Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 2, pp. 265-274, Warsaw 2013 ANALYTICAL SOLUTIONS OF ELECTRO-KINETIC FLOW IN NANO-FLUIDIC COMPONENTS BY USING HE’S HOMOTOPY PERTURBATION METHOD Mehran Khaki, Mohammad Taeibi-Rahni, Davood D. Ganji Islamic Azad University, Departments of Air space and Mechanical Engineering, Science and Research Branch, Tehran, Iran; e-mail: mehran.khaki@gmail.com; taeibi@sharif.edu; ddg davood@yahoo.com This paper aims to examine the electro-kinetic flow through nano-channels. The equations governing the fluid flow in a one dimensional channel are derived from the Poisson-Nernst- Planck theory. The boundary conditions for the governing equations are obtained from the electrochemical equilibrium requirements. The coupled equations are transformed into a single differential equation. The transformed equation is solved by He’s homotopy per- turbation method and an exact solution is achieved. The validity of results is verified by comparing with existing numerical results. The results are presented for velocity profiles, electrical potential distributions, mole fraction of cation and anion distributions and other physical properties. The results demonstrate reasonable agreement with those provided by other numerical methods and good accuracy of the obtained analytical solutions. Key words: electro-kinetic flow, nano-channels, homotopy perturbationmethod (HPM) Nomenclature φ∗ – electrical potential [V] φ – dimensionless electrical potential [–] φ0 – potential scale, φ=RT/F r∗ – vector of location [nm] εr – dielectric constant, εr =8.854 ·10−12C/(Vm) ε0 – permittivity of free space [–] ρi – number density, ρi =NAci [number/m 3] NA – Avogadro number, NA =6.022 ·1023 zi – valence of ion [–] ci – concentration of ion [mole/m 3] c – total concentration, c= ∑ ci+ csolvent Xi – mole fraction of ion, Xi = ci/c I – ionic strength, I = ∑ zic 2 i T – temperature [K] F – Faraday constant, F =96485.3415C/mole k – Boltzmann constant k=1.38065 ·1023 j/k R – universal ideal gas constant, R=8.3144j/(molek) λ – Debye length, λ=F−1 √ εERTI−1 u∗ – velocity [m/s] u0 – average electro-osmotic velocity [m/s] u – dimensionless velocity µ – viscosity [Kg/(ms)] Di – Diffusion coefficient [–] 266 M. Khaki et al. Re – Reynolds number, Re= ρu0H/µ Sc – Schmidt number, Sc=µ/ρDi x∗,y∗,z∗ – Cartesian coordinate x,y,z – dimensionless Cartesian coordinate Subscripts: − – anion, + – cation. 1. Introduction Miniaturization has been one of the swiftest revolutions in the scientific and industrial world during last century. The term “micro and nano fluidics” was invented about 40 years ago when micro-fabricatedfluidsystemsweredevelopedatStanford (gas chromatography) andat IBM(ink jet printer nozzles) (Zheng, 2003).Micro and nano fluidics, the study of fluid flow inmicrometer or nanometer sized devices, is one of the disciplines on which the operation of MEMS-NEMS depends. In recent years, scientists have studied the transport of fluids through micro and nano channels, analyzed aqueous solutions with a number of electrolytes, and calculated electric fields, flow fields and ion distributions. The derived differential equations are strongly nonlinear and coupled. Results of solving those nonlinear equations can help scientists to deeply know the describedprocess.The solutions of these nonlinear equations are normally obtained byusing, for example, traditional finite difference methods. In numerical methods, stability and convergence should be considered to avoid divergence or inappropriate results. It is often so hard to gain an analytical solution for these kinds of problemswhich include nonlinear terms. In recent decades, analytical solutions have developed for nonlinear differential equations. One of thesemethods is the Homotopy Perturbation Method (HPM). The homotopy perturbationmethod was introduced by Ji-HuanHe for the first time (1999, 2000a,b, 2001, 2003, 2006, 2009). This method has been used by many authors such as Ganji and Sadighi (2006,2007), Ganji et al. (2007a,b) and the corresponding ones included in authors’ references (Ariel et al., 2006; Ariel, 2010; Biazar and Ghazvini, 2007; Ghorbani and Saber- Nadjafi, 2007; Gorji, et al., 2007; Tari et al., 2007; Yusufoglu, 2007). These papers are published to handle a wide variety of scientific and engineering applications such as linear and nonlinear, homogeneous and inhomogeneous as well, because this method continuously transforms a diffi- cult problem into a simplest form, which is solvable. It has been shown by many authors that these methods provide improvements over existing numerical techniques. These methods give successive approximations of high accuracy of solutions. The aim of this paper is to analytically study the electro-kinetic flow in a nano-channel. The results are compared with numerical outcomes presented in pervious works. However, an analytical expression is more convenient for engineering calculations and is also the obvious starting point for a better understanding. 2. Homotopy perturbation method To explain this method, let us consider the following function A(u)−f(r)= 0 r∈Ω (2.1) with the boundary conditions B ( u, ∂u ∂n ) =0 r∈Γ (2.2) where A, B, f(r) and Γ are a general differential operator, a boundary operator, a known analytical function and the boundary of the domain Ω, respectively. Analytical solutions of electro-kinetic flow... 267 Generally speaking, the operator A can be divided into a linear part L and a nonlinear part N(u). Equation (2.1) can therefore be written as L(u)+N(u)−f(r)= 0 (2.3) By the homotopy technique, we construct a homotopy u(r,p) :Ω× [0,1]→R. This satisfies H(v,p) = (1−p)[L(v)−L(u0)]+p[A(v)−f(r)]= 0 p∈ [0,1] r∈Ω (2.4) or H(v,p) =L(v)−L(u0)+pL(u0)+p[N(v)−f(r)]= 0 (2.5) where p∈ [0,1] is an embedding parameter, while u0 is the initial approximation of Eq. (2.1), which satisfies the boundary conditions. Obviously, from Eqs. (2.4) and (2.5) we will have H(v,0)=L(v)−L(u0)= 0 H(v,1)=A(v)−f(r)= 0 (2.6) The changing process of p from zero to unity is just that of v(r,p) from u0 to u(r). In topology, this is called deformation, while L(v)−L(u0) and A(v)−f(r) are called homotopy. According to theHPM,we can first use the embedding parameter p as a “small parameter”, and assume that the solutions to Eqs. (2.4), (2.5) can be written as a power series in p v= v0+pv1+p 2v2+ . . . (2.7) Setting p=1 yields in the approximate solution to Eq. (2.4) u= lim p→1 v= v0+v1+v2+ . . . (2.8) The combination of the perturbationmethod and the homotopymethod is called the HPM, which eliminates the drawbacks of traditional perturbationmethodswhile keeping all its advan- tage. Series (2.8) is convergent for most cases. However, the rate of convergence depends on the nonlinear operator A(v). Moreover, He (1999) made the following suggestions: • The second derivative of N(v) with respect to v must be small because the parameter may be relatively large, i.e. p→ 1. • The norm of L−1(∂N/∂V must be smaller than one so that the series converges. 3. Mathematical model Water is an efficient solvent for most polar molecules and electrolytes, although it does not dissolve many organic substances. Many experiments discussed in this work are done using aqueous solution; however, the model is also effective for other solvents. It is also assumed that the solution in nano-channels is incompressible, which is generally accepted because of the properties of liquids. Furthermore, the electro-osmotic flow in nano-channels is a laminar flow, because the Reynolds number (Re) in this case will be very small. According to Poisson’s equation ∇2φ∗(r∗)=− 1 εrε0 ρE(r ∗)=− 1 εrε0 e ∑ i ziρi(r ∗) (3.1) 268 M. Khaki et al. The potential scales φ0 and εe were defined as φ0 = KT e = RT F εe = ε0εr (3.2) The dimensionless form of governing equations can be gained by applying the following para- meters φ= φ∗ φ0 x= x∗ L y= y∗ h z= z∗ W (3.3) Subsequently, we have ε21 ∂2φ ∂x2 + ∂2φ ∂y2 +ε22 ∂2φ ∂z2 =− FCh2 εeφ0 ∑ i ziXi =− β ε2 ∑ i ziXi (3.4) where ε1 = H L ε2 = h W ε= λ h β= c I λ= 1 F √ εeRT I For the steady state, the mass transport and momentum equations in the dimensionless form for an electro-chemical system are obtained as follows ∇2Xi+zi∇· (Xi∇φ)−ReSc∇· (Xiu)= 0 Re(v ·∇)v=−∇P − Fcφ0hziXi µu0 ∇φ+∇2v (3.5) For a one-dimensional channel demonstrated in Fig. 1 ε1 = H L ≪ 1 ε2 = H W ≪ 1 Re≪ 1 Therefore, the derivatives respected to the x and z directions were neglected so Eqs. (3.4) and (3.5) reduce to d2φ dy2 =− β ε2 ∑ i Xi d2Xi dy2 +zi d dy ( Xi dφ dy ) =0 d2U dy2 =− β ε2 ∑ i Xi (3.6) Fig. 1. Geometry of one dimensional channel; W ≫ h,L≫h In the nano-channel illustrated in Fig. 1, if the electrolyte consists of monovalent cation and monovalent anion, such as sodium chloride, governing Eqs. (3.6) are gained as follows d2φ dy2 =− β ε2 (X+−X−) d2u dy2 =− β ε2 (X+−X−) d dy (dX+ dy +X+ dφ dy ) =0 d dy (dX − dy −X − dφ dy ) =0 (3.7) The boundary conditions for described equations, Eqs. (3.7) are φ(0)=φ(1)= 0 u(0)=u(1)= 0 X − (0)=X − (1)=X0 − X+(0)=X+(1)=X 0 + (3.8) Analytical solutions of electro-kinetic flow... 269 4. Equation reduction Equations (3.7)1,2 are similar in the expression and boundary conditions. Therefore, we only consider Eqs. (3.7)1 to (3.7)4 for solution. From Eqs. (3.6)3 and (3.7)1 we have dX+ dy +X+ dφ dy = a=⇒ 1 ∫ 0 dφ dy dy= 1 ∫ 0 1 X+ ( a− dX+ dy ) dy =⇒φ(1)−φ(0)= a 1 ∫ 0 dy X+ − ln X+(1) X+(0) dX − dy −X − dφ dy = b=⇒ 1 ∫ 0 dφ dy dy= 1 ∫ 0 1 X − ( −b+ dX − dy ) dy =⇒φ(1)−φ(0)=−b 1 ∫ 0 dy X − +ln X − (1) X − (0) (4.1) With combining boundary conditions (3.8)1,2 with the equations above, we have a 1 ∫ 0 dy X+ =0=⇒A=0 b 1 ∫ 0 dy X − =0=⇒ b=0 (4.2) So Eqs. (3.6)3 and (3.7)1 could be rewritten in the following forms dX+ dy +X+ dφ dy =0=⇒ y ∫ 0 dφ=− X+ ∫ X0 + dX+ X+ =⇒φ(y)=− ln X+ X0+ dX − dy −X − dφ dy =0=⇒ y ∫ 0 dφ= X+ ∫ X0 + dX − X − =⇒φ(y)= ln X − X0 − ln X − X0 − =− ln X+ X0+ =⇒X+(y)X−(y)=X0+X 0 − (4.3) Finally, substitute Eqs. (4.3) into (3.7)1, which leads to X − d2X − dy2 − (dX − dy )2 + X0+X 0 − β ε2 X − − β ε2 X3 − =0 (4.4) The above ordinary differential equation governs the mole fraction of onion distribution. We applied the HPM on Eq. (4.4) to achieve an analytical solution. 5. Application of homotopy perturbation method We consider Eq. (4.4) with a monovalent electrolyte such as NaCl, and T =25◦C, h=20nm, X0+ = 0.00276, X 0 − = 0.00254, therefore: λ = 0.8nm, ε = λ/h = 0.04, I = 294.68mole/m3, β=188.679. According to the HPM, we can construct homotopy of Eq. (4.4) as follows H(u,P) = (1−p) [( X − (y) d2X − (y) dy2 − β ε2 X − (y)3 ) − ( X −0 (y) d2X −0 (y) dy2 − β ε2 X −0 (y)3 )] +p [( X − (y) d2X − (y) dy2 − β ε2 X − (y)3 ) + X0+X 0 − β ε2 X − (y)− (dX − (y) dy )2] =0 (5.1) 270 M. Khaki et al. Substituting X − = X −0 +PX −1 +P2X02 + . . . into Eq. (5.1) and rearranging the resultant equation based on powers of P-terms, we have P0 : [( X −0 (y) d2X −0 (y) dy2 − β ε2 X −0 (y)3 ) − ( X −0 (y) d2X −0 (y) dy2 − β ε2 X −0 (y)3 )] =0 X −0 (0)=X −0 (1)=X0 − =0.00254 (5.2) P1 : X −0 (y) d2X −1 (y) dy2 −3.53773125 ·105X −0 (y)2X −1 (y)+ d2X −0 dy2 X −1 (y) − (dX −0 dy )2 + d2X −0 dy2 X −0 (y)+0.8266498688X −0 (y)−1.179243750 ·105X −0 (y)3 =0 (5.3) X −1(0)=X−1(1)= 0.0 P2 : d2X −2 (y) dy2 X −0 (y)+ d2X −1 (y) dy2 .X −1 (y)+ d2X −0 (y) dy2 .X −2 (y) −2 dX −0 (y) dy dX −1 (y) dy −3.537731250 ·105X −0 (y)2X −2 (y) −3.537731250 ·105X −0 (y)X −1 (y)2+0.8266498688X −1 (y)= 0 X −2 (0)=X −2 (1)= 0.0 (5.4) P3 : d2X −3(y) dy2 X −0 (y)+ d2X −2 (y) dy2 X −1 (y)+ d2X −1 (y) dy2 X −2 (y)+ d2X −0 (y) dy2 X −3 (y) −2 dX −0 (y) dy dX −2 (y) dy − (dX −1 (y) dy )2 −3.537731250 ·105X −0 (y)2X −3 (y) −7.07546250 ·105X −0 (y)X −1 (y)X −2 (y)+0.8266498688X −2 (y) −1.17924375 ·105X −1 (y)3 =0 X −2 (0)=X −2 (1)= 0.0 (5.5) X − (y) can be written as follows by solving Eqs (5.2) to (5.5) X −0 =0.00254 X −1 (y)=−7.021198794 ·10−18e29.97638633Y −0.00007328084019e−29.97638633Y +0.00007328084019 X −2 (y)=−2.340399598 ·10−18e29.97638633Y −0.00002442694673e−29.97638633Y +0.00002442694673 X −3 (y)=−7.106629101 ·10−19e29.97638633Y −0.00007417248346e−29.97638633Y +0.00007417248346 (5.6) In the samemanner, the rest of components was found using theMaple package. According to the HPM, we can conclude that X − (y)= lim p→1 X − (y)=X −0 (y)+X −1 (y)+X −2 (y)+ . . . (5.7) Therefore, substituting the values of X −0 (y),X −1 (y),X −2 (y) etc. fromEqs. (5.6) intoEq. (5.7), yields X − (y)=−1.026419033·10−17e29.97638633Y−0.0001071282146e−29.97638633Y +0.002647128214 (5.8) Analytical solutions of electro-kinetic flow... 271 Substituting Eq. (5.8) into Eqs. (4.3)1,2, we found X+(y)= 7.01 ·10−6 ( −1.026419033 ·10−17e29.97638633Y −0.0001071282146e−29.97638633Y +0.002647128214 ) −1 φ(y)=u(y)= ln [( −1.026419033 ·10−17e29.97638633Y −0.0001071282146e−29.97638633Y +0.002647128214 ) 0.00254−1 ] (5.9) The shear stress is provided by derivation of Eq. (5.9)2 τ(y)= −1.211351711 ·10−13e29.97638633Y +1.264297932e−29.97638633Y −4.041019814 ·10−15e29.97638633Y −0.0421764244e−29.97638633Y +1.042176462 (5.10) 6. Results and discussion Theprocedureadopted in this paper for the current investigation of the electro-dynamic problem in a nano-channel can be described as follows: By using somemathematical calculations, the coupled differential equations, Eqs. (3.7), are reduced to Eq. (4.4). Equation (4.4) with the boundary conditions, Eq. (3.8)3, is solved by using the HPM. The detailed procedural steps are follows: • Firstly, the anion mole fraction distribution is computed by solving Eq. (4.4) given with the boundary conditions, Eq. (3.8)3. • Secondly, the electric potential, the velocity and the cation mole fraction distribution are solved by using equations (4.3)2 and (4.3)3, respectively. The shear stress is solved by derivation of velocity distribution. In this work, we found that the product of anion mole fraction into cation mole fraction at any point is constant. The comparisons are shown in Figs. 2 and 3 and Table 1, are in very good agreement with the solutions presented by Zheng (2003). Fig. 2. The comparison between the FDM by Zheng (2003) and HPM solutions for X − (y) (a) and for X+(y) (b) 272 M. Khaki et al. Fig. 3. The comparison between the FDM by Zheng (2003) and HPM solutions for φ(y) and u(y) (a) and for τ(y) (b) Table 1.The comparison between the FDM by Zheng (2003) and HPM solutions y X − (y) X+(y) φ(y) and u(y) FDM HPM FDM HPM FDM HPM 0 0.0025400000 0.0025400000 0.0027600000 0.002759842 0.0000000 −2.36224E-10 0.1 0.0026383916 0.0026417820 0.0026571803 0.0026533119 0.0379657845 0.0392896102 0.2 0.0026469254 0.0026468614 0.0026485279 0.0026484197 0.0411937238 0.0412104844 0.3 0.0026476253 0.0026471148 0.0026477585 0.0026481661 0.0414572920 0.0413062486 0.4 0.0026476661 0.0026471275 0.0026476733 0.0026481534 0.0414721722 0.0413110268 0.5 0.0026476635 0.0026471281 0.0026476606 0.0026481528 0.0414709923 0.0413112534 0.6 0.0026476661 0.0026471275 0.0026476733 0.0026481534 0.0414721722 0.0413110268 0.7 0.0026476253 0.0026471148 0.0026477585 0.0026481661 0.0414572920 0.0413062486 0.8 0.0026469254 0.0026468614 0.0026485279 0.0026484197 0.0411937238 0.0412104844 0.9 0.0026383916 0.0026417820 0.0026571803 0.0026533119 0.0379657845 0.0392896102 1.0 0.0025400000 0.0025400000 0.0027600000 0.002759842 0.0000000 −2.4733E-10 7. Conclusion In this paper, we successfully applied the homotopy perturbation method to solve a nonlinear differential equation with given boundary conditions for a nano-channel and showed graphical results of velocity, electric potential and ion mole fraction. The results were compared with the numerical solution available in the literature using the FDM, and a very good agreement was observed. The outcomes prove the effectiveness and accuracy of the HPM. The current work presents a new application of theHPMwhich could be used for similar problems in awide range of engineering situations. The key-factor of this paper is based on analytical solution of electro- kinetic problems in a nano-channel. Although these results obviously show that the product of anion mole fraction into cation mole fraction at any point is constant and also in the bulk of the channel, the mole fractions of cations and anions are the same, which implies that the electrolytic solution is neutral in the bulk, and the concentration difference between cation and anion species reaches its maximum at the wall. References 1. Ariel D.P., 2010, Homotopy perturbation method and the natural convection flow of a third grade fluid through a circular tube,Nonlinear Science Letters A, 1, 1, 43-52 Analytical solutions of electro-kinetic flow... 273 2. ArielD.P., HayatD., AsgharT.S., 2006,HomotopyPerturbationMethod andAxisymmetric Flow over a Stretching Sheet, International Journal of Nonlinear Sciences and Numerical Simula- tion, 7, 399-407 3. Biazar J.,GhazviniH., 2007,He’s variational iterationmethod for solvinghyperbolic differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 311-314 4. Ganji D.D., Afrouzi G.A., Talarposhti R.A., 2007a, Application of He’s variational itera- tionmethod for solving the reaction-diffusion equationwith ecological parameters,Computers and Mathematics with Applications 54, 1010-1017 5. Ganji D.D., Sadighi A., 2006, Application of He’s homotopy-perturbationmethod to nonlinear coupled systems of reaction-diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 411-418 6. GanjiD.D., SadighiA., 2007,Solutionof the generalizednonlinearboussinesq equationusingho- motopy perturbation and variational iterationmethods, International Journal of Nonlinear Scien- ces and Numerical Simulation, 8, 435-444 7. Ganji D.D., Tari H., Jooybari M.B., 2007b, Variational iterationmethod and homotopy per- turbationmethod fornonlinearevolutionequations,Computers andMathematics withApplications, 54, 1018-1027 8. Ghorbani A., Saberi-Nadjafi J., 2007, He’s homotopy perturbation method for calculating adomian polynomials, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 229-232 9. Gorji M., Ganji D.D., Soleimani S., 2007, New application of He’s homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 319-328 10. He J.H., 1999, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262 11. He, J. H., 2000a, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics, 35, 1, 37-43 12. He J.H., 2000b, Homotopy perturbation method: a new nonlinear analytical technique, Journal of Computational and Applied Mathematics, 135, 73-79 13. He, J. H., 2001, Bookkeeping parameter in perturbationmethods, International Journal of Non- linear Sciences and Numerical Simulation, 2, 3, 257-264 14. He J.H., 2003, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, 135, 73-79 15. He J.H., 2006, Addendum: new interpretation of homotopy perturbation method, International Journal of Modern Physics B, 20, 1141-1199 16. HeJ.H., 2009,An elementary introduction to the homotopyperturbationmethod,Computers and Mathematics with Applications, 57, 3, 410-412 17. Tari H., Ganji D.D., Rostamian M., 2007, Approximate solutions of K (2,2), KdV and mo- dified KdV equations by variational iterationmethod, homotopy perturbationmethod and homo- topy analysis method, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 203-210 18. Yusufoglu E., 2007, Homotopy perturbation method for solving a nonlinear system of second order boundary value problems, International Journal of Nonlinear Sciences and Numerical Simu- lation, 8, 353-358 19. Zheng, Zhi., 2003, Electrokinetic Flow in Mirco- and Nano-Fluidic Components, Ph.D. Thesis, Ohio State University, Ohio State 274 M. Khaki et al. Analityczne rozwiązania uzyskane perturbacyjną metodą homotopii He’go dla elektrokinetycznego przepływu przez nano-struktury Streszczenie Celem pracy jest przedstawienie wyników badań nad elektrokinetycznym przepływem cieczy przez nano-kanały. Równania opisujące przepływ w jednowymiarowym kanale wyprowadzono na podstawie teorii Poissona-Nernsta-Plancka.Warunki brzegowe uzyskano po spełnieniu wymogów równowagi elek- trochemicznej układu. Sprzężone równania przepływu przekształcono do postaci pojedynczego równania różniczkowego.Następnie rozwiązano go za pomocą perturbacyjnejmetody homotopii He’go, otrzymując wyrażenie analityczne i dokładne. Poprawność rezultatów sprawdzono, porównując je z istniejącymi wy- nikami symulacji numerycznych. Zaprezentowano profile prędkości przepływu, rozkłady potencjału elek- trycznego, molowe udziały frakcji anionów i kationów oraz inne parametry fizyczne układu. Wszystkie wyniki wykazały dobrą zgodność z obliczeniami opartymi na innych metodach badawczych, co potwier- dziło dokładność otrzymanych rozwiązań analitycznych. Manuscript received February 17, 2012; accepted for print May 23, 2012