Explicit Solution of Axisymmetric Stagnation. Flow towards a Shrinking Sheet by DTM-Padé

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1 Applied Mathematical Sciences, Vol. 4, 2, no. 53, Explicit Solution of Axisymmetric Stagnation Flow towards a Shrining Sheet by DTM-Padé Mohammad Mehdi Rashidi * Engineering Faculty of Bu-Ali Sina University, P.O. Box Hamedan, Iran Seied Amin Mohimanian Pour Engineering Faculty of Bu-Ali Sina University, P.O. Box Hamedan, Iran Abstract In this letter, the differential transform method (DTM) is applied to the axisymmetric stagnation flow towards a shrining sheet gives a system of nonlinear ordinary differential equations. The approximate solutions of these equations are calculated in the form of series with easily computable terms. Then, Padé approximant is applied to the solutions to accelerate the convergence of given series. The velocity and temperature profiles are shown. The results obtained in this study are compared with the numerical results. Keywords: Differential transform method, Padé Approximant, Axisymmetric stagnation flow, Shrining Sheet. Introduction Most phenomena in our world are essentially nonlinear and are described by nonlinear equations. Some of them are solved using numerical methods and some are solved using the analytic methods same as perturbation [, 2]. The numerical methods give discontinuous points of a curve and thus it is often costly and time consuming to get a complete curve of results and so in these methods, stability mm_rashidi@yahoo.com (M. M. Rashidi).

2 268 M. M. Rashidi and S. A. Mohimanian Pour and convergence should be considered so as to avoid divergence or inappropriate results. Besides, from numerical results, it is hard to have a whole and essential understanding of a nonlinear problem. Numerical difficulties additionally appear if a nonlinear problem contains singularities or has multiple solutions. In the analytic perturbation methods, we should exert the small parameter in the equation. Therefore, finding the small parameter and exerting it into the equation are deficiencies of the perturbation methods. Recently, much attention has been devoted to the newly developed methods to construct approximate analytic solutions of nonlinear equations without mentioned deficiencies. One of the semi-exact methods which doesn t need small parameters is the DTM, first proposed by Zhou [3], who solved linear and nonlinear problems in electrical circuit problems. Chen and Ho [4] developed this method for partial differential equations and Ayaz [5] applied it to the system of differential equations, this method is very powerful [6]. This method constructs an analytical solution in the form of a polynomial. It is different from the traditional higher order Taylor series method. The Taylor series method is computationally expensive for large orders. The DTM is an alternative procedure for obtaining analytic Taylor series solution of the differential equations. In recent years, the DTM has been successfully employed to solve many types of nonlinear problems [7-5]. Stagnation flow describe the fluid motion near the stagnation region, exists on all solid bodies moving in a fluid. The stagnation region encounters the highest pressure, the highest heat transfer and the highest rate of mass deposition. Problems such as the extrusion of polymers in melt-spinning processes, glass blowing, the continuous casting of metals and the spinning of fibers all involve some aspect of flow over a stretching sheet or cylindrical fiber. In stagnation point flow, rigid wall or a stretching or shrining surface occupies the entire horizontal x-axis, the fluid domain is y > and the flow impinges on the wall with different origin of stretching or shrining on the sheet. Hiemenz [6] was first to study two-dimensional stagnation flow. He used a similarity transform to reduce the Navier Stoes equations to non-linear ordinary differential equations. The axisymmetric case was solved by Homann [7]. Crane [8] found a closed form solution to two-dimensional stretching and Wang [9] obtained similarity solutions for the axisymmetric case. Mahapatra, Gupta [2, 2] and Chiam [22] considered the effect of stagnation point flow over stretching sheets. Wang [23] studied the flow pattern and temperature distribution in stagnation flow on a shrining sheet. He found that non-alignment of the stagnation flow and the shrining (or stretching) of the sheet destroys the symmetry and complicates the flow field. Finally, Loc [24] extended these results to oblique stagnation flow. In this study, the DTM is applied to find the totally analytic solution of axisymmetric stagnation flow towards a shrining sheet and have made a comparison with the numerical solution. M. Rahimpour [25] applied the homotopy analysis method (HAM) to this problem. In this way, the letter has been organized as follows. In Section 2, the flow analysis and mathematical formulation are presented. In Section 3, we extend the application of the DTM to

3 Explicit solution of axisymmetric stagnation flow 269 construct the approximate solutions for the governing equations. In Section 4, we explain about Padé approximation. Section 5 contains the results and discussion. The conclusions are summarized in Section 6. 2 Mathematical formulation Fig. shows an axisymmetric stagnation flow towards an axisymmetric shrining sheet. Non alignment occurs when the line of symmetry of the stagnation flow and that of shrining sheet are not aligned. Wang [23] introduced the following similarity transforms u = ax f ( ) + bch( ), v = ay f ( ), w = 2 υaf ( ), (2.) = z a υ, (2.2) in above equations u, vand w indicate the velocity components in the x, y and z directions, a is the strength of the stagnation flow, b is the stretching rate (shrining if b < ), c is the location of the stretching origin and υ is the inematic viscosity, is independent dimensionless parameter and primes denote differentiation with respect to. f ( ) and h( ) are the velocity similarity variables. Owing to possible non-alignment, it is more appropriate to use Cartesian axes instead of cylindrical axes. The velocities on the shrining (stretching) sheet are as follow u = b( x+ c), v = α a y, w =, (2.3) w w w where α is the ratio of the stretching rate to the strength of the stagnation flow. Upon substitution of Eqs. (2.) and (2.2.) into the Navier Stoes equations, a set of similarity non-linear ordinary differential equations are obtained and the boundary conditions f + f f + f = h + 2f h hf =, 2 2, (2.4) f () =, f () = b a = α, f ( ) =, h() =, h( ) =. (2.5) For the axisymmetric flow, Wang [22] found that solution is unique forα and there is no solution for α <. In this letter, the analytic solutions for α are considered.

4 262 M. M. Rashidi and S. A. Mohimanian Pour Fig.. Axisymmetric stagnation flow on an axisymmetric shrining sheet [24]. The energy equation for axisymmetric flow is T T T T T T u + v + w = κ , (2.6) x y z x y z where T is the temperature and κ is the thermal diffusivity. Let the temperature at infinity and that on the sheet be T, T, respectively. A dimensionless temperature θ is introduced T T θ =, T T (2.7) Eq. (2.6) become θ + 2Prƒ θ =, (2.8) where Pr = υ/ κ is the Prandtl number. The boundary conditions for Eq. (2.8) are θ () =, θ ( ) =. (2.9) 3 The differential transform method Basic definitions and operations of differential transformation are introduced as follows. Differential transformation of the function f ( ) is defined as follows d f ( ) F ( ) = [ ], = (3.)! dt in Eq. (3.) f ( ) is the original function and F( ) is transformed function which is called the T-function (it is also called the spectrum of the f ( ) at =, in the K domain). The differential inverse transformation of F ( ) is defined as

5 Explicit solution of axisymmetric stagnation flow 262 F (3.2) = f ( ) = ( )( ), combining Eqs. (3.) and (3.2), we obtain d f( ) ( ) f ( ) =, d! (3.3) = = Eq. (3.3) implies that the concept of the differential transformation is derived from Taylor s series expansion, but the method does not evaluate the derivatives symbolically. However, relative derivatives are calculated by iterative procedure that are described by the transformed equations of the original functions. From the definitions of Eqs. (3.) and (3.2), it is easily proven that the transformed functions comply with the basic mathematical operations shown in below. In real applications, the function f ( ) in Eq. (3.2) is expressed by a finite series and can be written as N f ( ) F( )( ), (3.4) = Eq. (3.4) implies that F( )( ) is negligibly small, where N is = N + series size. Theorems to be used in the transformation procedure, which can be evaluated from Eqs. (3.) and (3.2), are given below Theorem. If f ( ) = g( ) ± h( ), then F( ) = G( ) ± H ( ). Theorem 2. If f ( ) = cg( ), then F( ) = cg( ). where, c is a constant. n d g( ) ( + n)! Theorem 3. If f ( ) =, then F ( ) = G( + n). n d! Theorem 4. If f ( ) = g( ) h( ), then F ( ) = G( l) H ( l). n = n Theorem 5. If f ( ) =, then F( ) = δ ( n) where, δ ( n) = Taing differential transform of Eqs. (2.4) and (2.8), can be obtained [ ] r = l = [ ] ( + )( + 2)( + 3) F( + 3) + 2 ( + 2 r)( + r) F( r) F( + 2 r) r = ( r + )( + r) F( r + ) F( + r) + δ ( ) =, r= [ ] ( + 2)( + ) H( + 2) + 2 ( + r) F( r) H( + r) r= ( + r) H( r) F( + r) =, (3.5) (3.6) [ ] (3.7) ( + 2)( + ) Θ ( + 2) + 2Pr ( + r) F( r) Θ ( + r) =, r =

6 2622 M. M. Rashidi and S. A. Mohimanian Pour where F( ), H( ) and Θ ( ) are the differential transforms of f (), t h() t and θ (). t The transformed BCs are F() =, F() = b a = α, F(2) = β, H() =, H() = γ, (3.8) Θ () =, Θ () = κ, that β, γ and κ are constants. These constants are computed from the boundary condition. For α = 5, Pr =.7 and N = 22 the solutions of above equations (using the DTM) are as follows f ( ) , h( ) , θ ( ) (3.9) (3.) (3.) 4 Padé approximation Some techniques exist to accelerate the convergence of a given series. Among them, the so-called Padé technique is widely applied. Suppose that a function f ( ) is represented by a power series i = i i = i c, i so that i f ( ) = c. (4.)

7 Explicit solution of axisymmetric stagnation flow 2623 This expansion is the fundamental starting point of any analysis using Padé approximants. The notation ci, i =,,2, K is reserved for the given set of coefficients and f ( ) is the associated function. [ LM, ] Padé approximant is a rational fraction L a + a+ L+ al, M (4.2) b + b+ L+ bm which has a Maclaurin expansion which agrees with Eq.(4.) as far as possible. Notice that in Eq. (4.2) there are L + numerator coefficients and M + denominator coefficients [26]. So there are L + independent numerator coefficients and M independent denominator coefficients, maing L + M + unnown coefficients in all. This number suggests that normally [ LM, ] ought to fit the power series Eq. (4.) through the orders,, 2, K, L + M. In the notation of formal power series L i a + a+ L+ al L+ M + ci = + O( ), M (4.3) i = b + b+ L+ bm Baer [26] found that b b L b c c L a a L a O (4.4) ( M )( ) L ( L M M + + = L ). Equating the coefficients of, 2, K, bmcl M + + bm cl M L+ bcl+ =, bmcl M bm cl M L+ bcl+ 2 =, M b c + b c + L+ b c =. M L M L+ L+ M L+ L+ L+ M (4.5) If j <, we define c j = for consistency. Since b =, Eqs. (4.5) become a set of M linear equations for the M unnown denominator coefficients c b 2 3 M c L M + cl M + cl M + K cl L+ cl M + 2 cl M + 3 cl M + 4 K cl+ bm cl+ 2 cl M + 3 cl M + 4 cl M + 5 K cl+ 2 bm 2 = cl+ 3, M M M M M M c c c K c b c L L+ L+ 2 L+ M L+ M (4.6) from these equations, b i may be found. The numerator coefficients, a, a, K, a L, follow immediately from Eq.(4.4) by equating the coefficients of,, 2, K, L + M

8 2624 M. M. Rashidi and S. A. Mohimanian Pour a = c, a = c + bc a = c + bc + b c M min{ LM, } a = c + b c. L L i L i i =,, (4.7) Thus Eqs.(4.6) and (4.7) normally determine the Padé numerator and denominator and are called the Padé equations. The [ LM, ] Padé approximant is constructed i which agrees with c, i i through order L + M. For more detail the reader is = referred to [26]. The [,] Padé approximants of Eqs. (3.9)-(3.) are as follow f ( ) ( [,] )/( ), (4.8) h( ) ( [,] ) / ( ), (4.9) θ ( ) ( [,] ) / ( ). (4.)

9 Explicit solution of axisymmetric stagnation flow Results and discussion In this paper, the DTM is applied successfully to find analytical solutions of axisymmetric stagnation flow towards a shrining sheet. Graphical representation of results is very useful to demonstrate the efficiency and accuracy of the DTM for above problem. Figs. 2-4 show the velocity components ƒ( ), h ( ) and the dimensionless temperature distribution θ ( ) obtained by the DTM for different value of series size. Figs. 5-8 show the ƒ( ), ƒ ( ), h ( ) and θ ( ) obtained by the DTM and the DTM-Padé in comparison with the numerical solutions obtained by the fifth-order Runge Kutta method. In Figs. 5-8, we can see a very good agreement between the DTM and the numerical results, but these series diverge around infinity. One Padé approximant solve this problem and increase the convergence of given series. So, the solutions are obtained by DTM-Padé are more accurate than the DTM. In Figs. 9-2, the velocity components and the dimensionless temperature distribution are represented for different values of α DTM, N=5 DTM, N=5 DTM, N=2 Numerical f() Fig. 2. The velocity component f ( ) for the axisymmetric stagnation flow towards a shrining sheet obtained by the DTM for different value of N in comparison with the numerical solution, when α =.95.

10 2626 M. M. Rashidi and S. A. Mohimanian Pour h().4.2 DTM, N=5 DTM, N=5 DTM, N=2 Numerical Fig. 3. The velocity component h( ) for the axisymmetric stagnation flow towards a shrining sheet obtained by the DTM for different value of N in comparison with the numerical solution, when α = DTM, N=5 DTM, N=5 DTM, N=2 Numerical θ() Fig. 4. The dimensionless temperature distribution θ( ) for the axisymmetric stagnation flow towards a shrining sheet obtained by the DTM for different value of N in comparison with the numerical solution, when α =.95.

11 Explicit solution of axisymmetric stagnation flow f() DTM, α =-.25 DTM, α =5 DTM-Padé, α = -.25 DTM-Padé, α =5 Numerical Fig. 5. The velocity component f ( ) for the axisymmetric stagnation flow towards a shrining sheet obtained by the DTM ( N = 22) and the DTM-Padé ([,] Padé approximant) in comparison with the numerical solution f'() DTM, α =5 DTM, α = -.25 DTM-Padé, α =5 DTM-Padé, α =-.25 Numerical Fig. 6. The velocity component f ( ) for the axisymmetric stagnation flow towards a shrining sheet obtained by the DTM ( N = 22) and the DTM-Padé ([,] Padé approximant) in comparison with the numerical solution.

12 2628 M. M. Rashidi and S. A. Mohimanian Pour DTM, α = -.25 DTM, α =5 DTM-Padé, α = -.25 DTM-Padé, α =5 Numerical.6 h() Fig. 7. The velocity component h( ) for the axisymmetric stagnation flow towards a shrining sheet obtained by the DTM ( N = 22) and the DTM-Padé ([,] Padé approximant) in comparison with the numerical solution DTM, α =-.25 DTM, α =5 DTM-Padé, α = -.25 DTM-Padé, α =5 Numerical θ() Fig. 8. The dimensionless temperature distribution θ ( ) for the axisymmetric stagnation flow towards a shrining sheet obtained by the DTM ( N = 22) and the DTM-Padé ([,] Padé approximant) in comparison with the numerical solution.

13 Explicit solution of axisymmetric stagnation flow 2629 f() α =5 α =2 α = α =.5 α = α = -.25 α =-.5 α = -.75 α = Fig. 9. The velocity component f ( ) for different values of α obtained by the DTM-Padé ([,] Padé approximant). 5 f'() α =5 α =2 α = α =.5 α = α =-.25 α =-.5 α =-.75 α = Fig.. The velocity component f ( ) for different values of α obtained by the DTM-Padé ([,] Padé approximant).

14 263 M. M. Rashidi and S. A. Mohimanian Pour. h() α =5 α =2 α = α =.5 α = α =-.25 α =-.5 α =-.75 α = Fig.. The velocity component h( ) for different values of α obtained by the DTM-Padé ([,] Padé approximant). θ() α =5 α =2 α = α =.5 α = α = -.25 α =-.5 α = -.75 α = Fig. 2. The dimensionless temperature distribution θ ( ) for different values of α obtained by the DTM-Padé ([,] Padé approximant), when Pr =.7.

15 Explicit solution of axisymmetric stagnation flow Conclusion In this paper, the DTM was applied successfully to find the analytical solution of axisymmetric stagnation flow towards a shrining sheet. The results show that the differential transform method does not require small parameters in the equations, so the limitations of the traditional perturbation methods can be eliminated. The reliability of the method and reduction in the size of computational domain give this method a wider applicability. Therefore, this method can be applied to many nonlinear integral and differential equations without linearization, discretization or perturbation. References [] A.H. Nayfeh, Introduction to perturbation techniques, Wiley, 979. [2] R.H.Rand, D. Armbruster, Perturbation methods, bifurcation theory and computer algebraic, Springer, 987. [3] J.K. Zhou, Differential transformation and its applications for electrical circuits (in Chinese), Huazhong Univ. Press, Wuhan, China, 986. [4] C.K. Chen, S.H. Ho, Solving partial differential equations by two dimensional differential transform method, Applied Mathematics and Computation 6(999), [5] F. Ayaz, Solutions of the systems of differential equations by differential transform method, Applied Mathematics and Computation 47(24), [6] I.H. Abdel-Halim Hassan, Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos, Solitons and Fractals 36(28), [7] A.S.V. Ravi Kanth, K. Aruna, Solution of singular two-point boundary value problems using differential transformation method, Physics Letters A 372(28), [8] A. Arioglu, I. Ozol, Solution of differential difference equations by using differential transform method, Applied Mathematics and Computation 8(26), [9] A. Arioglu, I. Ozol, Solution of boundary value problems for integro-differential equations by using differential transform method, Applied Mathematics and Computation 68(25), [] N. Bildi, A. Konuralp, F. Oracı Be, S. Kucuarslan, Solution of different type of the partial differential equation by differential transform method and Adomian s decomposition method, Applied Mathematics and Computation 72(26),

16 2632 M. M. Rashidi and S. A. Mohimanian Pour [] F. Ayaz, Solutions of the system of differential equations by differential transform method, Applied Mathematics and Computation 47(24), [2] M.M Rashidi, Differential Transform Method for Solving Two Dimensional Viscous Flow, International Conference on Applied Physics and Mathematics (29). [3] M.M Rashidi, Differential Transform Method for MHD Boundary-Layer Equations: Combination of the DTM and the Padé Approximant, International Conference on Applied Physics and Mathematics (29). [4] M.M Rashidi, E. Erfani, A Novel Analytical Solution of the Thermal Boundary-Layer Over a Flat Plate with a Convective Surface Boundary Condition Using DTM-Padé, International Conference on Applied Physics and Mathematics (29). [5] M.M Rashidi, E. Erfani, New Analytical Method for Solving Burgers and Nonlinear Heat Transfer Equations and Comparison with HAM, Computer Physics Communications 8 (29) [6] K. Hiemenz, D. Grenzschicht, an einem in den gleichformingen Flussigeitsstrom eingetauchten graden Kreiszylinder, Dinglers Polytech. J., 326(9), [7] F. Homann, D. Einfluss, Zahigeit bei der Stromung um den Zylinder und um die Kugel, Z. Angew. Math. Mech., 6(936), [8] L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys., 2(97), [9] C.Y. Wang, The three-dimensional flow due to a stretching flat surface, Physics of Fluids 27(984), [2] T.R. Mahapatra, A. S. Gupta, Heat transfer in stagnation-point flow towards a stretching sheet, International Journal of Heat and Mass Transfer 38(22), [2] T. R. Mahapatra, A. S. Gupta, Stagnation-point flow towards a stretching surface, The Canadian Journal of Chemical Engineering 8(23), [22] T. Chiam, Stagnation-point flow towards a stretching plate, Journal of the Physical Society of Japan 63(994), [23] C.Y. Wang, Stagnation flow towards a shrining sheet, International Journal of Non-Linear Mechanics 43(28), [24] Y.Y. Lo, N. Amin, I. Pop, Non-orthogonal stagnation point flow towards a stretching sheet, International Journal of Non-Linear Mechanics 4(26), [25] M. Rahimpour, S.R. Mohebpour, A. Kimiaeifar, G.H. Bagheri, On the analytical Solution of axisymmetric stagnation flow towards a shrining sheet, International Journal of Mechanics 2(28). [26] G.A. Baer, P.G. Morris, P.A. Carruthers, Padé Approximants Part I: Basic Theory, Addison-Wesley Publishing Company (98). Received: February, 2

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