Explicit Analytic Solution for an. Axisymmetric Stagnation Flow and. Heat Transfer on a Moving Plate

Transcription

1 Int. J. Contep. Math. Sciences, Vol. 5,, no. 5, Explicit Analytic Solution for an Axisyetric Stagnation Flow and Heat Transfer on a Moving Plate Haed Shahohaadi Mechanical Engineering Departent, Engineering Faculty of Bu- Ali Sina University, Haadan, Iran haed_shah6@yahoo.co Ahad DoostHoseini Mechanical Engineering Departent, Engineering Faculty of Bu- Ali Sina University, Haadan, Iran a_doosthoseini@yahoo.co Abstract In this article, we use an efficient analytical ethod called hootopy analysis ethod (HAM) to derive an approxiate solution of an axisyetric stagnation flow on a oving solid boundary. Actually, we solved the Navier-Stocks and energy equations by the HAM. Unlike the perturbation ethod, the HAM does not require the addition of a sall physically paraeter to the differential equation. It is applicable to strongly and weakly nonlinear probles. Moreover, the HAM involves an auxiliary paraeter, h which renders the convergence paraeter of series solutions Controllable, and increases the convergence, and increases the convergence significantly. This article depicts that the HAM is an efficient and powerful ethod for solving nonlinear differential equations. Keywords: Nonlinear differential equations; axisyetric stagnation flow; Hootopy analysis ethod (HAM). Introduction Modeling of natural phenoena in ost sciences yields nonlinear differential

2 7 H. Shahohaadi, A. DoostHoseini equations the exact solutions of which are usually rare. Therefore, analytical ethods are strongly needed. For instance, one analytical ethod, called perturbation, involves creating a sall physically paraeter in the proble, however, finding this paraeter is ipossible in ost cases [, ]. Generally speaking, one siple solution for controlling convergence and increasing it does not exist in all analytical ethods. In 99, Liao [3] presented hootopy analysis ethod (HAM) based on fundaental concept of hootopy in topology [4-9]. In this ethod, we do not need to apply the sall paraeter and unlike all other analytic techniques, the HAM provides us with a siple way to adjust and control the convergence region of approxiate series solutions. HAM has been successfully applied to solve any types of nonlinear probles [-4]. In this work, the basic idea of HAM is described, and then we apply it to the stagnation flow equations. Stagnation flow on a oving solid boundary is basic in any convection-cooling processes. Stagnation flow towards a oving plate has been considered by Root [6], Wang [7], Libby [5] and extended by Weidan and Mahalinga [8]. These sources applied the conventional no slip condition on the solid boundary.. Basic idea of HAM Let us consider the following differential equation Nuτ [ ( )] =, () where N is a nonlinear operator, τ denotes in dependent variable, u ( τ ) is an unknown function that is the solution of the equation. We define the function φ(; τ p) = u (), τ p () where, p [,] and u ( τ ) is the initial guess which satisfies the initial or boundary condition and is li φ( τ; p) = u( τ). p (3) By eans of generalizing the traditional hootopy ethod, Liao [3] constructs the so-called zero- order deforation equation ( p) L[ φ( τ; p) u ( τ)] = ph H( τ) N [ φ( τ; p)], (4) where h is the auxiliary paraeter which increases the results convergence, H () τ is an auxiliary function and L is an auxiliary linear operator, p increases fro to, the solution φ(; τ p) changes between the initial guess u (; τ p) and

3 Explicit analytic solution 7 solution u () τ. Expanding φ(; τ p) in Taylor series with respect to p, we have φτ (; p) = u () τ + u () τ p, (5) where = u φτ ( ; p) ( τ ) =! p p =, (6) if the auxiliary linear operator, the initial guess, the auxiliary paraeter h, and the auxiliary function are so properly chosen, the series (5) converges at p =, and then we have u( τ ) = u ( τ) + u ( τ), (7) = which ust be one of the solutions of the original nonlinear equation, as proved by Liao [7]. It is clear that if the auxiliary paraeter is h = and auxiliary function is deterined to be H () τ =, Eq. (4) will be ( p) L[ φ( τ; p) u ( τ)] + p N[ φ( τ; p)] =, (8) this stateent is coonly used in hootopy perturbation ethod (HPM) procedure. Indeed, in HPM we solve the nonlinear differential equation by separating every Taylor expansion ter. Now we define the vector of u r as follows r r r r r u = u, u, u,..., u { n} 3. According to the definition Eq. (6), the governing equation and the corresponding initial condition of u () τ can be deduced fro zero-order deforation Eq. (4). Differentiating Eq. (4) for ties with respect to the ebedding paraeter p and setting p = and finally dividing by!, we will have the so-called th order deforation equation in the following fro r Lu [ ( τ ) x u ( τ)] = hh( τ) R ( u ), where (9) R r N[ φτ ( ; p)] ( u ) = ( )! p p =, () and x =, >. ()

4 7 H. Shahohaadi, A. DoostHoseini So by applying inverse linear operator to both sides of the linear equation, Eq. (9), we can easily solve the equation and copute the generation constant by applying the initial or boundary condition. 3. Application Now we consider the axisyetric stagnation flow. Fig. shows an axisyetic stagnation flow towards a oving plate with velocity U. We can align the x-axis with plate otion without loss of generality. The flow far fro the plate is given by the potential flow u = ax v = ay w = az p = p a x + y + a z (),,, ρ[ ( )/ ], where uvw,, are velocity coponents in the Cartesian x, y, z directions, a is the strength of the stagnation flow, ρ is the density, p is the pressure and p is the stagnation pressure. For viscous flow, let u = axf ( ) + Ug( ), v = a yh( ), w = aνf ( ), (3) p = p a x + y + w + w z (4) ρ[ ( )/ / ν ], where a/ vz and v is the kineatic viscosity. The subscript z denotes differentiation with respect to z. The two-diensional Navier- Stokes equations then reduce to the siilarity ordinary differential equations [9]. f ff f + ( ) + =, (5) g + fg f g =, (6) On the plate, Navier s condition gives u U = N ρv u, υ V = N ρνυ, (7) z z where N is a slip constant. The no slip condition is recovered when N=. For (no slip) the boundary conditions: f () =, f ( ) =, f () =, (8) g( ) =, g() =, (9) Let the teperature far fro the plate be T and teperature on the plate be T. Set T T θ ( ) =. T T () The energy equation becoes

5 Explicit analytic solution 73 θ + Pfθ =, () where P is the Prandtl nuber. A teperature slip condition siilar to Navier s condition is T T = ST z, () where S is a proportionality constant. Eq. () can be written as θ ( ) = + βθ ( ), (3) where a β S is the theral slip factor. At infinity the condition is v θ( ) =. (4) Now we solve the proble for no slip condition. In this work we consider β =. Fig.. Axisyetic stagnation flow. We choose the initial approxiation. e f ( ), = (5) g ( ) e, = (6) ( ) P, e θ = (7) π and the linear operator for equation (5) 3 φ ( ; p) L[ φ ( ; p)] =, 3 (8) and the linear operator for equation (6) φ ( ; p) L[ φ ( ; p)] =, (9)

6 74 H. Shahohaadi, A. DoostHoseini and the linear operator for equation () φ3 ( ; p) L[ φ3 ( ; p)] =, (3) we change equations (5), (6) and () to nonlinear for φ ( ; p) φ ( ; p) φ ( ; p) N p p p p 3 [ φ( ; ), φ( ; ), φ3( ; )] = + φ 3 ( ; ) ( ) +, (3) φ( ; p) φ( ; p) φ( ; p) 3 = + N [ φ (, p), φ (, p), φ ( ; p)] φ ( ; p) φ ( ; p), (3) φ3( ; p) φ3( ; p) N 3[ φ( ; p), φ( ; p), φ3( ; p)] = + Pφ ( ; p), (33) assuing H ( τ ) =, we use above definition to construct the zero-order deforation equations. ( ) 3 p L[ φ ( ; p) f ( )] = ph N [ φ ( ; p), φ ( ; p), φ ( ; p)], (34) ( ) 3 p L[ φ ( ; p) g ( )] = ph N [ φ ( ; p), φ ( ; p), φ ( ; p)], (35) ( ) p L[ φ ( ; p) θ ( )] = ph N [ φ ( ; p), φ ( ; p), φ ( ; p)], (36) obviously, when p= and p=, φ ( ;) = f ( ), φ ( ;) = f ( ), (37) φ ( ;) = g ( ), φ ( ;) = f ( ), (38) φ ( ;) = θ ( ), φ ( ;) = θ( ), (39) 3 3 Differentiating the zero-order deforation equations (34), (35) and (36) ties with respect to p. r r Lf [ x f ] = h R ( f, g, θ ), (4) r r Lg [ x g ] = h R ( f, g, θ ), (4) r r L[ θ x θ ] = h R ( f, g, θ ), (4) where 3 r r r f ( ) f ( ) f ( ) f ( ) R ( f, g, ) [ f ( ) ] ( x ), 3 n n n θ = + 3 n + n = r r r g ( ) g ( ) f ( ) R ( f, g, θ ) [ f ( ) g ( )], n n = + n n n = (43) (44)

7 Explicit analytic solution 75 r r r θ ( ) θ ( ) R ( f, g, ) P f ( ), n 3 θ = + n n = (45) and x, = >. (46) Fro (5)-(7) and (43) to (45), we now successively obtain the f ( ), g ( ) and θ ( ). The equations (4), (4) and (4) are linear and thus can be easily solved, especially by eans of sybolic coputation software such as Matheatica, Maple, MathLab and so on. We used ters in evaluating the approxiate solution. f = f ( ) = f ( ) + ( ), (47) g = g ( ) = g ( ) + ( ), (48) θ ( ) = θ ( ) + θ ( ). (49) = Note that this series contains the auxiliary paraeter h, which influence its convergence region and rate. We should therefore focus on the choice of h by plotting of h -curve. Fig., shows the h -curve of 3 g () 3, Fig.3, shows the θ (), h -curve of 3 3 h -curve of 3 f () 3. θ () Fig.4, shows the h -curve of 3 3 and Fig.5, shows the h h Fig.. The th-order approxiation of g ( ) versus h Fig.3. The th-order approxiation of θ ( ) versus h (P=.8)

8 76 H. Shahohaadi, A. DoostHoseini h h Fig.4. The th-order approxiation of θ ( ) versus h Fig.5. The th-order approxiation of f ( ) versus h (P=.) We should select optial h fro the region in which the diagra is quite horizontal. Horizontal region is the optial h region. Regarding figure () optial h equals, regarding figure (3) optial h equals -., regarding figure (4) optial h equals and regarding figure (5) optial h equals -. In this article we have obtained the values of f, g, θ by applying HAM rearkable ethod as well as by nuerical ethod and you will see the consequences of these ethods in figures No, 6, 7,8 and 9. These four diagras apparently show that quite analytic ethod of HAM is so close to nuerical solution with great exactness, which is a token of its high accuracy. Fig. 6. Coparison of nuerical results with HAM of f ( )

9 Explicit analytic solution 77 Fig. 7. Coparison of nuerical results with HAM of g ( ) Fig. 8. Coparison of nuerical results with HAM of θ ( ) for (P=.)

10 78 H. Shahohaadi, A. DoostHoseini Fig. 9. Coparison of nuerical results with HAM of θ ( ) for (P=.8) It should be noted that we obtain θ( ) for Prandtl nuber. and.8. We can understand that the HAM is a good ethod for varies value of Prandtl nuber. This is power of the HAM 4. Conclusions In this paper, we utilized the powerful ethod of hootopy analysis to obtain the stagnation flow equations. We achieved a very good approxiation with the nuerical solution of the considered proble. In addition, this technique is algorithic and it is easy to ipleentation by sybolic coputation software, such as Maple and Matheatica. Different fro all other analytic techniques, it provides us with a siple way to adjust and control the convergence region of approxiate series solutions. Unlike perturbation ethods, the HAM does not need any sall paraeter. It shows that the HAM is a very efficient ethod. References [] N. Ali Hasan, Introduction to Perturbation Techniques, John Wiley Sons, New York, 98, pp. 4. [] N. Ali Hasan, Probles in Perturbation, John Wiley & Sons, 985, pp..

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