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NUMERICAL SOLUTION OF BOUNDARY LAYER EQUATION FOR
NON-NEWTONIAN FLUID FLOW
Ph.D. Synopsis Submitted to
GUJARAT TECHNOLOGICAL UNIVERSITY
Ahmadabad, Gujarat
September-2018
For the Award of Degree of Doctor of Philosophy
In
Humanities and Science (Mathematics)
By
PATEL JAYSHRIBEN RAMJIBHAI Enrollment No: 139997673017
Under the Supervision of
Prof. (Dr.) Manisha P. Patel Assistant Professor
SCET, Surat
Doctoral progress committee (DPC) members
Prof. (Dr.) Mitesh S. Joshi Prof. (Dr.) Hema C. Surati
Associate Professor Assistant Professor
CKPCET, Surat SCET, Surat
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Content
1. Abstract……………………………………………………………….. 3
2. Brief description on the state of the art of the research topic………… 3
3. Definition of problem…………………………………………………. 6
4. Objectives …………………..………………………………………… 6
5. Scope of work………………….……………………………………… 6
6. Original contribution by the thesis…………………………….……… 7
7. Methodology of Research and Results………………………………... 7
8. Achievements with respect to objectives……………………………... 16
9. Conclusion…………………………………………………………….. 17
10. List of Publications..……………………………………………..….. 17
11. References…………………………………………………………… 18
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Title of thesis: Numerical Solution of Boundary Layer Equation for
Non-Newtonian Fluid Flow
1. Abstract:
The complex rehology of biological fluids has motivated investigations involving different non-
Newtonian fluids. In recent years, non-Newtonian fluids have become more and more important
industrially. Academic curiosity and practical applications have generated considerable interest in
finding the solutions of differential equations governing the motion of non-Newtonian fluids. The
governing equations of such non-Newtonian fluids are highly non linear partial differential
equations. These fluids flow problems present some interesting challenges to researchers in
engineering, applied mathematics and computer science.
Magneto Hydro Dynamic (MHD) and non-MHD laminar boundary layer flow of non-Newtonian
fluid is analyzed in the present work. Three non-Newtonian fluid models for the stress-strain
relationship - Sisko fluid model, Prandtl fluid model and Williamson fluid model- are considered for
the different flow geometry (semi-infinite flat & semi-infinite moving plates). Governing non linear
partial differential equations (PDE) are transformed into non linear ordinary differential equations
(ODE) using group theoretical methods. The quasi linearization method is then applied to convert
the non linear ODE into linear ODE. The numerical solutions of the resulting linear ordinary
differential equations are obtained using Finite difference method (method carried out with
Microsoft excel and MATLAB coding). The graphical presentation of the velocity profile is given
for each problem.
2. Brief description on the state of the art of the research topic:
In view of the fact that the earth is 75% covered with water and 100% with air. This air and the
water of rivers and seas are always moving. Also the flows of water, sewage and gas in pipes, in
irrigation canals, around rockets, aircraft, express trains, automobiles and boats etc… is the
resistance which acts on such flows. In this way, we are in so close a relationship to flow that the
fluid mechanics which studies flow is really a very familiar thing to us.
For a long time, there has been considerable interest in non-Newtonian fluids. Polymer solutions,
polymer melts, blood, paints and slurries, shampoo, toothpaste, clay coating and suspensions,
grease, cosmetic products, custard, are the most common examples of non- Newtonian fluids.
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Academic curiosity and practical applications have generated considerable interest in finding the
solutions of differential equations governing the motion of non-Newtonian fluids. The property of
these fluids is that the stress tensor is related to the rate of deformation tensor by some non-linear
relationship. These fluids present some interesting challenges to researchers in engineering, applied
mathematics and computer science.
One of the most successful, fascinating and useful applications of mathematics has been in the study
of motions of fluids (gases and liquids). The subject of Fluid Dynamics has made tremendous
advances since Euler gave his famous equations of fluid flow for perfect (non-viscous
incompressible and compressible) fluids in 1755 in his paper “General Principles of the motions of
Fluids”. At the end of the 19th century, Euler’s equations of motion and which had been developed
to great perfection.
Any disturbance created in the laminar flow in boundary layer is ultimately damped. This is known
as Laminar Boundary Layer. The boundary layer concept was first introduced by Ludwig Prandtl a
German aerodynamicist, in 1904 and derived the equations for boundary layer flow by correct
reduction of Navier-Stokes equations. He assumed that for fluids having relatively small viscosity,
the effect of internal friction in the fluid is significant only in a narrow region surrounding solid
boundaries or bodies over which the fluid flows. Thus, close to the body is the boundary layer
where shear stresses exert an increasingly larger effect on the fluid as one moves from free stream
towards the solid boundary. However, outside the boundary layer where the effect of the shear
stresses on the flow is small compared to values inside the boundary layer. With the help of this
concept, a very thin layer close to the body (boundary layer) where the viscosity is important, and
the remaining region outside this layer where the viscosity can be neglected.
It is the great achievement by Ludwig Prandtl, a high degree of correlation between theory and
experiment. He had led to unimagined successes in modern fluid mechanics. It was already known
that the great discrepancy between the results in classical hydrodynamics and reality was, in many
cases, due to neglecting the viscosity effects in the theory. Now the complete equations of motion of
viscous flows (the Navier Stokes equations) had been known for some time. However, due to the
great mathematical difficulty of these equations, no approach had been found to the mathematical
treatment of viscous flows (except in a few special cases). For technically important fluids such as
water and air, the viscosity is very small, and thus the resulting viscous forces are small compared to
the remaining forces (gravitational force, pressure force). For this reason it took a long time to see
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why the viscous forces ignored in the classical theory should have an important effect on the motion
of the flow.
From the literature survey, J. N. Kapur, (Kapur-1963) has derived two- dimensional boundary layer
equations for power-law fluids. He used stress-strain relationship for this fluid and derived
equations of motion by writing the order of various terms. Similarity solutions of laminar,
incompressible boundary layer equations of non-Newtonian fluids are derived using free-parameter
method (Hansen et al., 1968) in which the table of non-Newtonian models (Visco-inelastic) is
given. The Power-law and Ellis fluids have throughout studied, both experimentally and
theoretically (Tien, 1967, 1969). A theoretical study is made of laminar, incompressible flow of
non-Newtonian fluid near a suddenly accelerated plate where the fluid under consideration is
assumed to obey a proposed model of Powell-Eyring (Sirohi et al, 1987). Sujit Kumar (Sujit, 2006)
has derived boundary layer equation for a viscous elastic fluid flow with the assumption that the
normal stress is of the same order of magnitude as that of the shear stress, in addition to the usual
boundary layer approximations and obtained approximate analytical similarity solutions and then
numerical solutions of the similarity solutions. The flow of a power-law fluid is investigated in an
asymmetric channel caused by the movement of peristaltic waves with the same speed but with
different amplitudes and phases on the flexible walls of the channel by M.V. Subba Reddy et al.
(Subba Reddy, 2007). They have chosen the Ostwald–de Waele power-law model given in Bird et
al. (Bird, 1960). The laminar incompressible flow of a non-Newtonian fluid flow past 090 wedge
where the fluid is under consideration and assumed to obey a proposed model of Powell-Eyring
(Patel et al., 2009). The governing equation of motion is solved numerically by the satisfaction of
asymptotic boundary conditions. Yam et al. (2009) had presents a numerical analysis of the steady
boundary-layer flow of a Reiner–Philippoff fluid induced by a 90◦ stretching wedge in a variable
free stream. The governing partial differential equations are converted into a set of two ordinary
differential equations by the use of a similarity transformation. A complete analysis of all the Lie
point symmetries admitted by the equation describing the axis symmetric spreading under gravity
of a thin power-law liquid drop on a horizontal plane is performed (Neossi Nguetchue et al, 2009).
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3. Definition of the Problem:
There are several types of non –Newtonian fluid models of which are proposed by scientists
working in this area. Several empirical models are generally used to approximate the experimental
data are available for such fluids. It may not be possible to list all such models exhaustively.
However few of them are presented below:
Power law fluids, Reiner – Rivlin fluids, Bingham plastics, Ellis Fluids, Reiner-Philippoff fluid,
Prandtl fluids, Powell – Eyring fluids, Williamson fluids, Oldroyd fluid, Rivlin–Ericksen fluids,
Walters fluid, Maxwell fluids etc…
Magneto Hydro Dynamic (MHD) and non-MHD laminar boundary layer flow of non-Newtonian
fluid is analyzed in the present work. Three non-Newtonian fluid models for the stress-strain
relationship: Sisko fluid model, Prandtl fluid model and Williamson fluid model- are considered for
the different flow geometry (semi-infinite flat & semi infinite moving plates). Governing non linear
partial differential equations (PDE) are transformed into non linear ordinary differential equations
(ODE) using group theoretical methods. The quasi linearization method is then applied to convert
the non linear ODE into linear ODE. The numerical solutions of the resulting linear ordinary
differential equations are obtained using Finite difference method (method carried out with
Microsoft excel and MATLAB coding). The graphical presentation of the velocity profile is given
for each problem.
4. Objective:
From the literature survey it is found that, there are several types of non –Newtonian fluid models of
which are proposed by scientists working in this area. Calculations on non-Newtonian media present
a new challenge in flow analysis. Similarity techniques apply to find similarity solutions of
boundary layer flows of non-Newtonian fluids and it will be useful to treat the non-linear partial
differential equations and also to derive some closed form solutions.
5. Scope of work:
Calculations on non-Newtonian media present a new challenge in flow analysis. Simulating these
types of flows in order to calculate pipe and pump sizes presents a significant challenge to the
engineer. The stress-strain relationship for different type of viscous-inelastic fluids and similarity
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equations using these relationships will be helpful to many researchers and engineers for further
research.
Our aim is to modify similarity techniques and apply it to find similarity solutions of MHD and
non-MHD boundary layer flows of non-Newtonian fluids. Applications of these techniques are
useful for the treatment of engineering boundary value problems.
Our aim is to propose quite a new idea in the theory of similarity analysis and that is approximate
Similarity technique. We hope that our proposed approximate Similarity technique will be useful
to treat the non-linear partial differential equations.
Scopes of the new proposed technique are wide. It will be useful to solve non-similarity
equations and also hope to derive some closed form solutions.
It may be used for finding the exact solutions for defined fluid models.
6. Original contribution by the thesis:
Our contribution by this thesis has been presented as follows:
Two dimensional boundary layer equations of non-Newtonian fluid Sisko is taken. Its flow
considered with magneto hydrodynamic (MHD) field as well as without MHD field on semi
infinite flat surface and moving surface. Find the numerical solution and presented graphically
for different values of constants a, b, flow behavior constant n and MHD constant M0.
Two dimensional boundary layer equations of non-Newtonian fluid Prandtl is taken. Its flow
considered on semi-infinite flat surface and moving surface. Find the numerical solution and
presented graphically for different values of constants A and C.
Two dimensional boundary layer equations of non-Newtonian Williamson fluid is taken. Its flow
considered on semi-infinite flat surface. Find the numerical solution and presented graphically for
different values of constant λ.
7. Methodology of Research, Results:
The governing equations for two dimensional steady incompressible laminar boundary layer
equation of non-Newtonian fluid flow.
Page 8
0
u v
x y 2
'2
' ' 1 '' '
' ' '
e
dUu u uu v U
x y dx R y
With the boundary conditions:
At y=0, u=v=0
At y → ∞, u=U(x)
7.1 Laminar Boundary Layer Flow of Sisko Fluid
The governing equations of continuity and momentum of laminar boundary layer flow of Sisko
fluid past a semi-infinite flat plate are:
0
u v
x y
1n
u u dU u uu v U a b
x y dx y y y
With the boundary conditions:
y = 0: u=0, v=0
y →∞: u=U(x)
Appling one parameter group transformation we get similarity equation as,
12 2
1f ( ) 2 ( ) ( ) 3 ( ) 3 ( ) ( ) 0n
f f af nb f f c
With the boundary conditions:
0 0, 0 0, 1f f f
Now, By converting given non linear Ordinary Differential Equation (ODE) into linear ODE using
quasi linearization method then applying Finite Difference Method (FDM) in MATLAB. Find
numerical solution and presented velocity profile as graphically for the various values of constants a,
b and flow behaviour constant n.
Page 9
Figure-1: Velocity profile for a=b=0, Figure-2: Velocity profile for a=0, b=1
a=b=0.5 and varies n and varies n
7.2 Magneto hydro dynamic(MHD) flow of Sisko fluid past a semi-infinite flat
surface
The flow considered here is parallel to X-direction and Y-axis is the normal to it. The governing
equations of continuity and momentum of two dimensional MHD boundary layer flow of Sisko
fluid past a semi-infinite flat plate are:
0u v
x y
n 1
2
0
u u dU u uu v U a b B (U u)
x y dx y y y
With boundary conditions:
At y=0: u=0, v=0
At y →∞: u=U(x)
Appling one parameter group transformation we get similarity equation as,
n 12 2 2
1 0 1f ( ) 2f ( )f ''( ) c 3af '''( ) 3nb f ''( ) f '''( ) 3 B (c f '( )) 0
With the boundary conditions:
0 0, 0 0, 1f f f
2
0 0 1Put B M and C 1,
n 12
0f ' ( ) 2f ( )f ''( ) 3af '''( ) 3nb f ''( ) f '''( ) 3M (1 f '( )) 1 0
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Now, converting given non linear Ordinary Differential Equation (ODE) into linear ODE using
quasi linearization method then applying Finite Difference Method (FDM) in MATLAB. Find
numerical solution and presented velocity profile as graphically for the various values of constants
a, b, flows behaviour constant n and Magnetic parameter M0.
Figure 3: MHD Sisko fluid with a=b=0.5, Figure 4: MHD Sisko fluid with a=b=0.5,
n=0.5 and different values of Mo Mo = 4 and different values on n
Figure 5: MHD Power-Law Fluid with a=0, Figure 6: MHD Power-Law fluid for a=0,
b=1, n=0.5 and different Mo. b=1, Mo = 4 and different n.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
eta
f'
Mo = 1 Mo = 2 Mo = 4 Mo = 6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
eta
f'
n=0.5 n=1 n=1.5 n=2
0 0.1 0.2 0.3 0.4 0.5 0.60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
eta
f'
Mo = 1 Mo = 2 Mo =4 Mo = 6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
eta
f'
n=0.5 n=1 n=1.5 n=2
Page 11
Figure 7: MHD Newtonian fluid with a=1, Figure 8: All fluid with and without
b=0, n=1 and different Mo Magnetic field effect
7.3 Magneto hydro dynamic(MHD) flow of Sisko fluid past a semi-infinite
moving surface
The governing equations of continuity and momentum of two dimensional MHD boundary layer
flow of Sisko fluid are same as 7.2. If we take semi-infinite moving surface in place of past a semi-
infinite flat plate we get the following results:
Figure 9 Figure 10
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
eta
f'
Mo = 1 Mo = 2 Mo = 4 Mo = 6
0 0.05 0.1 0.15 0.2 0.25 0.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
eta
f'
MHD Sisko
non-MHD Sisko
MHD Power Law
non-MHD Power Law
MHD Newtonian
non-MHD Newtonian
Page 12
Figure 11 Figure 12
Figure 13: Non- MHD Sisko fluid Figure 14: MHD Power-Law fluid
Page 13
Figure 15: non-MHD Power-Law fluid Figure 16: MHD Newtonian fluid
Figure 17: non- MHD Newtonian fluid
7.4 Solution of boundary layer flow of Prandtl fluid past a flat surface
The governing equations of continuity and momentum of laminar boundary layer flow of
Prandtl fluid past a semi-infinite flat surface are:
0,u v
x y
,yx
u u dUu v U
x y dx y
Page 14
With boundary conditions:
At y=0: u=0, v=0 and y → ∞: u=U(x)
The Prandtl Fluid Model for 2-D flow problems is:
1 1sinyx
uA
C y
Appling one parameter group transformation we get similarity equation as,
2 2
3
3 32 1 0
2
A Af f f f f f
C C
With boundary conditions:
'(0) (0) 0f f
( ) 1f
Now, converting given non linear Ordinary Differential Equation (ODE) into linear ODE using
quasi linearization method then applying Finite Difference Method (FDM) in MATLAB. Find
numerical solution and presented velocity profile as graphically for the various values of constants
A and C.
Figure 18: velocity profile for Figure 19: velocity profile for
different values of A different values of C
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
eta
f '
f ' vs. eta for C=0.5 and different values of A
A=C=0.5
A=1.5,C=0.5
A=2.5,C=0.5
A=3.5,C=0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
eta
f '
f ' vs. eta for A=0.5 and different values of C
A=0.5,C=0.5
C=1, A=0.5
C=1.5,A=0.5
C=1.3, A=0.5
Page 15
7.5 Solution of boundary layer flow of Prandtl fluid past a moving surface
The governing equations of continuity and momentum of laminar boundary layer flow of Prandtl
fluid past a semi-infinite moving surface are as 7.4. If we take moving surface in place of past a
semi-infinite flat plate we get the following results:
Figure 20: velocity profile for Figure 21: velocity profile for
different values of A different values of C
7.6 A Williamson Fluid model past a moving surface (Nadeem et al. 2013)
2 2
2 22
u u u u uu v
x y y y y
The corresponding boundary conditions are:
0 : ; 0 w xat y u U B v
: 0 y u
By taking the transformations:
'( ) , ( ) , B
u Bxf v B f y
Making use transformations fluid flow equation takes the form:
2''' ' ' '' ''' 0 f f ff f f
Page 16
With the corresponding boundary conditions:
f=0, f ’=1 at η=0
Now, converting given non linear Ordinary Differential Equation (ODE) into linear ODE using quasi
linearization method then applying Finite Difference Method (FDM) in MATLAB. Find numerical
solution and presented velocity profile as graphically for the various values of constant lambda (λ).
Figure 22: velocity profile for different values of lambda (λ)
8. Achievements with respect to objectives:
Similarity techniques, one parameter group transformations are successfully apply for defined
flow problems.
After applying group transformation similarity analysis, highly non linear partial differential
equation converted into ordinary differential equation which is easy to solve compared to PDE.
Numerical method (Finite Difference Method - FDM) in MATLAB successfully applies for the
numerical solution of ODE.
Page 17
9. Conclusion:
The methods for obtaining similarity transformations are divided into two categories (i) Direct
methods and (ii) group theoretic methods. The direct methods such as, separation of variables do
not invoke group invariance. On the other hand, group theoretic methods are more elegant
mathematically. The main concept of invariance under a group of transformation is invoked
always. The governing non-linear partial differential equations of the laminar boundary layer flow
of non-Newtonian fluid are transformed into the non-linear ordinary differential equations. Then
the obtained non linear ODE is converted into linear ODE by quasi linearization method and is
solve by Finite Difference method using MATLAB solver. The graphical representations for
various fluid models are shown in Figure 1- 22.
From the figures we can say that the velocity profile increase more rapidly for Sisko fluid than it
is in Power-law fluid. Also if we increase the values of magnetic constant M0 the velocity profile
increases. When sisko fluid flow passing on the moving surface the velocity is decrease uniformly
with the increase in eta for variable fluid index, Magnetic induction and fluid parameters.
The velocity profile in the prandtl fluid model increases more rapidly when we increase the value
of constants A and C. In Williamson fluid flow model velocity profile and boundary layer
thickness decrease with increase the values of λ.
10. List of all publications :
1. Jayshri Patel, Manisha Patel and M.G. Timol, “Laminar Boundary Layer Flow of Sisko Fluid”,
Applications and Applied Mathematics (AAM), ISSN:1932- 9466,Vol. 10, Issue 2, December 2015,
pp. 909-918.
2. Jayshri Patel, Manisha Patel and M.G. Timol, “Magneto hydro dynamic flow of Sisko fluid”, ARS -
Journal of Applied Research and Social Sciences, ISSN: 2350-1472, Vol.3, Issue.14, July 2016,
pp.45-55.
3. Jayshri Patel, Manisha Patel and M.G. Timol, “On the Solution of Boundary Layer Flow of Prandtl
Fluid Past a Flat Surface”, Journal of Advanced Mathematics and Applications (JAMA), American
Scientific Publishers, 2156-7565, Vol. 6, 2017,pp.1-6.
4. Jayshri Patel, Manisha Patel and M.G. Timol, “Finite Difference Method on third order non-linear
Differential Equation: Magneto hydro dynamic flow of Sisko fluid”, Indian Journal of Industrial and
Applied Mathematics (IJIAN), ISSN: 0973-431 ,Vol. 8, No. 2, July–December 2017, pp. 191–200.
Page 18
5. Hemangini Shukla, Jayshri Patel, Hema C. Surati, Manisha Patel and M.G. Timol, “Similarity
Solution of Forced Convection Flow of Powell-Eyring & Prandtl-Eyring Fluids by Group-Theoretic
Method”, Mathematical Journal of Interdisciplinary Sciences, ISSN:2278-9561,Vol-5, No-2, March
2017, pp. 151–165.
Paper Communicated…
1. Jayshri Patel, Manisha Patel and M.G. Timol, “Numerical Solution of third order non-linear
differential equation using Finite Difference Method: MHD Flow of Sisko Fluid on a Moving
Surface”, Journal of the Brazilian Society of Mechanical Sciences and Engineering.
Paper Presented in Conference
1. Jayshri Patel, Manisha Patel and M.G. Timol, “Finite Difference Method on third order non-linear
Differential Equation: Magneto hydro dynamic flow of Sisko fluid”, International Conference on
Technology and Management (ICTM-2017), Feb.-2016, Sankalchand Patel University, Visnagar.
11. References:
1. Bird R.B., Stewart W.E. and Lightfoot E.M. (1960). Transport phenomena, John Wiley, New York.
2. Kapur J. N. (1963). Note on the boundary layer equations for power-law fluids, J. Phys. Soc. Japan,
18.
3. Hansen A.G. and Na T.Y. (1968). Similarity solution of laminar, Incompressible boundary layer
equations of non-Newtonian fluid, ASME Journal of basic engg.67, pp.71-74.
4. Tien C. (1967). Laminar natural convection heat transfer from vertical plate to power-law fluid,
Appl. Sci. Res., 17, pp.233-248.
5. Tien C., Tsuei H. S. (1969). Laminar natural convection heat transfer in Ellis fluids, Appl. Sci. Res.
20, pp.131-147.
6. Sirohi V., Timol M. G., Kalthia N. L. (1987). Powell-Eyring model flow near an accelerated plate,
Fluid Dynamic Research 2, pp. 193-204.
7. Sujit Kumar Khan (2006). Boundary Layer Viscoelastic Fluid Flow Over An Exponentially
Stretching Sheet Int. J. of Applied Mechanics and Engineering, vol.11, No.2, pp.321-335
8. Subba Reddy M.V., Ramachandra Rao A., Sreenadh S (2007). Peristaltic motion of a power-law
fluid in an asymmetric channel, International Journal of Non-Linear Mechanics 42, pp. 1153 – 1161.
Page 19
9. Patel Manisha, Timol M, G. (2009). Numerical treatment of Powell–Eyring fluid flow using method
of satisfaction of asymptotic boundary conditions (MSABC), J. of Applied Numerical Mathematics,
Applied Numerical Mathematics 59, pp. 2584–2592 (Elsevier).
10. Yam K. S., Harris S. D., Ingham D. B., Pop I (2009). Boundary-layer flow of Reiner–Philippoff
fluids past a stretching wedge, International Journal of Non-Linear Mechanics 44,pp. 1056 – 1062.
11. Neossi Nguetchue S. N., Abelman S., Momoniat E. (2009). Symmetries and similarity solutions for
the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane,
Applied Mathematical Modelling 33, pp. 4364–4377.
12. S. Nadeem, S. T. Hussain and Changhoon Lee (2013). Flow of a Williamson Fluid over a Stretching
Sheet, Brazilian Journal of Chemical Engineering, ISSN: 0104-6632, Vol. 30, No. 03, pp. 619 –
625.
