Freun Publishing House Lt., International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 Application of the homotopy perturbation metho to a magneto-elastico-viscous flui along a semi-infinite plate Mohame M. Mousa,, Aiarkhan Kaltayev Department of Basic Science, Benha High Institute of Technology, Benha University, 5, Egypt, Email: r.eng.mmmm@gmail.com Department of Mechanics, al-farabi Kazakh National University, 9/47 Masanchi 5, Almaty, Kazakhstan Abstract The equations governing the flow of an electrically conucting, incompressible viscous flui over an infinite flat plate in the presence of a magnetic fiel are investigate using the homotopy perturbation metho (HPM) with Paé approximants (PA) an 4 th orer Runge Kutta metho (4RKM). Approximate analytical an numerical solutions for the velocity fiel an heat transfer are obtaine an compare with each other, showing excellent agreement. The effects of the magnetic parameter an Prantl number on velocity fiel, shear stress, temperature an heat transfer are iscusse as well. Keywors: Electrically conucting elastico-viscous flui; symmetry solution; Homotopy perturbation metho; Paé approximation; 4 th orer Runge Kutta; Maple.. Introuction The bounary layer flow of an electrically conucting, incompressible viscous flui over a continuously flat plate is often encountere in many engineering an inustrial processes such as polymer technology, aeroynamic extrusion of plastic sheets an so on. The problem of a fluctuating flow of a magneto-elastico-viscous flui along an infinite flat plate uner the conition of very small elastic parameter was stuie in [ ]. This type of problems may be approximate to a problem of fluctuating flow of a magneto-viscous flui in case of consieration a very small elastic parameter. Frater [] pointe out that the solution for the velocity shoul ten to the Newtonian value when the elastic parameter vanishes. In this paper the flow of an electrically conucting, incompressible elastico-viscous flui along a flat plate coinciing with the plane y= is consiere, such that the flow is confine to the region y>. The magnetic fiel is assume to be normal to the plate on which the bounary layer is forme. The main purpose of this work is to investigate the effects of the magnetic fiel parameter an Prantl number on the velocity an shear stress of the flui analytically using the classical homotopy perturbation metho (HPM) with the enhancement of Paé approximants (PA) an using the evelope HPM as well; an numerically using the well-known 4 th orer Runge Kutta metho (4RKM). The classical homotopy perturbation metho, base on series approximation, is one among the newly evelope analytical methos for strongly nonlinear problems an has been proven successful in solving a wie class of nonlinear ifferential equations [5 9]. The evelope HPM can be achieve by introucing aition linear operator(s) with unknown parameter(s) that can be chosen suitably to fulfill certain esirable criteria an ientifie optimally [ ]. In this paper, we are intereste in applying the classical HPM with PA technique, evelope HPM an 4RKM for obtaining analytical an numerical solutions of the bounary layer flow of an electrically conucting elastico-viscous flui along an infinite flat plat with heat transfer
4 M.M. Mousa & A. Kaltayev: Homotopy perturbation metho to magneto-elastico-viscous flui along semi-infinite plate in presence of a magnetic fiel normal to the plate. The comparison of the analytical solutions with the numerical solution has been mae an excellent agreement note.. Governing equations In terms of the stream function ψ the governing equations of a steay twoimensional incompressible flow of an electrically conucting elastico-viscous flui over a semi-infinite flat plate coinciing with the plane y=, such that the flow is confine to the region y> uner the influence of a constant transverse applie magnetic fiel normal to the plate on which the bounary layer is forme are given in [, ]. The magnetic Reynols number is assume to be small an negligible in comparison to the applie magnetic file. The governing equations escribe flui motion an temperature are given by ψ M ψ ψ ψ ψ ψ + x y x x k ψ ψ ψ ψ ψ ψ ψ ψ = + xy x x xy () 4 4, 4 T ψ T ψ T =, () Pr x x where M is the magnetic parameter, k is a small elastic parameter representing the non- Newtonian character of the flui an Pr is the Prantl number. The bounary conitions of the problem are: ψ ψ y = : =, =, T = T, () x ψ y : ψ, T, (4) where T an ψ are constants. Because the elastic parameter k is small an may be neglecte, the solution of the problem escribe by Eqs. () an () may be approximate to the solution of the Newtonian flui escribe by the following equations: ψ M ψ ψ ψ ψ ψ x x x + =, T ψ T ψ T + Pr, = x x (5) () uner the same bounary conitions. This approximation gives excellent results in case of small values of k as we will see in the next sections.. Invariant transformation Using one-parametric group transformation inclue in PDEtools package of Maple software, the two-inepenent variables PDEs (5) an () will be transforme into ODEs in only one-inepenent similarity variable... The complete set of invariants The invariants set obtaine by Maple are: gx ( ) ψ( xy, ) y η( xy, ) = x, F( η ) =, θη ( ) = Txy (, ), gx ( ) ( xy, ) = x, F( η) ψ ( xy, ), g( xt ) ( xy, ) θ ( η) = gx ( ) ( xy, ) = x, F( η) = ψ( xy, ), η = y, (7) (8) η y (9) θ ( η) = Txy (, ) exp, g( x) y gx ( ) ( xy, ) η( xy, ) = x, F( η) = ψ, x x x () θη= T xy,, ( ) ( ) where η is the similarity variable, F an θ are invariants of the epenent variables ψ an T respectively an g is an arbitrary function. From the invariants set (7) (), it is clear that the invariants in Eq. () are the only ones which make both of ψ an T a function in x an y.
ISSN: 55-9 International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 5 Therefore, we use Eq. () for oing the similarity transformation of PDEs (5) an ()... The orinary ifferential equations invariant transformation Substituting Eq. () into PDEs (5) an () yiels the following system of ODEs: F( η) + F( η) F( η) M F( η ) =, () θ( η) + Pr F ( η) θ( η) =. () By examining invariants in Eq. () an bounary conitions () an (4), function g (x) shoul be equal to zero in orer to make the left bounary point constant at y=. Therefore, the suitable similarity invariants of this problem are: η ( xy, ) = c, F( η) ( xy, ) y ψ + =, θη ( ) = T( x, y), x x () where c is an arbitrary constant (left bounary point of the similarity bounary problem). Hence, the appropriate corresponing conitions are: ( η ) F η = c: =, F ( η) =, θ( η) = T, (4) η ( ) F η η η :, θ( η). η ψ (5) It is obvious that Eq.() is the Blasius equation in the case of M = [4]. For convenience an comparison with results in [], let ψ = T =, c= an η =. 4. Analytical solution using the classical HPM with PA technique Following the stanar proceures of the HPM escribe in [5 9], the system () an () shoul be written in the classical homotopy form, F ( p) U( η ) + ( ) ( ) p U η U η U( η ) M U( ), η η η + = η () ( p) V( η ) θ + p V + Pr U V = where ( η ) ( η ) ( η ), ( ) ( ) ( ) θ( ) U F, V θ, F = U c = F c an θ = V c = c. (7) One can now try to obtain a solution of system () an () in the form of, ( η) ( η) ( η) ( η) U = U + pu + p U +..., (8) ( η) ( μ) ( η) ( η) V = V + pv + p V +..., (9) where U n an V n, n=,,, are functions yet to be etermine. Substituting Eqs. (8) an (9) into system () an (7), an arranging the coefficients of "p" powers yiels: p : U =, V =, p : U + U U MU =, V + PrU V =, p : U MU + UU + UU =, V + PrUV + PrUV =, () with corresponing initial conitions, ( ) ( ) ( ) V ( ) =, V ( ) = β, U =, U =, U = α, Un = U n = U n = Vn = V n =, at η =, () for n =,,,..., where unknown initial values α an β can be calculate using the bounary conitions in Eq. (5) after obtaining a close form expression to the solution.
M.M. Mousa & A. Kaltayev: Homotopy perturbation metho to magneto-elastico-viscous flui along semi-infinite plate We continue solving system () corresponing to initial conitions () for U n an V n, n=,,, until n= an hence obtaine a six-term approximation: F ( η ) = U an θ ( η) n= n = V. n= It is known that Paé approximations (PA) [] have the avantage of manipulating the polynomial approximation into a rational function of polynomials. This manipulation provies us with more information about the mathematical behavior of the solution. Besies that, a power series solution is not useful for large value of η. Therefore, the combination of the series solution through HBM or any other series solution metho with the Paé approximation provies an effective tool for hanling bounary value problems on semi-infinite omains. It is a known fact that Paé approximation converges on the entire real axis if the solution is free of singularities on the real axis. So, the more accurate analytical solutions will be obtaine after application of PA [M/N] to both F an θ such that M+N (highest power of η in the series solution). We have applie PA [/] to obtain the analytical solution for the problem, say F [/] an θ [/]. n 5. Analytical solution using evelope HPM Accoring to the evelope HPM [ ], a homotopy of the system () an () may be written as ( ) F η + a + () p F( η) F( η) M F( η) a =, θη ( ) b p ( ) ( ) P F η θη b + + r =, () where a an b are unknown constants to be further ientifie. Using p as an expaning parameter as that in the classic perturbation metho, we have : + FF MF =, ( ) = F ( ) = F ( ) =, ( ) ( ) ( ) ( ), θ ( ), p : F + a=, F = F =, F =, θ + b =, θ = = p F a F θ + Pr Fθ b =, θ( ) = θ( ) =. (4) Solving the system (4) an setting p =, we obtain a first-orer approximate solution which reas 7 9 7 9 F F F M M Ma 48 44 8 5 7 Ma + a + + a η + a + a η a η, 4 84 8 5 7 7 9 θ ( η) = θ( η) + θ( η) = + bapr+ Pra + Prb + Pr η 8 4 Prbaη + Pr+ Pr a Prb bapr η 78 9 9 4 ( η ) = ( η) + ( η) = + a a Ma η + η (5) + ba a + b η 44 48 5 Pr Pr Pr. () There are many approaches for ientification of the unknown parameters in the obtaine solution. One of those methos is weighte resiuals, especially the least squares metho [ ]. For the present problem, we set RF RF =, an Rθ Rθ =, (7) a b to ientify the unknown constants a an b,where R F an R θ are the resiuals RF = F + FF MF, an R θ = θ + Pr F θ. (8)
ISSN: 55-9 International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 7. Results an iscussion With the analytical solution given by F [/] an θ [/] using the classical HPM with PA technique, approximate values of α= F () an β=θ () can be calculate using the conitions in Eq. (5). Some numerical results of α an β that are obtaine from F [/] (η )=ψ an θ [/] (η )= are presente in Table for ifferent values of M an Pr when η = an ψ =. M α Table Numerical values of α= F () an β=θ ()for ifferent values of M an Pr β Pr=.5 Pr=.7 Pr=. Pr=. Pr=...459.4445.958.957.449497.49975.5.848.8484.944.455.8795.55599..577.747.797.847.9945.4 (a) (b) (c) Fig. : Profiles of (a) stream function; (b) velocity an (c) shear stress using analytical results of F [/] ; F [/] an F [/] respectively an numerical results of 4RKM for various values of M at Pr=.7 (a) (b)
8 M.M. Mousa & A. Kaltayev: Homotopy perturbation metho to magneto-elastico-viscous flui along semi-infinite plate Fig. : Profiles of (a) temperature an (b) heat transfer using analytical results of θ [/] an θ [/] respectively an numerical results of 4RKM for various values of M at Pr=.7 (a) (b) Fig. : Profiles of (a) temperature an (b) heat transfer using analytical results of θ [/] an θ [/] respectively an numerical results of 4RKM for various values of Pr at M= (a) (b) (c) () Fig. 4: Profiles of (a) stream function; (b) velocity; (c) temperature an () heat transfer using evelope HPM analytical results an 4RKM numerical results for M= an M=.5 at Pr=.7
ISSN: 55-9 International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 9 With the first-orer approximate solution arising in Eqs. (5) an (), approximate values of unknown constants a an b are optimally ientifie using Eqs. (7) an (8) an presente in Table for M= an M=.5 at Pr=.7. Table. Numerical values of a an b for M= an M=.5 at Pr=.7 M a b..7984.8574.5.757794. 47 In orer to obtain a numerical solution, we have solve the initial value problem of Eqs. () an () corresponing to conitions in Eq. (4) an the numerical values arise in Table using the well-known 4RKM. Figs. (a), (b) an (c) show the variations of the flui stream function, velocity an shear stress with η. As shown in Figs. (a) an (b), the stream function F an flui velocity F ecrease an come near to each other as the magnetic parameter M increases. In aition, Fig. (b) shows that the smaller the value of M, the faster it reaches the maximum value of F. From Fig. (c), it is clear that the behavior of the shear stress F epens on the magnetic parameter an the istance. In case of M=, the shear stress starts with the high value, an then ecreases with increasing istance. Oppositely for M>, the shear stress starts with a lower value, an then increases with the istance. Figs. (a) an (b) show the variations of the temperature an heat transfer with η. As shown in Fig. (a), the temperature θ increases with the increasing of M. For M=, the temperature almost linearly epens on η. From Fig. (b), it is clear that the heat transfer θ starts with a higher value for the lower values of M an then ecreases. In aition, for the higher values of M, the behavior of the heat transfer with η tens to be uniform an takes a horizontal shape. Figs. (a) an () illustrate the effect of Prantl number Pr on the temperature an heat transfer at M=. The results are obtaine for Pr =.5,, an. Form Fig., it clear that the temperature an heat transfer rapi ecrease as the Prantl number increases. Moreover, the rapi ecrease of θ an θ becomes more obvious for larger values of Pr. To emonstrate the acceptability an accuracy of evelope HPM results, even though we use only the first-orer approximate solution, the behaviors of the flui stream function, velocity, temperature an heat transfer using the close form solutions in Eqs. () an (), with the values in Table, are illustrate in Figs. 4(a), (b), (c) an () in a comparison with 4RKM results. It is obvious that the results of α an β obtaine by the classical HPM with PA technique are use for obtaining the numerical solution using 4RKM by converting the bounary value problem to an initial value one. Moreover, the analytical solutions using the classical HPM with PA technique an evelope HPM in great agree with the numerical solution using the 4 th orer Runge Kutta metho. The results obtaine in this investigation, in case of the elastic parameter k=, agree with that obtaine in [] in case of k=.. Hence, the problem of fluctuating flow of a magneto-elastico-viscous flui over a semi-infinite flat plate uner the conition of a very small elastic parameter k can be approximate to the problem of fluctuating flow of a magneto-viscous flui, i.e. k=. The present results of F for M= agree with that obtaine in [4] as well. 7. Conclusions The homotopy perturbation metho is applie to the system of nonlinear ifferential equations that escribe a magneto-viscous flui along a semi-infinite flat plate in presence of a magnetic fiel. The excellent agreement of the analytical solution with the 4RKM numerical one shows the reliability an efficiency of the HPM. The behaviors of flui stream function, velocity, shear stress, temperature an heat transfer illustrate by the graphs are consistent with the graphs obtaine in [, 4] an therefore further establish the reliability an effective-ness of the HPM. It has been emonstrate that the HPM can be applie avantageously even when
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