next to solid surfaces. Although the Debyescreening layer is only a few nanometers in thickness, the potential drop across this layer can be signicant.

The boundary conditions associated with Eqs. (5) and (7) consist of zero normal ux of each of the species at insulatingwalls; given ionic concentrations at the conduits inlet; normal ux dominated by convective ux at the outlet of the conduit(outow boundary condition); and BulterVolmer equation (Bard and Faulkner, 2000) at the surfaces of the electrodes.

The NP Eq. (5) and the local electroneutrality condition (7) constitute a well-dened and widely used approximation forelectrochemical transport phenomena. Witness that the models for the uid motion (Eqs. 1 and 2) and the ionic mass trans-ck is the molar concentration, Dk is the diffusion coefcient, and zk is the valance of the kth ionic species. F is FaradaysNk uck Dkrck zk RT FckrV u B; k 1; . . . ;N; 4for the various ionic species. The ionic ux density of species k is

Dkprovides us with a relationship between the current ux and the electric potential V. In the above, r is the electric conduc-tivity of the solution. The second term in RHS of Eq. (3) represents current induction caused by the motion of a conductor inan electric eld. Strictly speaking, Eq. (3) applies only to liquid metals in which the current is transported by electrons.

In the case of electrolyte solutions, a more accurate model for the current ux consists of the NernstPlanck (NP) equa-J rrV u B; 3qot

u ru J Brp lr u; 2

where q and l are, respectively, the liquids density and viscosity; t is time; J is the electric current ux; p is pressure; and Bis the magnetic eld intensity. In the above, we assume that the liquids magnetic permeability is sufciently small so thatthe magnetic eld inside the uid can be approximated with B. The Ohms law,opt here the notation that bold letters represent vectors. The momentum equation is

ou

22. Theory

We consider an incompressible, viscous uid. The velocity vector u satises the continuity equation

r u 0: 1In this paper, we review the basic theory of MHD as applied to low conductivity solutions and describe various applica-tions of MHD such as pumps, integrated uidic networks, stirrers, liquid chromatographs, thermal cyclers, and microcoolers.etric body force which scales unfavorably as the conduits dimensions are reduced. Thus, MHD is appropriate mostlyderate conduit sizes with characteristic dimensions on the order of 100 lm or larger.The advantage of MHD compared to electroosmosis is operation at relatively small electrode potentials, typically below1 V, and much higher ow rates as long as the conduits dimensions are not too small. The disadvantage of MHD is that it is aoreover, AC operation induces parasitic eddy currents that may lead to excessive heating. DC operation with RedOxs that undergo reversible electrochemical reactions alleviates many of the disadvantages of DC MHD (Qian and Bau,; Arumugam et al., 2006; Kabbani et al., 2007).et al., 2004; Qian and Bau, 2005a), networks (Bau et al., 2002, 2003), heat exchangers (Sviridov et al., 2003; Singhal et al.,2004; Duwairi and Abdullah, 2007), and analytical and biomedical devices (Leventis and Gao, 2001; West et al., 2002,2003; Bao and Harrison, 2003a; Lemoff and Lee, 2003; Eijkel et al., 2004; Clark and Fritsch, 2004; Homsy et al., 2007; Gaoet al., 2007; Panta et al., 2008) operating under either DC or AC electric elds have been designed, modeled, constructed,and tested. The DC operation is often adversely impacted by the electrodes electrochemistry leading to bubble formationand electrode corrosion. These problems are partially solved with the use of AC elds. AC operation requires, however,the use of electromagnets instead of the permanent magnets that are used in DC operation, which increases power consump-

port (Eqs. 5 and 7) are strongly coupled. The ow eld affects the mass transport due to the presence of the convective ux inexpression (4). On the other hand, the ionic mass transport affects the current density J, which, in turn, affects the ow eldthrough the Lorentz body force J B. Therefore, one needs to solve simultaneously the full mathematical model, which con-sists of the continuity and NavierStokes equations, the set of NP equations, and the local electroneutrality condition withthe appropriate boundary conditions, to obtain the ow eld, the ionic species concentrations, and the potential of the elec-trolyte solution. Due to space limitations, we will not be able to discuss here the solution of the full mathematical model inany detail. The numerical solutions of the 2D and 3D systems were reported, respectively, in Qian and Bau (2005b) and Kab-bani et al. (2007).

The energy equation is

qcpoTot u rT

jr2T J J

rU; 8

where j and cp are, respectively, the thermal conductivity and the heat capacity of the uid. U = 2l e e is the viscous dis-sipation, where e 12 ru ru T is the strain rate tensor.

We non-dimensionalize the equations of motion using the conduits height H as the length scale in the x and z directionsand the widthW as the length scale in the y-direction (Fig. 1). The magnetic eld intensity scale is the maximum eld inten-sity B. The electric potential scale is the maximum voltage difference in the solution V. It is perhaps important to point out

and

cover(e 1bound

In the

12 S. Qian, H.H. Bau /Mechanics Research Communications 36 (2009) 1021y

W

H JB

Fig. 1. A schematic diagram of the MHD pump. Two electrodes with a potential difference DV are deposited along the opposing walls of the conduit. Theright gure depicts a cross-section of the conduit. The conduit is lled with an electrolyte solution and exposed to a uniform magnetic eld of intensity B.zWhen the Hartman number is large, the velocity scales like Ha . In other words, the dimensional velocity is on the order ofV/(BW). The magnitude of the velocity is dictated by a balance between the applied and induced (Hartman break) electriccurrents. The velocity prole uxz H2a e dVdy dpdx

is nearly at along most of the conduits height with two boundary

Vuxz H2a edVdy

dpdx

1 cosh Haz cosh Ha2

12< z