World Academy of Science, Engineering and Technology

International Journal of Mathematical and Computational Sciences

Vol:7, No:7, 2013

Numerical Solution of a Laminar Viscous Flow

Boundary Layer Equation Using Uniform Haar
Wavelet Quasi-linearization Method
Harpreet Kaur, Vinod Mishra, and R. C. Mittal

Unfortunately, since Blasius equation is non-linear, there is

Abstract—In this paper, we have proposed a Haar wavelet quasi- not known analytic solution in closed form. The Blasius
linearization method to solve the well known Blasius equation. The problem models the behavior of two-dimensional steady state
International Science Index, Mathematical and Computational Sciences Vol:7, No:7, 2013 waset.org/Publication/16506

method is based on the uniform Haar wavelet operational matrix laminar viscous flow of an incompressible fluid over a semi-
defined over the interval [0, 1]. In this method, we have proposed the
infinite flate plate, provided the boundary layer assumptions
transformation for converting the problem on a fixed computational
domain. The Blasius equation arises in the various boundary layer are verified is governed by the continuity and the Navier-
problems of hydrodynamics and in fluid mechanics of laminar Stokes equations of motion for classical Blasius flate plate
viscous flows. Quasi-linearization is iterative process but our flow prob [4] and governing equations are simplified to
proposed technique gives excellent numerical results with quasi-
linearization for solving nonlinear differential equations without any
∂u ∂v
iteration on selecting collocation points by Haar wavelets. We have + =0 (1)
solved Blasius equation for 1 ≤ α ≤ 2 and the numerical results are ∂x ∂y
compared with the available results in literature. Finally, we ∂u ∂v ∂ 2u
conclude that proposed method is a promising tool for solving the u +v =µ 2 (2)
∂x ∂y ∂ y
well known nonlinear Blasius equation.
u ( x, 0) = v ( x, 0) = 0 or at u = v = 0 at x = 0 or
Keywords—Boundary layer Blasius equation, collocation points,
quasi-linearization process, uniform haar wavelets. where y = 0, U = U ∞ at x = 0 .
I. INTRODUCTION The boundary conditions for this case are that both
components of the velocity are zero at the wall due to no slip,

T HE solutions of the one-dimensional third order boundary

value problem described by the well known blasius
equation is similarity solution of the two dimensional
and that the horizontal velocity approaches the constant free
stream velocity at some distance away from the plate.
Assuming that the leading edge of the plate is x0 and the
incompressible laminar boundary layer equations. Tsou et al.
plate is infinity long. To make this quantity dimensionless, it
[1] made a numerically and theoretically experiment on this
problem to prove that a blasius flow is physically realizable. A can be divided by y to obtain where fu the dimensionless
recent study by Boyd [2], [3] pointed out how this particular stream function is. The velocity component u can be
problem of boundary layer theory has arisen the interest of expressed as follows: u ( x, y ) → ∞ as y → ∞ for the Blasius
promient scientist. In fluid mechanics, the problems are flate plate flow introducing a similarity variable and a
usually governed by systems of partial differential equations. If dimensionless stream function f (η ) as;
somehow, a system can be reduced to a single ordinary
differential equation, this constitutes a considerable
U y
mathematical simplification of the problem. If the number of η=y = Re x (3)
independent variables can be reduced, then partial differential vx x
equations can be replaced by ordinary differential equation. In u 1 Uv
the modeling of boundary layer, this is sometimes possible and = f ; v= (η f '− f ) (4)
U 2 x
in some cases, the system of partial differential equations
reduces to a system involving a third order differential
where Re x is the local Reynolds number  = Ux  . We obtain by
equation.  
 v 
applying (3) and (4).
H. Kaur, corresponding author, is with the Mathematics Department,
SLIET, Longowal, India (e-mail: maanh57@gmail.com). ∂u Uη ∂v U η
V. Mishra Author is with the Mathematics Department, SLIET, Longowal, =− f '' ; = f '' (5)
India (e-mail: mishrasliet560@gmail.com).
∂x 2 x ∂y 2 x
R. C. Mittal Author is with the Mathematics Department, IIT Roorkee,
India (e-mail: rcmmmfma@iitr.ernet.in).

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 World Academy of Science, Engineering and Technology
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Vol:7, No:7, 2013

And the equation of continuity is satisfied identically on the with arbitrary smoothness. Wavelets generalize readily to
other hand, we get several dimensions.
The Haar wavelet function was introduced by Alfred Haar
∂u U ∂ 2u U 2 in 1910 [16] in the form of a rectangular pulse pair function.
= Uf '' ; = f ''' (6) After that many other wavelet functions were generated and
∂y vx ∂y 2 vx
introduced. Those include the Shannon, Daubechies [15] and
Legendre wavelets. Among these forms, Haar wavelet is the
Note that in (3)-(6), U = U ∞ represents Blasius whereas only real valued wavelet that is compactly supported,
U = uw indicates sakiandis flow, respectively. By inserting symmetric and orthogonal. The basic and simplest form of
(4)-(6) in (2), this system can be simplified further to an Haar wavelet is the Haar scaling function that appears in the
ordinary differential equation. To do this, we have an equation form of a square wave over the interval t ∈ [0,1] , generally
that reads written as;

α f '''(η ) + f (η ) f ''(η ) = 0 (7)

1 t ∈ [0,1)
h1 (t ) =  (8)
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Transformed boundary conditions for the momentum (7) 0 elsewhere

are f = f ' = 0 at η = 0 and f ' → 1 as η → ∞.
The above expression, called Haar father wavelet, is the
In Blasius equation, the second derivative of f (η ) at zero zeroth level wavelet which has no displacement and dilation of
plays an important role. Howarth [5] solved the Blasius unit magnitude. The following definitions illustrate the
equation numerically and found f ''(0) = 0.33206 . Asaithambi translation dilation of wavelet function.
[6] solved the Blasius equation more accurately and obtained There are many excellent accounts of multiresolution and
this number as f ''(0) = 0.332057336 . Some researchers have wavelet theory.
The sequence {ψ i }i = 0 is a complete orthonormal system in
∞
solved the problem numerically and some analytically.
However, the solutions obtained were not very accurate. A
homotopy perturbation solution to this problem was presented L2 [0,1] and by using the concept of multiresolution analysis
by Fang [7], He et al. [8] Ahmad [9], [10] also obtained the (MRA) as an example the space V j can be defined like
solution of Blasius problem using approximate analytical
method. Liao [11], [12] obtained an analytic solution for the
Blasius equation which is valid in the whole region of the V j = sp {ψ j , k }
j = 0,1,2,....,2 j −1
problem. He constructed a five-term approximate-analytic
= W j −1 ⊕ V j −1 = W j −1 ⊕ W j − 2 ⊕ V j − 2 ⊕ ....... = ⊕ Jj =+11W j ⊕ V0 , (9)
solution for the Blasius using the variational iteration method
[13] and Abbasbandy [14] obtained numerical solution of
Blasius equation by adomians decomposition method. Many The linearly independent functionsψ j , k (t ) spanning W j are
calculations should be done to construct the resulting semi- called wavelets. Original signal can be expressed as a linear
analytic solutions and this increases considerably the CPU
combination of the box basis functions in V j .The functions
time especially when a large number of terms of solutions are
to be used. From the review of the proposed schemes, two ψ (t ) and ψ j , k (t ) are all orthogonal in [ 0,1] , with
general limitations may be observed: The proposed
approximate-analytic methods cannot yield accurate solutions 1 1

when a rather small number of solution terms are used. ∫ψ (t )ψ

0
j,k (t ) dt = 0 and ∫ψ
0
j ,k (t )ψ l , m (t )dt = 0 (10)
Our main goal here is to show how to solve numerically the
blasius problem by Haar wavelet approximation. In this work,
the Haar wavelet quasilinearization (HWQ) process is For ( j, k ) ≠ ( 0,0) in the first case and ( j, k ) ≠ ( l, m ) in
proposed to solve the classical Blasius flat-plate problem. The the second.
numerical results are obtained via proposed method for The Haar mother wavelet is the first level Haar wavelet and
α = 1,1.2,1.5,1.8 and 2 and compared with the available can be written as the linear combination of the Haar scaling
results in literature. function by using

II. STRUCTURE OF HAAR WAVELETS BASED ON MULTI h2 (t ) = h1 (2t ) + h1 (2t − 1) (11)

RESOLUTION ANALYSIS (MRA)
Wavelets were ripe for discovery in the 1980s. The great Similarly, the other wavelets can be generated with two
impetus came from two discoveries: the multiresolution of operations of translation and dilation. Each Haar wavelet is
Mallat or Meyer and most of all the discovering by composed of a couple of constant steps of opposite sign during
Daubechies [15] compactly supported orthogonal wavelets its subinterval and is zero elsewhere. The term wavelet is used

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 World Academy of Science, Engineering and Technology
International Journal of Mathematical and Computational Sciences
Vol:7, No:7, 2013

to refer to a set of orthonormal basis functions generated by d2 f  d2 f df  π 

dilation and translation of a compactly supported scaling = cos 2 (π t / 2) +  −2 cos(π t / 2)sin(π t / 2).  
dη 2  dt 2 dt  2 
function h1 (t ) (father wavelet) and a mother wavelet h2 (t )
4 cos 2 (π t / 2)
associated with an multiresolution analysis of L2 ( R) . Thus we (14)
π2
can write out the Haar wavelet family as
d3 f 8 d3 f 4 ( 4 cos(π t / 2) + 1)
 k k + 0.5 = cos 6 (π t / 2) − cos 4 (π t / 2)sin(π t )
≤t< dη 3 π 3 dx 3 π2 (15)
1 2 j
2j
 d2 f 2 df
+ cos 2 (π t / 2) ( −2 cos 2 (π t / 2) cos(π t ) + sin 2 (π t ))
 k + 0.5 k +1 dt 2 π dt
h(t ) = h(2 t − k ) =  −1
j
≤t ≤ j (12)
 2j 2
0 elsewhere The proposed technique is based on operational matrix at
 collocation points. The operational matrix is derived from

integration of Haar wavelet family by Chen and Hsiao in 1997
[17]. The Haar basis has the very important property of
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For i ≥ 2, i = 2 j + k + 1, j ≥ 0, 0 ≤ k ≤ 2 j − 1 and collocation

multiresolution analysis that V j +1 = V j ⊕ W j . The orthogonality
1
l− property puts a strong limitation on the construction of
points are defined as tl = 2 , l = 1, 2,...., 2m.
2m wavelets and allows us to transform any square integral
Here m is the level of the wavelet, we assume the maximum function on the interval time [ 0,1) into Haar wavelets series as
level of resolution is index J , then m = 2 j , ( j = 0,1, 2,...., J );
∞ 2 j −1
in case of minimal values m = 1, k = 0 then i = 2 . For any
f (t ) = c0 h0 (t ) + ∑ ∑ c2 j + k h2 j + k (t ), t ∈ [0,1] (16)
fixed level m , there are m series of i to fill the interval j = 0 k =0

[0,1) corresponding to that level and for a provided J , the

index number i can reach the maximum value M = 2J +1 ,when Similarly the highest derivative can be written as wavelet
∞
including all levels of wavelets. series ∑ a h (t ) .
i i In applications, Haar series are always
We can find the required derivatives in terms of operational i =−∞

matrix. The operational matrix pi , n (t ) of order 2m × 2m can be 2m

obtained by integration of Haar wavelet. Integrals can be

truncated to 2m terms, that is ∑ a h (t )
i =0
i i [17], then we have

evaluated from (12) and the first two of them are given below. used the quasi-linearization process. The quasi-linearization
Also for the ease of implementation, we have used the same process is an application of the Newton Raphson Kantrovich
notations for Haar wavelets and their integrals as [17]. approximation method in function space [18]. The idea and
advantage of the method is based on the fact that linear
1  2 equations can often be solved analytically or numerically while
k  k k + 0.5 
 t −  tε  ,  there are no useful techniques for obtaining the general
2  m  m m 
solution of a nonlinear equation in terms of a finite set of
 2
 1 − 1  k + 1 − t   k + 0.5 k + 1 
tε  , 
particular solutions. Consider an nth order nonlinear ordinary
pi ,2 (t ) =  4m 2 2  m   m m  (13) differential equation

 1  k +1 
tε  ,1 L( n ) f (t ) = g ( f (t ), f (1) (t ), f (2) (t ), f (3) (t )......, f ( n −1) (t ), t ) (17)
 4m 2  m 

0 elsewhere
with the initial conditions
III. TRANSFORMATION OF BLASIUS EQUATION AND SOLUTION
PROCEDURE TO SOLVE THE PROBLEM f (0) = λ0 , f (1) (0) = λ1 , f ( 2) (0) = λ2 ,......, f ( n −1) (0) = λn −1 (18)

We begin now the development of the numerical procedure

for solving the Blasius problem. The transformation Here L( n ) is the linear nth order ordinary differential
 πt  operator, g is nonlinear function of f (t ) and its ( n − 1)
η = tan   and a collocation method with orthogonal Haar
 2 derivatives are f ( s ) (t ), s = 0,1, 2,......, n − 1.
wavelets are introduced to solve numerically the third order The quasi-linearization prescription determines the (r + 1)th
nonlinear Blasius differential (7).
iterative approximation to the solution of (17) and its
Under the transformation η = tan  π t  , derivatives are derived
 2
linearized form is given by (19).
as

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 World Academy of Science, Engineering and Technology
International Journal of Mathematical and Computational Sciences
Vol:7, No:7, 2013

L( n ) f r +1 (t ) = g ( f r (t ), f r(1) (t ), f r(2) (t ), f r(3) (t ),....., f r( n −1) (t ), t ) + 0.35

n −1 (19) f(η) by HWQM

∑( f
s =0
(s)
r +1 (t ) − f r( s ) (t )) gu( s ) ( f r (t ), f r(1) (t ), f r(2) (t ),....., f r( n −1) (t ), t ) 0.3 f(η) by HAM[8]
f'(η) by HWQM
f'(η) by HAM[8]
0.25 f''(η) by HWQM
where f r(0) (t ) = f r (t ) .The functions g = ∂gs are functional (s)
f''(η) by HAM[8]
u
∂f
0.2

f(η),f'(η),f''(η)
derivatives of the functions. The zeroth approximation f 0 (t ) is
chosen from mathematical or physical considerations. 0.15

We linearize the nonlinear (7) by using quasi-linearization

process and followed by simplification yields 0.1

f (t ) f '' (t ) = f r (t ) f r'' (t ) + ( f r +1 (t ) − 2 f r (t )) f ''r (t ) + f r''+1 (t ) f r (t ) (20) 0.05

0
Then by following Haar wavelet quasilinearization method
International Science Index, Mathematical and Computational Sciences Vol:7, No:7, 2013 waset.org/Publication/16506

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

[19]-[21], equation can easily be written as the system. η

Fig. 1 Plot of Comparison of Present Method for α = 2

 16 2m 2m

α
 π3 cos 6
(π tl ) ∑ ai hi + ∑ ai pi ,3 
 i =0 i =0
 IV. NUMERICAL RESULTS AND DISCUSSION
 4 2t 
 2 cos (π tl / 2) − cos (π tl / 2) sin(π tl )  
4 2
In blasius equation, the second derivative of f (η ) at zero
 π π 
plays an important role. Numerical solutions by applying the
 4  proposed technique to several values of η and α = 2 for
 π 2 cos (π tl / 2)(−2 cos (π tl / 2) cos(π tl ) + sin (π tl )) 
4 2 2
2m
+ ∑ ai pi ,2   f (η ) and its derivatives are given in Table I and comparison
i =0  − 1 t 2 cos 2 (π t / 2) sin(π t ) 
  is shown in Fig. 1. For different values of α = 1,1.2,1.5,1.8
 π
l l l

 −8  f ''(η ) are computed and dipicted in Table II and Fig. 2.
 π 2 cos (π tl / 2)(2 cos(π tl ) sin(π tl / 2) + sin(π tl )) + 
4
2m
d2 f
+ ∑ ai pi ,1  2 + Further the quantity for η = 0 has been computed i.e
i =0  2tl  dη 2
 2 cos (π tl / 2)
4

π  f ''(0) = 332057 . Computational work is computed by C++
 8  and MATLAB R2007b for wavelet mode m = 32.
 π 2 cos (π tl / 2)(2 cos(π tl ) sin(π tl / 2) + sin(π tl ))
4

 
2m
 4tl  0.335
∑
i=0
ai pi ,21  − 2 cos (π tl / 2)(−2 cos (π tl / 2) cos(π tl ) + sin (π tl )) 
π
2 2 2
α =1
α =1.2
 2
 α =1.5
 4tl 2tl 3 
 − cos 4
(π t / 2) − cos 2
(π t / 2) sin(π t )  0.33
α =1.8
 π2 π
l l l
 α =2

8 4t
= 2 cos 4 (π tl / 2) sin(π tl )(2 cos(π tl ) + 1) − 2l cos 2 (π tl / 2)
π π (21) 0.325
2tl 2
(−2 cos (π tl / 2) cos(π tl ) + sin (π tl )) − 2 cos (π tl / 2)
2 2 4
f''(η)

π
0.32

0.315

0.31
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
η

Fig. 2 Plot of f ''(η ) to the Blasius Flow for Different Values of α

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Vol:7, No:7, 2013

TABLE I V. CONCLUSION
WAVELET SOLUTION f (η ) AND ITS DERIVATIVES FOR DIFFERENT VALUES OF
In this work Haar wavelet method is applied to solve
η AND α = 2 nonlinear Blasius equation. To the best of our knowledge, the
η f (η ) f '(η ) f ''(η ) method of quasi-linearization has not been used for above
0.0156 4.053e-005 0.005188 0.332057 nonlinear problem with Haar wavelets. The advantage of
0.0469 0.0003648 0.015565 0.332054 quasi-linearization is that one does not have to apply iterative
0.0781 0.0010133 0.025941 0.332046
0.1094 0.0019861 0.036318 0.332037
procedure. The results of the comparison with other method
0.1719 0.0049046 0.057067 0.331954 indicate that the proposed method is feasible. Also the effect
0.2031 0.0068503 0.067440 0.331893 of constant parameters on response of system for Haar wavelet
0.2344 0.0091202 0.077810 0.331805 method is also shown by figures. It is also shown that the use
0.2656 0.0117144 0.088178 0.331698
0.2969 0.0146329 0.098541 0.331563 of the quasi-linearization process and proposed transformation
0.3281 0.0178756 0.108901 0.331396 makes easier by Haar wavelet method to handle nonlinearity in
0.3594 0.0214426 0.119254 0.331193 a shorter time of computations. We observed that f ''(η ) at any
0.3906 0.0253339 0.129643 0.330951
0.4219 0.0295495 0.139938 0.330666 point near the η = 0 decreases when α increases.
International Science Index, Mathematical and Computational Sciences Vol:7, No:7, 2013 waset.org/Publication/16506

0.4531 0.0340893 0.150266 0.330337

0.4844 0.0389535 0.160583 0.329959
0.5156 0.0441419 0.170888 0.329533 ACKNOWLEDGMENT
0.5469 0.0496545 0.181179 0.329049
0.5781 0.0554915 0.191453 0.328516
Author Harpreet Kaur is thankful to Sant Longowal Institute
0.6094 0.0616527 0.201712 0.327911 of Engineering and Technology (SLIET), Longowal,
0.6406 0.0681382 0.211947 0.327246 (Established by Govt. of India) for providing financial support
0.6719 0.0749479 0.222162 0.326515 as a senior research fellowship.
0.7031 0.0820826 0.232354 0.325713
0.7344 0.0895403 0.242519 0.324838
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