# Differential Transformation Method for solving Falkner-Skan equation to Boundary Layer Flow

**ABSTRACT** n this paper we apply the differential transformation method to solve the third –order boundary value problems characterized by the Falkner-Skan equation over a semi-infinite domain.

The Falkner-Skan equation has two coefficients. Four types of flow in the boundary layer along a surface of revolution near the stagnation point are studied by giving different specific values for coefficients of Falkner-Skan equation. These types of flow are called Blasius flow, Homann axisymmetric stagnation flow, Hiemenz flow, and Homann steady flow.

The effectiveness of the method is illustrated by applying it successfully to various instances of the Falkner-Skan. The differential transformation method show good agreements of results for solving nonlinear BVPs.

**2**Bookmarks

**·**

**327**Views

- Citations (16)
- Cited In (0)

- [Show abstract] [Hide abstract]

**ABSTRACT:**We present a computational method for the solution of the third-order boundary value problem characterized by the well-known Falkner–Skan equation on a semi-infinite domain. Numerical treatments of this problem reported in the literature thus far are based on shooting and finite differences. While maintaining the simplicity of the shooting approach, the method presented in this paper uses a technique known as automatic differentiation, which is neither numerical nor symbolic. Using automatic differentiation, a Taylor series solution is constructed for the initial value problems by calculating the Taylor coefficients recursively. The effectiveness of the method is illustrated by applying it successfully to various instances of the Falkner–Skan equation.Journal of Computational and Applied Mathematics 01/2005; 176(1):203-214. · 0.99 Impact Factor - Annals of Mathematics · 3.03 Impact Factor
- SourceAvailable from: damtp.cam.ac.ukPhilosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 01/1960; 253(1023):101-136. · 2.89 Impact Factor

Page 1

Differential Transformation Method for solving

Falkner-Skan equation to Boundary Layer Flow

Dr. Anwar Ja’afar Mohamad Jawad

Al-Rafidain University College , Iraq anwar_jawad2001@yahoo.com

Dr. Thair Saddon Slibi

Al-Rafidain University College , Iraq.

Abstract

In this paper we apply the differential transformation method to solve the third –

order boundary value problems characterized by the Falkner-Skan equation over a

semi-infinite domain.

The Falkner-Skan equation has two coefficients. Four types of flow in the

boundary layer along a surface of revolution near the stagnation point are studied

by giving different specific values for coefficients of Falkner-Skan equation. These

types of flow are called Blasius flow, Homann axisymmetric stagnation flow,

Hiemenz flow, and Homann steady flow.

The effectiveness of the method is illustrated by applying it successfully to

various instances of the Falkner-Skan. The differential transformation method show

good agreements of results for solving nonlinear BVPs.

1. Introduction

The nonlinear third-order Falkner-Skan equation is a famous example of

the BVPs on infinite intervals arisen in many branches of sciences, e.g. applied

mathematics, physics, fluid dynamics and biology. Falkner-Skan equation[1] is:

( ) ( ) ( ) , ( ( )) -

( ) ( ) ( ) (1)

Which arises in the study of viscous flow past a wedge of angle ( ) as shown in

figure (1). Where are constants. It is known that a smooth solution

( ) exists and is unique for each .

Figure (1) Illustrates of Falkner-Skan flow past a wedge

Page 2

Previous work on this equation includes the mathematical treatments

due to Weyl [2] , Copper[3], and Rosenhead [4]. These works have mainly

focused on obtaining existence and uniqueness results. Smith [5] , Cebeci and

Keller [6] , and Na [7] have considered other numerical methods. These

approaches have used shooting and invariant imbedding. Salama [8] developed

a one-step method of order 5.

In this paper, the Differential Transformation Method (DTM) is applied

to solve the nonlinear third-order Falkner-Skan equation. The DTM is considered

another technique used for solving BVPs near singular points. The power and

strength of the Differential Transformation Method (DTM) is to solve the

nonlinear third-order Falkner-Skan equation for which is not

discussed in previous literatures.

In section 2 we try to explain all types of flow included from the general

Falkner-Skan equation. Section 3 illustrates the outline of the Differential

Transformation Method (DTM). In section 4 we apply the Differential

Transformation Method (DTM) to solve the nonlinear third-order Falkner-Skan

equation. Section 5 presented the numerical results.

2. Types of flow

The flow scheme in Figure (2) is the result of the interaction of a

current that is spatially uniform for large negative X (left part of diagram) with a

solid plate (thin shaded rectangle), which is idealized as being infinitely thin and

extending infinitely far to the right as X → ∞. Because all fluid flows must be

zero at a solid boundary, the velocity must slow rapidly to zero in a “boundary

layer,” which thickens as X→∞. The region of velocity change (“shear”) is

called a “boundary layer” because in fluids of low viscosity, such as air and water,

the shear layer is very thin [9].

Figure (2) flow scheme

The Blasius problem has developed a vast bibliography [10]. Even

though the problem is almost a century old, recent papers that employ the

Blasius problem as an example include [11-13].

The boundary layer equations for the incompressible viscous flow over

a flat plate with zero pressure gradient are:

,

(2)

Page 3

If we assume similarity between boundary layers and introduce the

stream function, these two partial differential equations (PDE’s) reduce to a

single 3rd order ordinary differential equation (ODE) as shown below.

( ) ( ) ( ) (3)

which is called Blasius’ Equation. This equation can be solved for the

parameter ( ) which is nothing more than a modified stream function whose

first derivative is the ratio of local horizontal velocity (u) to free stream

velocity( ) , i.e.

( ) . Further, the new independent variable in this

equation is a scaled distance from the wall given by:

√

(4)

Since this is a third order ODE, we need to have three boundary conditions to

solve. The boundary conditions for this case are that both components of the

velocity are zero at the wall due to no slip, and that the horizontal velocity

approaches the constant free stream velocity at some distance away from the

plate.

( ) ( )

Unfortunately, since Blasius’s equation is non-linear (it includes the product of

the dependant function with one of its derivatives), there is not known analytic

solution. Instead, it is common to solve this type problem numerically though a

marching method: i.e. start at an initial point, say = 0 , with initial conditions

and march in space to a desired vale of . Since we only have 2 initial conditions

and one condition at the other boundary, , we will have to supply an

arbitrary 3rd initial condition on ( ) , say ( ) . If, after marching the

solution to a suitably large value of , the other boundary condition of

( ) is not satisfied, we correct our initial guess for and begin

iterating till the far boundary condition is satisfied.

In general the two dimensional constant viscous flow over a semi-

infinite flat plate is modeled by the Blasius problem

( ) ( ) ( )

( ) ( ) ( ) , ) (5)

Here represents a similarity variable introduced by Blasius [14] to transform

a pair of partial differential Navier-Stokes equations into a single ordinary

differential equation contained in Eq.(5). For a thorough discussion of this

problem see Schlichting and Gersten [15]. If the third condition in Eq.(5) is

replaced by ( ) where > 0, then the initial-value problem

( ) ( ) ( )

( ) ( ) ( ) , ) (6)

will be called the Blasius initial value problem. It is well-known that any solution

of Eq.(6) corresponding to a fixed has the property that as , the first

derivative of ( ) approaches a constant limit. There is a unique value of , say

, for which the solution of the initial-value problem Eq.(6) is also the solution

( )

Page 4

of the Blasius problem Eq.(5). Several authors have devised numerical

algorithms to find a good approximate value of this number, see Asaithambi [1]

and references therein. For

(7)

Recently Wang [16] developed an analytical method for finding . Fang et al.

[17] showed that for arbitrary ,

√ (8)

Steven Finch [18] generalizes the discussion to clear up any confusion and

considered

( ) ( ) ( ) ( ) ( ) ( )

(9)

where > 0, b > 0.

When ( )

Then

√ (10)

, and thus for

where ξ = 0.4695999883

In view of the above result, it is sufficient to consider the Blasius problem for a

fixed , say

Let , then the boundary-value problem

( )

( ) ( )

( ) ( ) ( ) , ) (11)

will be called the modified Blasius problem and the related problem

( )

( ) ( )

( ) ( ) ( ) , ) (12)

will be named the modified Blasius initial value problem. Physically problem

Eq.(11) now models a boundary layer flow over a moving plate with constant

velocity . Every solution of the problem Eq.(6) increases from 0 to 1 remaining

convex throughout until ( ) approaches a constant limit. On the other hand a

solution of the problem in Eq.(12) first decreases attains a minimum and then

increases to infinity. A striking difference between the two problems is that

whereas the Blasius problem Eq.(5) possesses a unique solution, the modified

Blasius problem in Eq.(12) possesses two solutions when

(13)

these solutions coalesce into a unique solution for and there is no solution

beyond this critical value [11]. The evidence in favor of this remarkable

phenomenon's based on numerical results and it is not clear why it should

appear at and then disappear at .

it is found that

( )

: = ξ / √2 = 0.3320573362...,

. We shall do this in this paper.

Page 5

Finally it is concluded that Blasius' equation is a special case of Falkner-

Skan equation when = 1/2, in which the wedge reduces to a flat

plate. If = 1, = 1, it describes Hiemenz flow; If = 1, = 0.5, it describes

Homann axisymmetric stagnation flow; if = 2, = 1, it represents the

problem of Homann, describing the steady flow in the boundary layer along a

surface of revolution near the stagnation point. Sometimes, the Falkner-Skan

equation specifically refers to = 1.

Physically relevant solutions also exist for negative , more precisely, in the

range −0.19883768... = μ ≤ < 0. (Positive corresponds to flow toward the

wedge; negative corresponds to flow away from the wedge.) By “physically

relevant”, we mean that a solution ( ) further satisfies [11]:

( ) for all ( )

3. Outline of Differential Transformation Method

The differential transformation DTM of the k-ith derivatives of function ( )

is defined as follows [19, 20]:

, ( )

and is the differential inverse transformation of

( ) ∑ ( )( )

(15)

for finite series of k = N , Eq.(15) can be written as:

( ) ∑ ( )( )

(16)

The following theorems can be deduced from Eqs. (14) and (16) are given

below [20]:

Theorem 1. If ( ) ( ) ( ) ,then ( ) ( ) ( ) (17)

Theorem 2. If ( ) ( ) ,then ( ) ( ). (18)

Theorem 3. If ( ) ( )

Theorem 4. If If ( ) ( )

Theorem 5. If ( ) ( ) ( ) ,then ( ) ∑

Theorem 6. If ( ) ,then ( ) ( ) {

( )

- (14)

)(xy

)(kY

defined as follows:

,then ( ) ( ) ( ) (19)

,then ( ) ,( ) - ( ) (20)

( )

( ) (21)

(22)

4. Numerical Application

The solution of Falkner-Skan equation Eq.(1) is ( ) which will be calculated

by applying the Differential Transformation Method then the recurrence

equation is:

( ) ( )( )( ) ( )

* ∑

( ) ( )( ) ( )+ -

The boundary condition at ( ) is replaced with a free boundary

condition as Asaithambi did in [1]:

( ) ( ) , at (24)

( )( )( ), ∑

(23)

Page 6

in which is the unknown free boundary (truncated boundary). Then the

original problem Eq.(1) becomes the free boundary problem with Eq.(24)

defined on a finite interval, where is to be determined as a part of the

solution. Then, a shooting algorithm is used to find the solution of the boundary

value problem . We add another initial value condition

( ) (25)

We will use Runge-Kutta method repeatedly for the solution of the IVP

represented by Eq.(1) and Eq.(25) until we find the and such that the

condition Eq.(24) is satisfied.

The solution ( ) is:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

(26)

The three boundary conditions are ( ) ( ) ( )

, and then applying the Differential Transformation Method to them we get:

( ) , ( ) and ( )

Where is from Eq.(8). Substituting in

Eq.(23) then:

( )

( )

. (27)

,

,

, * ( )+ * ( ( ) )+-,

( )

( )

, * ( ) ( )+ * ( ( ) ( ) )+- ,

, * ( ) ( ) ( )+ * ( ( ) ( ) ( ) )+-

, * ( ) ( ) ( ) ( )+ * ( ( ) ( )

( ) ( ))+-

( )

( ( ) ( ) ( ) ( ) ( ))+-

( )

( ) ( ) ( ) ( ) ( ) ( ) )-

(28)

5. Numerical Results

In this part, we will present our numerical results corresponding to

various instances of flow cases. We plot in Figures the solution curves of the

Falkner-Skan equations.

Case 1 : (Blasius equation) For

( )

( )

, * ( ) ( ) ( ) ( ) ( )+ *

, * ( ) ( ) ( ) ( ) ( ) ( )+ (

, .

√

( ) , ( ) and ( ) 0.1660286681 .

( ) , ( ) , ( ) ,

( ) , ( ) , ( )

Page 7

( ) , ( ) , ( )

We get the solution of Blasius equation ( ) as in the following equation :

( )

(29)

Figure (3) represents Blasius flow for its function ( ) and derivatives

( ) ( ) for .

Figure (3) Blasius flow

Case 2 : ( Hiemenz flow), For , .

√

( ) , ( ) and ( ) .

( ) , ( ) , ( ) ,

( ) , ( )

( ) , ( ) ,

( ) .

( )

(30)

Figure (4). represents Hiemenz flow for its function ( ) and derivatives

( ) ( ) for .

Page 8

Figure (4) Hiemenz flow

Case 3 ( Homann axisymmetric stagnation flow ), if = 1, = 0.5.

( ) , ( ) and ( ) .

( ) , ( )

( ) , ( ) ,

( ) , ( ) ,

( ) , ( )

( )

. (31)

Figure (5) represents Homann axisymmetric stagnation flow

( ) and derivatives ( ) ( ) for .

flow for its function

Figure (5) Homann axisymmetric stagnation flow

Page 9

Case 4 : it represents the problem of (Homann steady flow) in the boundary

layer along a surface of revolution near the foreard stagnation point if

= 2, = 1.

( ) , ( ) and ( ) .

( ) 0.1666667, ( ) ,

( ) , ( ) ,

( ) , ( ) ,

( ) , ( ) .

( )

–

(32)

Figure (6) represents Homann steady flow

derivatives ( ) ( ) for .

flow for its function ( ) and

Figure (6) Homann steady flow

Figures (7-9) illustrates the comparisons between f ( ) and derivatives

( ) ( ) for the four types of flow in .

Figure(10) ( ) Corresponding to different , for = 1.

Figure (10) show that a reverse flow from the wedge can be occur due to

( ) which is one of the more important result in this paper.

Page 10

Figure (7) ( ) Corresponding for different flows

Figure (8) ( ) Corresponding for different flows

Page 11

Figure (9) ( ) Corresponding for different flows

Figure (10) ( ) Corresponding to different , = 1.

6. Conclusion

The Differential Transformation Method is presented for the boundary value

problems of a class of nonlinear third-order differential equation on semi-infinite

intervals. Using small values of , can gave good approximation result

for ( ). We successfully compute several instances of the Falkner-Skan

Page 12

equation by the DTM with different values of the coefficients that show different

types of flow. One of the more important result in this paper is the reverse flow

from the wedge that can be occur due to ( ) . Our results are in excellent

accordance with those already reported in literatures.

References

[1] Asai Asaithambi; Solution of the Falkner-Skan equation by recursive

evaluation of Taylor coefficients, J. Comput. Appl. Math. 176 (2005) pp. 203-

214.

[2] H. Weyl ; On the differential equations of the simplest boundary-layer

problem, Ann of Math. 43 (1942) 381-407.

[3] W.A. Coppel ; On a differential equation of boundary layer theory , Philos.

Trans. Roy. Soc. London Ser A 253 (1960) 101-136.

[4] L. Rosenhead ; Laminar boundary layers; Clarendon Press, Oxford, England ,

(1963).

[5] A. M. O. Smith ; Improved Solutions of the Falkner and Skan boundary-layer

equation; Fund paper FF-10 , J Aero. Sci. (1954)

[6] T. Cebeci, T. Keller , Shooting and parallel shooting methods for solving the

Falkner-Skan boundary layer equation. J. Comput. Phs. 7 (1971) 289-300.

[7] T. Y. Na , Computational methods in Engineering boundary value problems,

Academic press, New York , (1979).

[8] A. A. Salama; Higher order method for solving free boundary problem;

Numer. Heat transfer , part B Fundamentals 45 (2004) 385-394.

[9] John P. Boyd ; The Blasius Function: Computations Before Computers, the

Value of Tricks, Undergraduate Projects, and Open Research Problems,

SIAM ,Vol. 50, No. 4,(2008), pp. 791–804.

[10] J. P. Boyd, The Blasius function in the complex plane, J. Experimental Math.,

8 (1999), pp. 381–394.

[11] F. M. Allan and M. I. Syam, On the analytic solutions of the nonhomogeneous

Blasius problem, J. Comput. Appl. Math., 182 (2005), pp. 362–371.

[12] C. M. Bender, A. Pelster, and F. Weissbach, Boundary-layer theory, strong-

coupling series, and large-order behavior, J. Math. Phys., 43 (2002), pp.

4202–4220.

[13] N. Bildik and A. Konuralp, The use of variational iteration method,

differential transform method and Adomian decomposition method for

solving different types of nonlinear partial differential equations, Internat. J.

Nonlinear Sci. Numer. Simul., 7 (2006), pp. 65–70.

[14] FAIZ AHMAD , DEGENERACY IN THE BLASIUS PROBLEM , Electronic Journal

of Differential Equations, Vol. No. 79, (2007), pp. 1–8.

[15] H. Schlichting, K. Gersten; Boundary Layer Theory, 8-th edition, Springer,

Berlin, (2000).

[16] L. Wang; A new algorithm for solving classical Blasius equation, Appl. Math.

Comput. 157(2004) pp. 1-9.

Page 13

[17] T. Fang, F. Guo and C. F. Lee; A note on the extended Blasius problem,

Applied Mathematics Letters 19 (2006) 613-617.

[18] Steven Finch, Prandtl-Blasius Flow, November 12, (2008)

[19] Vedat Suat E. , Shaher Momani , Differential transformation method for

obtaining positive solutions for two points Nonlinear Boundary Value

Problems , Mathematical Manuscripts Vol.1 No.1 (2007) pp. 65-72.

[20] Ravi Kanth A.S.V. Aruna K. ,Solution of singular two-point boundary value

problems, physics letters A 372 (2008) pp. 4671-4673.

Page 14

زٌكلاف ( تلداعه لح ًف ًلضافخلا لٌىحخلا تقٌزط- توخاخولا تقبطلا ىاٌزجل ) ىاكس

داىج ذوحه زفعج رىًأ .د

تعهاجلا يٌذفازلا تٍلك - ثابساحلا ثاٌٍقح تسذٌه نسق

س زئاث .دع

تعهاجلا يٌذفازلا تٍلك-

ًبٍلص ىوذ

ثابساحلا ثاٌٍقح تسذٌه نسق

تصلاخلا

زٌكلاف( تلداعه لحل ًلضافخلا لٌىحخلا تقٌزط ماذخخسا اٌحزخقا ثحبلا اذه ًف– ىاٌزجل ) ىاكس

ةزخفلو توخاخولا تقبطلا زٍغ

زٌكلاف( تلداعه يىخحح

هو يٍخباثلل تٌٍعه نٍق داوخعاب دى

و

Homann

.زقخسولا ىاٌزجلل

.ةدذحه

–

تطقً يه بزقلاب ىاٌزجلل عاىًا تعبرا تسارد نح . يٍخباث ىلع ) ىاكس

Homann

، يرىحولا زظاٌخولا ىاٌزجلل كزلا ًه عاىًلاا ٍذ

Blasius

،

Hiemenz

,

كلح عه اهجئاخً تًراقه نح )ًلضافخلا لٌىحخلا تقٌزط ( لحلل تهذخخسولا تقٌزطلا ةءافك حضىخل

.اهاوح تقباطخه جئاخٌلا جًاكو تقباسلا قزطلاب تجزخخسولا