ANALYTIC APPROXIMATE SOLUTIONS FOR FLUID-FLOW IN THE PRESENCE OF HEAT AND MASS TRANSFER

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1 Kilicman, A., et al.: Analytic Aroximate Solutions for Fluid-Flow in the Presence... THERMAL SCIENCE: Year 8, Vol., Sul.,. S59-S64 S59 ANALYTIC APPROXIMATE SOLUTIONS FOR FLUID-FLOW IN THE PRESENCE OF HEAT AND MASS TRANSFER by Adem KILICMAN a*, Yasir KHAN b, Ali AKGUL c, Naeem FARAZ d, Esra Karatas AKGUL e, and Mustafa INC f a Deartment of Mathematics and Institute for Mathematical Research, University Putra Malaysia, Serdang, Selangor, Malaysia b Deartment of Mathematics, Zheiang University, Hangzhou, China c Deartment of Mathematics, Art and Science Faculty, Siirt University, Siirt, Turkey d Deartment of Mathematics, Shanghai University, Shanghai, China e Deartment of Mathematics, Faculty of Education, Siirt University, Siirt, Turkey f Deartment of Mathematics, Science Faculty, Firat University, Elazig, Turkey Original scientific aer htts://doi.org/.98/tsci779k Introduction This aer outlines a comrehensive study of the fluid-flow in the resence of heat and mass transfer. The governing non-linear ODE are solved by means of the homotoy erturbation method. A comarison of the resent solution is also made with the existing solution and excellent agreement is observed. The imlementation of homotoy erturbation method roved to be extremely effective and highly suitable. The solution rocedure exlicitly elucidates the remarkable accuracy of the roosed algorithm. Key words: homotoy erturbation method, heat and mass transfer, fluid-flow Most of the fluid-flows roblems articularly heat and mass transfers are modeled for the non-linear PDE. There are several non-linear PDE in the literature and which can not be solved by analytical methods and hence need to be solved by using numerical methods which lead the researchers to observe the behavior of the system. There are several methods to aroximate the solutions and the most commonly exercised methods are finite difference methods, Runge-Kutta methods, and finite element methods. In ractical use, some of these methods are not easy and also require comlex calculation []. Among these methods, the finite-difference methods are known as effective tools to solve several tyes of PDE []. Further, in the conditional stability analysis of exlicit finite-difference schemes it is also necessary to ut a severe constraint on the time arameter, while the imlicit finite-difference schemes are observed that comutationally exensive [3]. On the other side, these methods can be made highly effective and accurate, but require a structured grids. In this study, we emloy the homotoy erturbation method (HPM) to solve the non-linear ODE which arise in the heat generation and chemical reactions. The HPM was first introduced by He [4]. Well known remarkable feature of the HPM is that a few erturbation terms will be sufficient to obtain reasonable accurate solutions. The technique has been em- * Corresonding author, akilic@um.edu.my

2 S6 Kilicman, A., et al.: Analytic Aroximate Solutions for Fluid-Flow in the Presence... THERMAL SCIENCE: Year 8, Vol., Sul.,. S59-S64 loyed by several researchers in order to solve a large variety of linear and non-linear roblems [5-]. Problem formulation Consider basic governing equations of the roblem with boundary conditions is: with boundary conditions: ( ) f + Re f ff Grθ Gcφ = () θ f δθ fθ Pr + Ec + Re = () Re f S φ γφ φ = (3) f =, f =, θ =, φ = at y = f =, f =, θ =, φ = a t y = According to HPM [4], the homotoy construction of eqs. ()-(3) can be exressed in the form: 3 ( ) ( L ) ( f f) + f + Re f f ff Grθ Gcφ = ( L ) ( θ θ ) + θ + Pr Ec f + Prδθ Pr Re fθ = ( ) ( ) ( L ) ( φ φ ) + φ Sγφ SRe fφ = (4) (5) f = f + f + f +,... θ = θ + θ + θ +,... φ = φ + φ + φ + (6)... Assuming Lf =, Lθ =, and L3φ =, making substitution f, θ, and φ from eq. (6) into eq. (5) and by using simle algebraic simlification and arrangement on owers of -terms, we obtain the following sets of equations: where L, L, and L 3 are defined: () : Lf =, Lθ=, L3φ= f() =, f () =, θ() =, φ() = f (/) =, f (/) =, θ (/) =, φ (/) = (7) 4 = 4, L = L, and L3 = (8) On solving eq. (7) we get initial guess: f 3 θ ( ) =, and φ ( ) = (9) 3 ( ) =,

3 Kilicman, A., et al.: Analytic Aroximate Solutions for Fluid-Flow in the Presence... THERMAL SCIENCE: Year 8, Vol., Sul.,. S59-S64 S6 ( ) : + Re Gr Gc = () Lf ff ff θ φ f () =, f () =, f (/) =, f (/) = : Lθ + Pr Ec f + Prδθ Pr Re f θ =, θ () = θ (/) = () : Re =, () = (/) = () Lφ Sγφ S fφ φ φ : + Re Gr Gc = f () =, f () =, f (/) =, f (/) = ( ) Lf f k f k fk f k θ φ k= k= ( ) Lθ f k f k δθ fkθ k θ θ k= k= : + Pr Ec + Pr Pr Re =, () = (/) = : Re f φ =, φ () = φ ( / ) = ( ) L3φ Sγφ S k = k k On solving eqs. () and () One can use one of the software such as MATHEMAT- ICA, MAPLE or MATLAB. Then we write first order aroximations: 3 Gr Gc Re 3 Gr 3 Gc 3 3Re 3 Gr 4 Gc 4 Re 7 f = EcPr 9Pr Re Prδ 9EcPr δ Pr RePr 3 δ Pr 3 θ = EcPr 4 RePr 5 6EcPr Re S Re S 3 Re S 5 Sλ Sλ Sλ 3 φ = Results and discussion By looking at the grahical reresentation of the results we notice that a very useful demonstration of the efficiency and accuracy of the method HPM for considered roblems. In order to verify the accuracy of the resent method, we have comared HPM results with the numerical and HAM results. The tabs. -4 clearly reveal that resent solution method namely HPM shows excellent agreement with the HAM and numerical solution. This analysis shows that HPM suits for boundary-layer flow roblem in the resence of heat and mass transfer. In figs. -4 we show the velocity, f ( ), temerature, θ, ( ) and concentration rofiles φ ( ) obtained by the HPM. The effect of Grashof number, Gc, (is also known as the local / solutal) on the velocity is shown in the fig.. It is noted from fig. that initially f increases but after the center of the channel it decreases as Grashof number increases. Figures and 3 illustrate the effect of δ and Eckert number on temerature θ. Figure shows that those ositive values of δ increases temerature θ and the negative values of δ decreases temerature θ. From fig. 3 it is found that θ is an increasing function of Eckert number. Figure 4 deicts the influence of chemical reaction arameter γ on the concentration rofiles φ ( ). It is noticed that φ ( ) decreases when γ increases. () ()

4 S6 Kilicman, A., et al.: Analytic Aroximate Solutions for Fluid-Flow in the Presence... THERMAL SCIENCE: Year 8, Vol., Sul.,. S59-S64 Table. Comarison between the HPM, HAM, and numerical solutions of f (), θ (), φ () for different values of Reynolds numbers f / () HPM HAM Numerical θ / () HPM HAM Numerical φ / () HPM HAM Numerical Table. Comarison between the HPM, HAM, and numerical solutions of θ () for different values of Eckert number θ / () HPM HAM Numerical Table 3. Comarison between the HPM, HAM, and numerical solutions of θ () for different values of δ θ / () HPM HAM Numerical Table 4. Comarison between the HPM, HAM, and numerical solutions of φ () for different values of chemical reaction arameter φ() HPM HAM Numerical Re =, λ =, Gr =, S =, Pr =, Ec =, δ =.6.4. Gc = Gc = Gc = Gc = Figure. Variation of local solutal Grashof number on the velocity f..8.6 Re =, λ =, Gr =, S =, Pr =, Ec =, Gc =.4 δ =. δ = 5 δ = δ = Figure. Variation of δ on the temerature θ

5 Kilicman, A., et al.: Analytic Aroximate Solutions for Fluid-Flow in the Presence... THERMAL SCIENCE: Year 8, Vol., Sul.,. S59-S64 S63.75 θ().5.5. Conclusions Re =, λ =, Gr =, S =, Pr =, Gc =, δ = Re =, δ =, Gr =, S =, Pr =, Ec =, Gc = φ() Ec = γ =.5 Ec = γ =. Ec = γ =.5 Ec = 3 γ = Figure 3. Variation of Eckert number on the temerature θ In this aer, the non-linear ODE which result from the similarity solutions of a steady viscous fluid with heat generation and chemical reaction were solved by using an analytical solution method as known the HPM. Comarison of the results obtained using the develoed HPM with numerical and HAM results. The variations of various emerging arameters on the velocity, temerature as well as concentration rofiles are also discussed through the grahs and tables, resectively. Then we can easily make the following observations. y The HPM is an effective and easy to use if one comares with HAM and numerical solution method. y The tangential velocity at the wall is an increasing function of Reynolds number. y Behaviors of δ and Econ temerature θ ( ) are similar. y Concentration rofile decreases by increasing chemical reaction arameter. The roosed analytical aroach for this roblem might have many more alications and thus ossible to aly in similar ways to the other boundary-layer flows to get accurate series solutions..6 Figure 4. Variation of γ on the temerature φ Nomenclature Ec Eckert number, [ ] f dimensionless velocity rofile, [ ] Gc the local solutal Grashof number, [ ] Gr the local thermal Grashof number, [ ] Pr Prandtl number, [ ] Re Reynolds numbers, [ ] S Schmidt number, [ ] Greek symbols γ chemical reaction arameter, [ ] θ dimensionless temerature rofile, [ ] ϕ dimensionless concentration, [ ] Acronym HAM homotoy analysis method References [] Dehghan, M., Shokri, A., A Numerical Method for KdV Equation Using Collocation and Radial Basis Functions, Nonlin. Dynam., 5 (7), -,. - [] Dehghan, M., Tatari, M., Determination of a Control Parameter in a One Dimensional Parabolic Equation Using the Method of Radial Basis Functions, Mathematical and Comuter Modeling, 44 (6), -, [3] Dehghan, M., Finite Difference Procedures for Solving a Problem Arising in Modeling and Design of Certain Otoelectronic Devices, Math. Comut. Simul., 7 (6),,. 6-3 [4] He, J. H., Homotoy Perturbation Technique, Com. Meth. Al. Mech. Engrg., 78 (999), 3-4, [5] Khan, Y., Wu, Q., Homotoy Perturbation Transform Method for Nonlinear Equations Using He s Polynomials, Comut. Math. Al., 6 (), 8,

6 S64 Kilicman, A., et al.: Analytic Aroximate Solutions for Fluid-Flow in the Presence... THERMAL SCIENCE: Year 8, Vol., Sul.,. S59-S64 [6] He, J. H., A Note on the Homotoy Perturbation Method, Thermal Science, 4 (),, [7] Khan, Y., et al., The Effects of Variable Viscosity and Thermal Conductivity on a Thin Film Flow over a Shrinking/Stretching Sheet, Comut. Math. Al., 6 (),, [8] He, J. H., Analytical Methods for Thermal Science An Elementary Introduction, Thermal Science, 5 (), Sul.,. S-S3 [9] Khan, Y., et al., A Series Solution of the Long Porous Slider, Tribology Transactions, 54 (),, [] Khan, Y., Two-Dimensional Boundary Layer Flow of Chemical Reaction MHD Fluid over a Shrinking Sheet with Suction and Inection, J. Aeros. Eng., 7 (4), 5, 449 [] Khan, Y., Latifizadeh, H., Alication of New Otimal Homotoy Perturbation and Adomian Decomosition Methods to the MHD Non-Newtonian Fluid Flow over a Stretching Sheet, Inter. J. Numer. Methods Heat Fluid Flow, 4 (4),, Paer submitted: November 7, 7 Paer revised: December, 7 Paer acceted: December 7, 7 8 Society of Thermal Engineers of Serbia Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This is an oen access article distributed under the CC BY-NC-ND 4. terms and conditions