The analysis of the suction/injection on the MHD Maxwell fluid past a stretching plate in the presence of nanoparticles by Lie group method

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1 Open Phs. 25; 3:35 4 Research Article Open Access Limei Cao Xinhui Si* Liancun Zheng and Huihui Pang The analsis of the suction/injection on the MHD Maxwell fluid past a stretching plate in the presence of nanoparticles b Lie group method Abstract: In this paper the magnetohdrodnamic (MHD) Maxwell fluid past a stretching plate with suction/injection in the presence of nanoparticles is investigated. The Lie smmetr group transformations are used to convert the boundar laer equations into non-linear ordinar differential equations. The dimensionless governing equations are solved numericall using Bvp4c with MATLAB which is a collocation method equivalent to the fourth order mono-implicit Runge-Kutta method. The effects of some phsical parameters such as the elastic parameter K the Hartmann number M the Prandtl number Pr the Brownian motion Nb the thermophoresis parameter Nt and the Lewis number Le on the velocit temperature and nanoparticle fraction are studied numericall especiall when suction and injection at the sheet are considered. Kewords: Lie group; Maxwell fluid; MHD; stretching surface; suction/injection PACS: 2.2.Sv 2.6.Lj 47.5.Cb DOI.55/phs-25-7 Received September 5 24; accepted November 7 24 Introduction During last few ears the boundar laer flow behaviours of different tpes of fluids attracted the interest of man *Corresponding Author: Xinhui Si: School of Mathematics and Phsics Universit of Science and Technolog Beijing Beijing 83 China sixinhui_ustb@26.com Limei Cao: School of Mathematics and Phsics Universit of Science and Technolog Beijing Beijing 83 China caolimei@ustb.edu.cn Liancun Zheng: School of Mathematics and Phsics Universit of Science and Technolog Beijing Beijing 83 China lianchunzheng@63.com Huihui Pang: College of Science China Agricultural Universit Beijing 83 P.R. China phh2@63.com 25 Limei Cao et al. licensee De Gruter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3. License. researchers. Especiall the interest of non-newtonian fluids is increasing substantiall due to the large number of practical applications in industrial and manufacturing processes. Because of the complexit of these fluids man tpes of constitutive equations have been constructed to exhibit the properties of the non-newtonian fluids [ 9]. For example Maxwell model a subclass of non-newtonian fluids can predict the stress relaxation and therefore has become more popular [ ]. Noor [2] presented analsis of the MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reactions. Liu et al. [3] studied the time periodic electroosmotic flow (EOF) of generalized Maxwell fluids between two microparallel plates and an analtical solution of EOF velocit was presented. Haat et al. [4] constructed an analtic solution for unstead MHD flow in a rotating Maxwell fluid through a porous medium. Nadeem et al. [5] studied numericall two dimensional boundar-laer flows and the heat transfer of a Maxwell fluid past a stretching sheet. The investigation of flow due to a stretching sheet also has received great attention due to the various industrial applications such as in the manufacturing of polmer sheets filaments and wires. During the manufacturing process the moving sheet is assumed to stretch on its own plane and the stretched surface interacts with the ambient fluid both mechanicall and thermall. Initiall Sakiadis [6] introduced the concept of a boundar laer flow over a stretching surface. Crane [7] modified the idea introduced b Sakiadis and extended this idea for both linear and exponentiall stretching sheets. In addition a few recent investigations have been made for the low flow over a stretching surface including the various effects described in Refs. [8 2]. For example Nadeem et al. [8] investigated the stagnation point flow of a viscous fluid towards a stretching sheet and obtained the analtical solution of the boundar laer equation b homotop analsis method. Chen [2] analzed magneto-hdrodnamic mixed convective flow and heat transfer of an electricall conducting Download Date /6/8 5:8 PM

2 36 Limei Cao et al. power-law fluid past a stretching surface in the presence of heat generation/absorption and thermal radiation. Lie group analsis can be used to obtain similarit transformations that can reduce a sstem of governing partial differential equations and associated boundar conditions to a sstem of ordinar differential equations [22 24]. This technique has been applied b man researchers to solve some partial differential problems [25 29]. Motivated b above works in this paper we present a general procedure for appling the one-parameter group of transformations to the MHD Maxwell fluid past a stretching plate in the presence of nanoparticles. The present stud is to extend the work done b Nadeem et al. [5] to the case of the flow past a porous plate with suction/injection. As a result we obtain the similarit transformation b Lie group analsis. Finall the results for some phsical parameters of the velocit temperature distribution and nanoparticle fraction are investigated numericall when the suction or injection at the wall are considered. 2 Preliminaries Consider a two-dimensional stead incompressible fluid flowing past a stretching plate with suction/injection. Here we assume that x-axis is measured along the horizontal stretching surface and the flow is assumed to be confined to ȳ >. The sheet is stretched with the linear velocit ū( x) = a x where a > is constant. A uniform constant magnetic surface field is applied normal to the stretching surface. The effects of the induced magnetic field are negligible. The boundar laer equations of the Maxwell fluid with nanoparticles are [5]: ū x + v = (2.) ȳ ū ū ( ) ( ū 2ū + v x ȳ =ν ȳ 2 + κ ū 2 2 ū x ū v2 ȳ 2 + 2ū v 2 ū 2 ) x ȳ ū T x { + τ σb2 ρ (ū + κ v ū ) (2.2) ȳ ( T 2 + v ȳ = α T x T ȳ 2 D B ( C x T x + C T ȳ ȳ ) + ) ( DT T ) [ ( ) 2 T + x ū C ( C 2 ) ( ) ( + v x ȳ = D C B x C DT 2 T ȳ 2 + T ( ) ] } 2 T ȳ x T ȳ 2 (2.3) ) (2.4) where ū and v denote the velocities in the x and ȳ directions respectivel ν is the kinematic viscosit of the fluid κ is the relaxation time of the Upper-Convected Maxwell fluid σ is the electrical conductivit B is the magnetic induction ρ is the densit of the fluid α is the thermal diffusivit T is the fluid temperature C is the nanoparticle faction T w and C w are the temperature of fluid and nanoparticle faction at the wall respectivel D B is the Brownian diffusion D T is the thermophoretic diffusion coefficient and τ is the ratio between the effective heat capacit of the nanoparticle material and heat capacit of the fluid. When ȳ tends toward infinit the ambient values of T and C are denoted b T and C respectivel. The corresponding boundar conditions are: ū = a x v = v w T = T w C = C w at ȳ = ū = T = T C = C as ȳ (2.5) where v w is the injection or suction velocit at the wall. The following non-dimensional variables can be introduced x = Θ = x ν/a = T T T w T Φ = ȳ ν/a u = C C C w C ū aν v = v aν (2.6) and the stream function Ψ defined b u = Ψ and v = Ψ x leads to Eqs.(2.)-(2.4) taking the following nondimensional form Ψ Ψ x Ψ x Ψ Ψ K ( (Ψ ) 2 Ψ xx + (Ψ x ) 2 Ψ 2Ψ x Ψ Ψ x ) + M 2 (Ψ KΨ x Ψ ) = (2.7) Ψ Θ x Ψ x Θ Pr (Θ xx + Θ ) Nb(Φ x Θ x + Φ Θ ) Nt((Θ x ) 2 + (Θ ) 2 ) = (2.8) LePr(Ψ Φ x Ψ x Φ ) (Φ xx +Φ ) Nt Nb (Θ xx+θ ) = (2.9) where K = aκ ( ) is the elastic parameter M 2 = σb 2 /ρa is the Hartmann number Pr = ν/α is the Prandtl number Nb = τd B (C w C )/ν is Brownian motion parameter Nt = τd T (T w T )/(νt ) is the thermophoresis parameter and Le = α/d B is the Lewis number. The boundar conditions can be written as Ψ = x Ψ x = S Θ = Φ = at = Ψ = Θ = Φ = at (2.) where S = v w / av (S > corresponds to suction and S < corresponds to injection). Download Date /6/8 5:8 PM

3 The analsis of the suction/injection on the MHD Maxwell fluid 37 3 Lie point smmetries of the problem Let us consider a one-parameter Lie group of infinitesimal transformations x * = x + εξ(x Ψ Θ Φ) + O(ε 2 ) * = + εζ (x Ψ Θ Φ) + O(ε 2 ) Ψ * = Ψ + εψ(x Ψ Θ Φ) + O(ε 2 ) Θ * = Θ + εθ(x Ψ Θ Φ) + O(ε 2 ) Φ * = Φ + εϕ(x Ψ Θ Φ) + O(ε 2 ) (3.) where ε is a small parameter. A sstem of partial differential equations (2.7)-(2.9) is said to admit a smmetr generated b the vector field Γ ξ x + ζ + ψ Ψ + θ Θ + ϕ Φ (3.2) if it is left invariant b the transformation (x Ψ Θ Φ) (x * * Ψ * Θ * Φ * ) given b (3.). The vector Γ given b (3.2) is said to be a Lie point smmetr vector field for (2.7)-(2.9) if Γ [3] (Ψ Ψ x Ψ x Ψ Ψ K((Ψ ) 2 Ψ xx + (Ψ x ) 2 Ψ 2Ψ x Ψ Ψ x ) + M 2 (Ψ KΨ x Ψ )) = (3.3) Γ [3] (Ψ Θ x Ψ x Θ Pr (Θ xx + Θ ) Nb(Φ x Θ x + Φ Θ ) Nt((Θ x ) 2 + (Θ ) 2 )) = (3.4) From Eqs. (3.3)-(3.5) the following sstem of linear partial differential equations is given ψ Ψ x + ψ x Ψ ψ x Ψ ψ Ψ x ψ K(2ψ Ψ Ψ xx + ψ xx (Ψ ) 2 + 2ψ x Ψ x Ψ + ψ (Ψ x ) 2 2ψ x Ψ Ψ x 2ψ Ψ x Ψ x 2ψ x Ψ x Ψ ) + M 2 (ψ Kψ x Ψ Kψ Ψ x ) = ψ Θ x + θ x Ψ ψ x Θ θ Ψ x Pr (θxx + θ ) Nb(ϕ x Θ x + θ x Φ x + ϕ Θ + θ Φ ) Nt(2θ x Θ x + 2θ Θ ) = LePr(ψ Φ x + ϕ x Ψ ψ x Φ ϕ Ψ x ) (ϕ xx + ϕ ) (3.7) (3.8) Nt Nb (θxx + θ ) =. (3.9) The components ψ x ψ ψ x ψ ψ xx ψ x ψ θ x θ θ xx θ ϕ x ϕ ϕ xx ϕ can be determined from the following expressions ψ S = D S ψ Ψ x D S ξ Ψ D S ζ θ S = D S θ Θ x D S ξ Θ D S ζ ϕ S = D S ϕ Φ x D S ξ Φ D S ζ ψ JS = D S ψ J Ψ Jx D S ξ Ψ J D S ζ θ JS = D S θ J Θ Jx D S ξ Θ J D S ζ ϕ JS = D S ϕ J Φ Jx D S ξ Φ J D S ζ (3.) where SJ stand for x and the total derivatives D x D are Γ [3] (LePr(Ψ Φ x Ψ x Φ ) (Φ xx +Φ ) Nt Nb (Θ xx +Θ )) = where Γ [3] ξ x + ζ + ψ Ψ + θ Θ + ϕ Φ + ψx Ψ x + ψ + ψ x + ψ + ψ xx Ψ Ψ x Ψ Ψ xx + ψ x + ψ + θ x + θ Ψ x Ψ Θ x Θ + θ xx + θ + ϕ x + ϕ Θ xx Θ Φ x Φ + ϕ xx + ϕ Φ xx Φ is the third prolongation of the vector Γ. (3.5) (3.6) D x x + Ψ x Ψ + Θ x Θ + Φ x Φ + Ψ xx Ψx + Θ xx Θx + Φ xx Φx + Ψ x Ψ +... D + Ψ Ψ + Θ Θ + Φ Φ + Ψ Ψ + Θ Θ + Φ Φ + Ψ x Ψx (3.) Substituting (3.) into (3.7) and then assuming the coefficients of Ψ x Ψ x Ψ Ψ Ψ x (Ψ ) 2 Ψ xx (Ψ ) 3 Ψ xx Ψ x Ψ to be zero one obtains ξ = ζ x = ξ Ψ = ζ Ψ = ψ Ψ ξ x =. (3.2) Then substituting (3.2) and (3.) into (3.7) and letting the coefficients of (Ψ ) 2 Ψ Ψ x Ψ be zero one can obtains the following results again. ψ xψ = ψ x ψ Ψ + M 2 ζ x = (M 2 + K)ψ + ξ + M 2 ξ = ψ + 3ξ + M 2 ψ xψ = (M 2 + K)ψ x + ψ Ψ =. (3.3) Download Date /6/8 5:8 PM

4 38 Limei Cao et al. Similarl substituting (3.) into the energ and nanoparticle fraction equations (3.8) and (3.9) and then assuming the coefficients of Ψ x and Ψ to be zero ields θ x = θ = ϕ x = ϕ =. (3.4) From (3.2)(3.3) and (3.4) we have ξ = a x ζ = a 2 ψ = a Ψ + a 3 θ = a 4 ϕ = a 5. (3.5) If one assumes a = a 2 = a 3 = a 4 = a 5 = the characteristic equations for the similarit transformations will be dx x = d = dψ Ψ = dθ = dφ (3.6) then the similarit variable and functions can be obtained from (3.6) Ψ = xg() Θ = Θ() Φ = Φ(). (3.7) Substituting (3.7) into (2.7)-(2.9) obtains the following ordinar equations d 3 G d 3 + d 3 G KG2 d 3 ( dg d d 2 Θ d 2 ) 2 M 2 dg d + ( + M2 K)G d2 G d 2 dθ dθ dφ + Pr[G + Nb d d d + Nt d 2 Φ d 2 dg d 2 G 2KG d d 2 = (3.8) dφ + LePrG d + Nt Nb The boundar conditions (2.3) become ( ) 2 dθ ] = (3.9) d d 2 Θ =. (3.2) d2 dg() = G() = S Θ() = Φ() = d dg( ) = Θ( ) = Φ( ) =. d 4 Numerical solutions and discussion (3.2) Since equations (3.8)-(3.2) together with the boundar conditions (3.2) are coupled nonlinear boundar value problem a numerical treatment would be more appropriate. Thus these equations are solved numericall b Bvp4c with MATLAB. Since the velocit changes sharpl in the boundar laer near the plate this region with sharp change makes this boundar value problem a relativel difficult one. In order to resolve better the boundar laer and obtain a more accurate solution the relative error tolerance on the residuals is defined to be 6 (i.e. RelTol= 6 ) during the process of numerical computation. The results are presented graphicall and in tables. For the verification of accurac of the applied numerical scheme a comparison of the present results corresponding to Θ () and Φ () with the ones obtained b Nadeem et al. [5] and Khan et al. [3] are presented in Table which shows a favorable agreement. Table 2 presents the numerical values of Θ () and Φ () for various values of the Brownian motion parameter Nb and the thermophoresis parameter Nt when other parameters are fixed. It can be observed that Θ () is an increasing function of the thermophoresis parameter Nt and the Brownian motion parameter Nb respectivel. However Φ () is the increasing function of the thermophoresis parameter Nt. When we consider the influence of the Brownian motion parameter Nb on Φ () the case is different. Φ () is the decreasing function of the Brownian motion parameter Nb. In the following section we will investigate the influence of different phsical parameters on the velocit temperature distribution and nanoparticle fraction. To begin with we discuss the effects of the elastic parameter K and the Hartmann number M on the velocit G (). Figure shows that the velocit boundar laer for injection is thicker than the one for suction. For the case of injection the boundar laer thickness reduces with the greater elastic parameter K. However the boundar laer thickness has increasing behavior with increasing the elastic parameter K when there exists suction. Figure also reveals that the velocit G () is sensitive to the tin change of the elastic parameter K as there is suction at the wall. Figure 2 illustrates the effects of the Hartmann number on the velocit. The boundar laer thickness decreases with the increasing Hartmann number M for both injection and suction. Phsicall the magnetic field is normal to the fluid so for greater values of the Harmann number it increases the resistance of the fluid flow. Figures 3 4 illustrate the effects of the thermophoresis parameter Nt and the Brownian motion parameter Nb on the temperature distribution and the nanoparticle fraction. No matter there exists suction or injection at the wall the values of temperature Θ() and nanoparticle fraction Φ() tend to zero as both of them are far awa from the wall. Furthermore temperature Θ() is the increasing function of the thermophoresis parameter Nt and the Brownian motion parameter Nb. In phsical meaning the enhancement of the Brownian motion can improve the heat transfer. Figure 5 shows the influence of Prandtl number on the temperature distribution. When there is injection at Download Date /6/8 5:8 PM

5 The analsis of the suction/injection on the MHD Maxwell fluid 39 Table : The comparison of Θ () and Φ () for Pr = Le = and Nb =. Present results K = M = S = K = M = Nadeem et al. [5] Khan et al. [3] Nt Θ () Φ () Θ () Φ () Θ () Φ () Table 2: The values of Θ () and Φ () for S = M = K = Pr = 3 Le = 5 Nt= Nt= Nt=.6 Nt= Nb Θ () Φ () Θ () Φ () Θ () Φ () Θ () Φ () M=. Pr=2 Nb= Nt= Le=3 K= Pr=2 Nb= Nt= Le=3.6.6 G () G () K=. M=.6. S= S=. S= S= Figure : Effects of K on the velocit G (). Figure 2: Effects of M on the velocit G (). the wall the temperature boundar laer becomes thinner and is a decreasing function of Prandtl number. However when there is suction at the wall the temperature is increasing function of Prandtl number. Phsicall the increase of Prandtl number also means the enhancement of the heat transfer which leads to the thermal boundar laer becomeing thinner. The effects of Lewis number Le Prandtl number Pr and thermophoresis parameter Nt on the nanoparticle fraction Φ() are illustrated in Figures 6 8. It is observed that the effects of above parameters exhibit different trends as there is suction or injection at the plate. The nanoparticle fraction is the decreasing function of Lewis number and the Prandtl number but the increasing function of the thermophoresis parameter when there is suc- tion at the wall. However when there is injection at the wall the nanoparticle fraction changes its trends as the variable changes from zero to the infinit. 5 Conclusion Using group-theoretical methods we have investigated the similarit solutions of MHD Maxwell fluid past a stretching plate with suction/injection in the presence of nanoparticles. B determining the transformation group we can obtain the information about the invariants and smmetries of these equations. In turn with the assistance of these invariants and smmetries the governing equa- Download Date /6/8 5:8 PM

6 4 Limei Cao et al. S= M=.6 K= Pr=2 Nb= Le=3 M= K= Pr=2Nb= Nt=.6.6 Nt=.6. Θ() Φ() Φ(). Le= S= S= Figure 3: Effects of Nt on the temperature Θ() and nanoparticle fraction Φ(). Figure 6: Effects of Le on the nanoparticle fraction Φ()..6. S= M= K= Pr=2 Nt= Le=4 Nb= Θ() Φ().5 2 Φ().6. M= K= Nb= Nt= Le=. Pr= S= S= Figure 4: Effects of Nb on the temperature Θ() and nanoparticle fraction Φ(). Figure 7: Effects of Pr on the nanoparticle fraction Φ(). Θ().6. M=.6 K= Nb= Nt= Le=3 Pr= S= S= 5 5 Figure 5: Effects of Pr on the temperature Θ(). Φ().6. M= K=Pr=.5 Nb= Le=2 Nt=.6 S= S= Figure 8: Effects of Nt on the nanoparticle fraction Φ(). Download Date /6/8 5:8 PM

7 The analsis of the suction/injection on the MHD Maxwell fluid 4 tions are reduced to a sstem of coupled nonlinear differential equations. The phsical effects of different parameters such as the Hartmann number Mthe elastic parameter K the Prandtl number Pr the Lewis number Le the Brownian motion parameter Nb and the thermophoresis parameter Nt on velocit temperature and nanoparticle fraction are analzed b graphs and discussed in detail. Furthermore one important conclusion can be drawn that the influence of these parameters on the velocit temperature distribution and nanoparticle fraction is obviousl different when there is suction or injection at the plate. Acknowledgement: This work is supported b the National Natural Science Foundations of China (No.3224) the Fundamental Research Funds for the Central Universities (Nos.FRF-BR-2-5FRF-TP-2-8AFRF-TP-4-7A2) and Beijing Higher Education Young Elite Teacher Project(Nos. YETP387YETP322) Research Foundation of Engineering Research Institute of USTB (No.Yj2-5). References [] T. Haat A. M. Siddiqui S. Asghar Int. J. Eng. Sci (2) [2] K. Sadegh M. Sharifi Int. J. Non-linear Mech (24) [3] B. Serdar M. Salih Dokuz Int. J. Eng. Sci (26) [4] M. H. Haroun Nonlinear Sci. Numer. Simul (27) [5] X. H. Si L.C. Zheng Y. Chao Cent. Eur. J. Phs (2) [6] A. M. Siddiqui A. Zeb Q. K. Ghori A. M. Benharbit Chaos Soliton. Fract (28) [7] M. Sajid I. Ahmad T. Haat M. Aub Commun. Nonlinear Sci. Numer. Simul (29) [8] T. Haat T. Javed Z. Abbas Int. J. Heat Mass Tran (28) [9] T. Haat M. Nawaz M. Sajid S. Asghar Comput. Math. Appl (29) [] K. Sadegh H. Hajibegi S. M. Taghavi Int. J. Non-Linear Mech (26) [] M. S. Abel J. V. Tawade M.M. Nandeppanavar Meccanica (22) [2] N. F. M. Noor World Acad. Sci. Eng. Technol (22) [3] Q. S. Liu Y.J. Jian L. G. Yang J. Non-Newton. Fluid Mech (2) [4] T. Haat C. Fetecau M. Sajid Phs. Lett. A (28) [5] S. Nadeem R. U. Haq Z. H. Khan J. Taiwan Inst. Chem. Eng (24) [6] B. C. Sakiadis J. Am. Inst. Chem. Eng (96) [7] L. Crane Zeit. Angew. Math. Phs (97) [8] S. Nadeem A. Hussain Commun. Nonlinear Sci. Numer. Simul (2) [9] S. Nadeem R. U. Haq C. Lee Int. J. Nonlinear Mech (29) [2] C. H. Chen Int. J. Nonlinear Mech (29) [2] N. F. M. Noor S. A. Kechil I. Hashim Commun. Nonlinear Sci. Numer. Simul (22) [22] J. Olver Application of Lie groups to differential equations (Springer New York 989) [23] L. V. Ovsiannikov Group analsis of differential equations (Academic Press New York 982) [24] Y. Zhang F. X. Mei Chin. Sci. Bull (2) [25] F. S. Ibrahim M. A. Mansour M. A. A. Hamad Electron. J. Differ. Equ. 39 (25) [26] Y. Z. Boutros M. B. Abd-el-Malek N. A. Badran H. S. Hassan Appl. Math. Modell (27) [27] M. Pande B. D. Pande V. D. Sharma Appl. Math. Comput (29) [28] M. A. A. Hamad M. Ferdows Commun. Commun. Nonlinear Sci. Numer. Simul (22) [29] Y. Z. Boutros M. B. Abd-el-Malek N. A. Badran H. S. Hassan J. Comput. App. Math (26) [3] W. A. Khan I. Pop Int. J. Heat. Mass. Tran (2) Download Date /6/8 5:8 PM