Magnetohydrodynamic Electroosmotic Flow with Patterned Charged Surface and Modulated Wettability in a Parallel Plate Microchannel∗

Na Hao (郝娜) and Yong-Jun Jian (菅永军)

School of Mathematical Science,Inner Mongolia University,Hohhot 010021,China

Abstract This paper investigates the magnetohydrodynamic (MHD) electroosmotic flow (EOF) of Newtonian fluid through a zeta potential modulated parallel plate microchannel with patterned hydrodynamic slippage.The driven mechanism of the flow originates from the Lorentz force generated by the interaction of externally imposed lateral electric field Ey and vertical magnetic field Bz and electric field force produced by an externally applied electric field Ex.It is assumed that the wall zeta potential and the slip length are periodic functions of axial coordinate x,an analytical solution of the stream function is achieved by utilizing the method of separation of variables and perturbation expansion.The pictures of streamlines are plotted and the vortex configurations produced in flow field due to patterned wall potential and hydrodynamic slippage are discussed.Based on the stream function,the velocity field and volume flow rate are obtained,which are greatly depend on some dimensionless parameters,such as slip length ls,electrokinetic width λ,the amplitude δ of the patterned slip length,the amplitude m of the modulated zeta potential and Hartmann number Ha.The variations of velocity and volume flow rate with these dimensionless parameters are discussed in details.These theoretical results may provide some guidance effectively operating micropump in practical nanofluidic applications.

Key words: magnetohydrodynamic (MHD) flow,electroosmotic flow (EOF),modulated hydrodynamic slippage,patterned charged surface

1 Introduction

Nowadays,microfluidic device applications are increasing in the fields of chemical separation,medical diagnostics,material synthesis,energy harvesting,and others.[1−4]Normally,surface tension,pressure gradient,surface acoustic wave,electric field,magnetic field and their suitable combination have been implemented to manipulate microfluidic flow.[5−8]Electroosmotic flow(EOF),defined as the motion of fluids in narrow confinements under the action of an axial electric field in the presence of an electric double layer (EDL) near the walls,has wide spectrum of applications.[9−11]In order to minimize the Joule heating effects in EOF,magnetodydrodynamic (MHD) micropump has been proposed.MHD flow is driven by Lorentz force generated by imposing a lateral electric field and a vertical magnetic field.[12]Over recent years,many researchers have investigated the MHD flow in microchannel.Rivero and Cuevas[13]considered the influence of slip condition on the MHD flow in a rectangular microchannel.Jianet al.[14]analytically studied the transient rotating MHD flow for both DC and AC electric fields through a parallel microchannel.Si and Jian[15]studied MHD flow of non-Newtonian fluids through two corrugated walls.In order to obtain augmented flow rates,the more general requirement of studying the interaction between EOF and MHD flow is needed.MHD EOF is produced by combining the Lorentz force and electric force in axial direction.Chakraborty and Paul[16]analyzed the MHD EOF in narrow microchannels and obtained optimized flow rates.Driven by combination of electroosmotic,pressure and magnetic forces,unsteady MHD EOF of incompressible viscous fluid through a circular tube was developed with the Caputo Fabrizio time fractional derivative by Abdulhameedet al.[17]Chakrabortyet al.[18]analyzed thermal behavior of MHD EOF in narrow channels taking viscous dissipation and Joule heating into account under constant wall heat flux.Abdulhameedet al.[19]investigated the MHD EOF flow acted by an arbitrary time-dependent pressure gradient,electric field force and Lorentz force and heat transfer through a cylindrical microchannel.Jian[20]investigated heat transfer and entropy generation of MHD EOF flow through a slit microchannel.Xie and Jian[21]studied entropy generation of two-layer MHD EOF flow through microparallel channels.Sarkaret al.[22]studied heat transfer of MHD EOF with interfacial slip.

Recently,many researchers have investigated the influences of patterned wettability and modulated surface charges on micro-and nanofluidic flows.[23−24]Hendyet al.[25]investigated the effect of patterned variations in slip length on micro-and nanofluidic flows.Shitet al.[26]investigated EOF of non-Newtonian power-law fluid in a micro-channel in the presence of Joule heating effects and examined the effects of thermal radiation and velocity slip conditions.Zhao and Yang[27]analyzed the EOF of electrolytic solutions in a microchannel with patterned hydrodynamic slippage on channel walls.Pereira[28]studied the effect of patterned slip on the flow of pressure driven power-law fluids.It was shown that for shear-thickening fluids,patterned slip would induce significant transverse flows comparable in size to those produced for Newtonian fluids.Datta and Choudhary[29]investigated the EOF through zeta potential patterned nanochannels under slip boundary conditions with constant slip length.Mayuret al.[30]analyzed the generation of flow vortices in nanoscale confinements under the combined effects of patterned surface charge and substrate wettability.By using molecular dynamics simulations,they theoretically achieved the most efficient microfluidic mixing based upon an optimum patterning frequency for a desired volumetric flow rate.Chaudhuryet al.[31]analyzed the consequences of substrate wettability on the convective heat transfer characteristics in microchannel flows.Ghoshet al.[32]reported the influence of finite EDL thickness on the EOF over patterned charged surfaces by solving the fully coupled Poisson-Nernst-Planck and Navier-Stokes equations.The effect of imposing shear flow on a charge-modulated EOF was theoretically investigated by Wei.[33]The flow structures exhibit either saddle points or closed streamlines,depending on the relative strength of an imposed shear to the applied electric field.Ghosh and Chakraborty[34]focused on evaluating the induced streaming electric field along the orthogonal directions in a narrow fluidic confinement under patterned surface wettability and modulated surface charges.They further demonstrated that the presence of anisotropic modulations on the channel walls gave rise to considerable off-diagonal effects.Ghosh and Chakraborty[35]studied the alterations of EOF because of interactions between patterned wettability gradients on microfluidic substrates and modulated surface charge distributions.They proved that it is possible to control pattern vortices occurring in the fluidic confinement by using such intricate coupling.Shit and Ranjit[36]studied effects of different zeta potential and wall slip on MHD EOF through a peristaltically induced micro-channel.They observed that the different zeta potential plays an important role in controlling fluid velocity.

Comparing the above recent studies,we can know that some researchers studied patterned charge surface or modulated wettability on EOF or on MHD,some scholars analyzed both conditions on EOF but no one considers the patterned charge surface and modulated wettability simultaneously on MHD EOF.Therefore,motivated by the concept of multiple field driven microfluidic flow with charge and slip modulated surfaces,the objective of this article is to study the MHD EOF with patterned charged surface and modulated wettability through a parallel plate microchannel.The analytical solutions of stream function,flow velocities and volume flow rates are obtained by using the method of separation of variables and superposition principle.The dependence of velocities and volume flow rates on several nondimensional parameters will be determined graphically.These theoretical results may provide some guidance effectively operating micropump in practical nanofluidic applications.

2 Formulation of the Problem

2.1 Physical Model and Explanation of the Problem

As shown Fig.1(a),we investigate MHD EOF in rectangular microchannels taking patterned surface potential and modulated slip length on walls into consideration.A Cartesian coordinate system (x∗,y∗,z∗) is established with the original point is located at the bottom of channel whose height ish,width is W and length isL.The upper plate is stipulated atz=hand the lower plate is stipulated atz=0.An external electric fieldExis imposed along positivex∗-axis,electric fieldEyis applied alongy∗-axis direction and a uniform magnetic fieldBzis implemented perpendicular to upper and lower walls of the rectangular channel,i.e.,along the positivez∗-axis direction.The MHD EOF is driven by the axial Lorentz force generated by the interaction ofEyandBzand electric field force resulted fromEx.Assuming both the height of the channelhand the widthWto be much smaller than the lengthL,i.e.,L ≫handL ≫W,and the ratioW/hof width to height is very large,so the flow in the rectangular channel can be simplified as the one in two parallel plate microchannels(seen Fig.1(b)).Moreover,patterned zeta potentials on the two walls are the same and can be written as

whereζis constant wall electrical potential,mis amplitude of the modulated part of the surface potential andq∗is patterned frequency.Furthermore,modulated hydrodynamic slip conditions on the upper and lower walls respectively yield

whereu∗is axial velocity component,is constant slip length on walls,δis patterned amplitude of the slip length andnis asymmetry factor of patterned interfacial slip.It needs to note that we have assumed the patterned frequencies of zeta potential and slippage on the walls are identical.

Fig.1 Schematic of the physical modeling.(a)3D view of the MHD EOF micropump and(b)Duct’s cross-section of the MHD EOF micropump.

2.2 Electrical Potential Distribution

In the present work,it is notable that the lateral forces (i.e.,the Lorentz and electroosmotic forces inydirection) are neglected under the small magnitude of lateral electric field.So the influence of transverse flow weakens,i.e.v ≈0.[21]Firstly,considering a symmetric electrolyte solution,the electrical potentialψ∗satisfies Poisson-Boltzmann equation.It is assumed that the electrical potential is much smaller than the thermal potential,i.e.|ψ∗| ≤kbTa/zve,the Debye – Huckel approximation sinh(zveψ∗/kbTa)≈zveψ∗/kbTacan be used to simplify the Poisson-Boltzmann equation as[37−38]

where

is the inverse of the Debye length,εis the permittivity of the electrolyte solution,n0is ion density of bulk solution,zvis the ion valence,eis the electron charge,kbis the Boltzmann constant andTais the absolute temperature.Introducing the following dimensionless variables

the dimensionless form of Eq.(4) is

Since Eq.(4) is linear,so the boundary condition (1) can be decompose into constant and modulated zeta potential portions based on the superposition principle.The constant wall zeta potential condition is given by

and the modulated zeta potential condition can be written as

Corresponding dimensionless boundary conditions of Eqs.(7) and (8) are respectively

By utilizing the method of separation of variables,the solutions of Eq.(6) under the boundary conditions of Eqs.(9) and (10) are given by

for the constant wall zeta potential condition and

for the modulated zeta potential condition,hereB=(λ2+q2)1/2.

2.3 Streaming Function and Flow Fields of the MHD EOF

In present analysis,the flow is two-dimensional due to the patterned zeta potential and modulated wettability.Affected by the magnetic field and electric fields,the momentum and continuity equations of incompressible viscous electrolyte solution can be expressed as[39]

whereρis density of the fluid,µis dynamic viscosity of the fluidis velocity,is Lorentz force andis electric current density obtained from Ohm’s law.[40]The expression of electrical conductivity of the electrolyte solution isσe=Kcell/R,Kcellis conductivity constant,Ris resistance.Andρe=−ε∆∗ψ∗is charge density.Therefore,the component form of twodimensional continuity equation and Cauchy momentum equation can be expressed as

Except for the patterned slip boundary conditions(2)and(3) yielded by axial velocityu∗,vertical velocity componentw∗of Eqs.(15),(16),and (17) should satisfy the following no-penetration conditions

We would adopt the following non-dimensional variables

dimensionless governing equations and boundary conditions are rewritten as

where the parametersls,SandHaare non-dimensional slip length,normalized electric field strength in theydirection and Hartmann number respectively.From Eqs.(21)and(22),eliminating the pressure by taking the derivatives of them regarding tozandxrespectively,and then subtracting the two derived equations,we have

Defining a stream function based on Eq.(20)

Eq.(26)and boundary conditions(23),(24),and(25)can be expressed as

Similar to the solving process of electrical potential,the form solution of stream function can be expressed as

whereϕ0andmϕ1are the solutions of Eq.(28) when its right hand side term is replaced by the constant and modulated zeta potential solutions (11) and (12).Evidently,Eq.(32)is the stream function while the electrical boundary condition is replaced by the combination of Eqs.(9)and 910).Furthermore,taking the amplitude of the patterned slip lengthδas a small parameter,andϕ0areϕ1expanded in the form of power series by using a regular perturbation approach as

When she came back, she had hundreds of things to talk about; but the most beautiful, she said, was to lie in the moonlight, on a sandbank, in the quiet sea, near the coast, and to gaze on a large city11 nearby, where the lights were twinkling like hundreds of stars; to listen to the sounds of the music, the noise of carriages, and the voices of human beings, and then to hear the merry bells peal24 out from the church steeples; and because she could not go near to all those wonderful things, she longed for them more than ever

In the following subsections,the solutions ofϕ0andϕ1will be derived in detail.

(i)Case with constant surface potential and patterned interfacial slip

Substituting Eq.(11) andϕ0into Eq.(28) yields

Inserting the expanded form Eq.(33) ofϕ0into Eq.(35)and related boundary conditions(29),(30),and(31),then collecting terms of equal powers ofδ,governing equations of,andand related boundary conditions can be achieved.In order to describe the process concisely,the details of these problems of each order are ignored.The final results can be written as

The coefficients ofc22andc24can be expressed as

andfi(i=0,1) are given in appendix A.Based upon the expression Eq.(33) of stream function,subtracting the values of stream function estimated atz=1 andz=0 and then eliminating the periodic terms averaged within a period,the volume flow rate is obtained as

(ii)Case with modulated surface potential and patterned interfacial slip

Substituting Eq.(12) andϕ1into Eq.(28) yields

Similarly,the expanded terms ,,andin Eq.(34)of the stream functionϕ1are given by

The coefficients ofc12andc14are

fi(i=2,3,4,5)in Eqs.(44),(45),and(46)are also given in Appendix.In the same way,volume flow rate is given by

here

Finally,from Eqs.(40) and (48),the whole volume flow rate can be gained as

The flow velocities can be computed by Eq.(27)if stream function is known.

3 Results and Discussion

In previous sections,we have obtained the stream function and the velocities of MHD EOF through two patterned charged micro-parallel plates with modulated wettability.The flow strongly depends on some dimensionaless parameters.For typical microfluidic MHD EOF,although Hartmann numberHashould be smaller than 1 in microfluid,we broaden the range ofHain order to obtain effect of magnetic field apparently.[18,41]Besides,the dimensionless slip length changes from 0.01 to 0.1,[35]and the non-dimensional electrokinetic widthλmeans ratio between the height of microchannel and the thickness of electric double layer,is considered from 10 to 50[42−43]although it may be very large in our present problem.The influences of the above mentioned parameters on streamline and flow field are investigated below.

Figures 2(a)and 2(b)clearly show the influence of amplitude of the periodic partmon streamlines with other parameters remaining unaltered.At some locations,the vortex occurs with the increase ofm.The reason why the vortex structure in Fig.2(b) is stronger than that in Fig.2(a)is that the termmcos(qx)plays a dominant role in boundary conditions.

Figure 3 demonstrates the variations of streamline with patterned frequencyq.It is clearly show that the periodicity of the vortex will decrease with an increasingq.This result is similar to that obtained in the analysis of EOF.[35]

Fig.2 Streamlines for different m(ls=0.05, δ =0.1, Ha=3,λ=50, q =2, n=2),(a) m=5,(b)m=12.

In order to describe the variations of flow field directly,the velocity and the volume flow rate are also discussed in detail.Figure 4 illustrates velocity distribution for different electrokinetic widthλ.It can be observed from Fig.4 that all horizontal velocity componentsushows a configuration with M-shape when the slip lengths of two walls is small.At the same time,the vertical velocity componentwbehaves reverse trends whenz <0.5 andz >0.5 in Fig.4(b),which is an elementary condition to produce vortex.Also,the magnitudes of the velocitiesuandwbecome larger with the increase ofλ,especially approaching to the EDL.Mathematically,the reason is that the leading order term in stream function is dominant and is proportional toλ.

Figure 5 depicts the effect of Hartmann numberHaon velocities.From Figs.5(a)and 5(b),it is interesting to note that asHagets bigger,the velocities of both directions decreasing monotonously,which demonstrates theHaacts as an brake to the velocity field.Unlike the trend of velocity in one dimensional flow increase and then decrease as Ha goes up.[21]The reason is that the influence of aid termO(HaS) within the Lorentz forces in Eq.(21)is eliminated when the flow field is expressed by stream function in Eq.(28).

Fig.4 Velocity distributions for differentλ(x=2,ls=0.01,m=3,δ=0.1, Ha=1,q=3,n=2),(a) axial velocityu; (b) vertical velocity w.

Fig.5 Velocity distributions for different Ha (x=2, ls =0.01, m=3, δ =0.1, q =3, λ=15, n=2),(a) axial velocity u; (b) vertical velocity w.

The influence of non-dimensional slip length on velocity distributions is depicted in Fig.6(a).It can be seen from Fig.6(a) that the velocity magnitudes increase with the increase ofls.It is worth noting thatls=0 will lead to no slip condition.The influence of m on the velocity is described in Fig.6(b).The velocity is going to augment with an increasing ofm.Besides,the minimal value ofm=0 is associated with the case of uniform zeta potential.The effect ofδon the velocity is given in Fig.6(c).As is shown,the velocity increases withδ.

Figure 7(a)demonstrates the variations in volume flow rate with electrokinetic widthλfor four different values of the amplitude of patterned slip length.δ=0 means the case of constant slip on the walls.For the bigger value ofδ,the net flow rate tends to grow.For a prescribedδ,the flow rate increases withλdue to the same trend of the velocity.It is observed from Fig.7(b)that an increasingHaresults in a decreasing volume flow rate due to its negative role on the velocity.

Fig.6 Velocity distributions for ls, m and δ (q=3, n=2),(a) x=0, m=3, δ=0.2, λ=10, Ha=0.5; (b)x=2, ls =0.05, δ =0.2, λ=10, Ha=1; (c) x=2, ls =0.05, m=3, λ=10, Ha=0.5.

Fig.7 Variations of volume flow rates with λ for different δ and Ha (m=3, q=3),(a) (ls=0.01, Ha=0.1,n=3); (b) (ls =0.05, δ =0.2, n=2).

4 Conclusions

In the present work,we have studied two-dimensional MHD EOF with effects of patterned slip boundary conditions and charge-modulate surfaces through parallel plate microchannels.The analytical solution of stream function has been achieved by using the method of separation of variables and perturbation expansion.Numerical results show thatqdetermines the periodicity of the vortex andmaffects the strength of vortex.The velocities of both directions decrease monotonously asHabecomes larger.This situation is different from one-dimensional MHD EOF where there exists a critical value of theHa.Moreover,the velocities and volume flow rates increase as the increase of the magnitude of patterned zeta potentialm,modulated amplitude of the slip lengthδand electrokinetic widthλ.

Appendix

All the coefficients withinfiare obtained by solve the following linear system of equations

Wheni=0,2,4,j=1,and wheni=1,3,j=2,whilei=5,j=3.DandXiare expressed as