Differential Transformation Method for solving FalknerSkan 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 FalknerSkan equation over a semiinfinite domain.
The FalknerSkan 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 FalknerSkan 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 FalknerSkan. The differential transformation method show good agreements of results for solving nonlinear BVPs.
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ABSTRACT: We present a computational method for the solution of the thirdorder boundary value problem characterized by the wellknown Falkner–Skan equation on a semiinfinite 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):203214. · 0.99 Impact Factor  SourceAvailable from: damtp.cam.ac.ukPhilosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 01/1960; 253(1023):101136. · 2.89 Impact Factor

Article: Shooting and parallel shooting methods for solving the FalknerSkan boundarylayer equation
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ABSTRACT: We present three accurate and efficient numerical schemes for solving the FalknerSkan equation with positive or negative wall shear. Newton's method is employed, with the aid of the variational equations, in all the schemes and yields quadratic convergence. First, ordinary shooting is used to solve for the case of positive wall shear. Then a nonlinear eigenvalue technique is introduced to solve the inverse problem in which the wall shear is prescribed and the pressure distribution is to be determined. With this approach the reverse flow solutions (i.e., negative wall shear) are obtained. Finally, a parallel shooting method is employed to reduce the sensitivity of the convergence of the iterations to the initial estimates.Journal of Computational Physics 01/1971; 7(2):289300. · 2.14 Impact Factor
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Differential Transformation Method for solving
FalknerSkan equation to Boundary Layer Flow
Dr. Anwar Ja’afar Mohamad Jawad
AlRafidain University College , Iraq anwar_jawad2001@yahoo.com
Dr. Thair Saddon Slibi
AlRafidain University College , Iraq.
Abstract
In this paper we apply the differential transformation method to solve the third –
order boundary value problems characterized by the FalknerSkan equation over a
semiinfinite domain.
The FalknerSkan 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 FalknerSkan 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 FalknerSkan. The differential transformation method show
good agreements of results for solving nonlinear BVPs.
1. Introduction
The nonlinear thirdorder FalknerSkan 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. FalknerSkan 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 FalknerSkan 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 onestep method of order 5.
In this paper, the Differential Transformation Method (DTM) is applied
to solve the nonlinear thirdorder FalknerSkan 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 thirdorder FalknerSkan equation for which is not
discussed in previous literatures.
In section 2 we try to explain all types of flow included from the general
FalknerSkan 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 thirdorder FalknerSkan
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 [1113].
The boundary layer equations for the incompressible viscous flow over
a flat plate with zero pressure gradient are:
,
(2)
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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 nonlinear (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 NavierStokes 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 initialvalue problem
( ) ( ) ( )
( ) ( ) ( ) , ) (6)
will be called the Blasius initial value problem. It is wellknown 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 initialvalue 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 boundaryvalue 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.
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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 FalknerSkan
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 kith 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 FalknerSkan 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)
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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 RungeKutta 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
FalknerSkan equations.
Case 1 : (Blasius equation) For
( )
( )
, * ( ) ( ) ( ) ( ) ( )+ *
, * ( ) ( ) ( ) ( ) ( ) ( )+ (
, .
√
( ) , ( ) and ( ) 0.1660286681 .
( ) , ( ) , ( ) ,
( ) , ( ) , ( )
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( ) , ( ) , ( )
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 .
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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
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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 (79) 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.
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Figure (7) ( ) Corresponding for different flows
Figure (8) ( ) Corresponding for different flows
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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 thirdorder differential equation on semiinfinite
intervals. Using small values of , can gave good approximation result
for ( ). We successfully compute several instances of the FalknerSkan
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 FalknerSkan 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 boundarylayer
problem, Ann of Math. 43 (1942) 381407.
[3] W.A. Coppel ; On a differential equation of boundary layer theory , Philos.
Trans. Roy. Soc. London Ser A 253 (1960) 101136.
[4] L. Rosenhead ; Laminar boundary layers; Clarendon Press, Oxford, England ,
(1963).
[5] A. M. O. Smith ; Improved Solutions of the Falkner and Skan boundarylayer
equation; Fund paper FF10 , J Aero. Sci. (1954)
[6] T. Cebeci, T. Keller , Shooting and parallel shooting methods for solving the
FalknerSkan boundary layer equation. J. Comput. Phs. 7 (1971) 289300.
[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) 385394.
[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, Boundarylayer theory, strong
coupling series, and largeorder 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, 8th edition, Springer,
Berlin, (2000).
[16] L. Wang; A new algorithm for solving classical Blasius equation, Appl. Math.
Comput. 157(2004) pp. 19.
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[17] T. Fang, F. Guo and C. F. Lee; A note on the extended Blasius problem,
Applied Mathematics Letters 19 (2006) 613617.
[18] Steven Finch, PrandtlBlasius 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. 6572.
[20] Ravi Kanth A.S.V. Aruna K. ,Solution of singular twopoint boundary value
problems, physics letters A 372 (2008) pp. 46714673.
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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