Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 4, pp. 775-787, Warsaw 2003

SIMILARITY SOLUTIONS TO BOUNDARY LAYER
EQUATIONS FOR THIRD-GRADE NON-NEWTONIAN FLUID

IN SPECIAL COORDINATE SYSTEM

Muhammet Yürüsoy

Department of Mechanical Education Afyon Kocatepe University, Afyon, Turkey

e-mail: yurusoy@aku.edu.tr

Two dimensional equations of steady motion for third order fluids are
expressed in a special coordinate system generated by the potential flow
corresponding to an inviscid fluid. For the inviscid flow around an ar-
bitrary object, the streamlines are the φ-coordinates and the velocity
potential lines are ψ-coordinates which form an orthogonal curviline-
ar set of coordinates. The outcome, boundary layer equations, is then
shown to be independent of the body shape immersed into the flow. As
the first approximation, it is assumed that the second grade terms are
negligible compared to the viscous and third grade terms. The second
grade terms spoil scaling transformation which is the only transforma-
tion leading to similarity solutions for a third grade fluid. By using Lie’s
groupmethods, infinitesimal generators of boundary layer equations are
calculated. The equations are transformed into an ordinary differential
system.Numerical solutions to the outcoming nonlinear differential equ-
ations are found by using a combination of the Runge-Kutta algorithm
and a shooting technique.

Key words: boundary layer equations, Lie’s groups, third grade fluids

1. Introduction

As a non-Newtonian fluid model, Rivlin-Ericksen fluids gained much ac-
ceptance from both theorists and experimenters. Special cases of the model,
which is the fluid of the third grade, are extensively used, and a lot of works
have been done on the subject. Several boundary layer equations are derived
for differentnon-Newtonianmodels. For the sake of brevity,wementioned only



776 M.Yürüsoy

a few examples. Acrivos et al. (1960) and Pakdemirli (1996) derived bounda-
ry layer equations for power-law fluids. For the rate type of fluids, the works
due to Beard and Walters (1964) and Astin et al. (1973) are of significant
importance. The multiple deck boundary layer concept has been applied to
the second and third grade fluids by Pakdemirli (1994). Yürüsoy and Pakde-
mirli (1999) considered boundary layer equations for third grade fluids over a
stretching sheet in Cartesian coordinates.

Wechoose aconvenient coordinate systemfirstpurposedbyKaplun(1954),
whichmakes the equations independent of the body shape immersed into the
flow. The coordinate system is an orthogonal curvilinear system in which φ-
coordinates are the streamlines and ψ-coordinates are the velocity potentials
of the inviscid flow past a two-dimensional arbitrary profile. The boundary
layer equations of Newtonian fluids in this coordinate system are given byKe-
vorkian and Cole (1981). The boundary layer equations of the second-grade
fluids in this coordinate systemare derived byPakdemirli and Suhubi (1992a),
and the general symmetry groups for the equations are calculated using exte-
rior calculus by the same authors (1992b). They showed that second-grade
boundary layer equations accept only scaling transformation, and theypresen-
ted a similarity solution corresponding to this transformation. For the fluids
of grade three Pakdemirli (1992) showed that the additional term, due to the
third grade, prevent the applicability of the scaling transformation, hence no
similarity solutions exist.

First of all, in this article, in deriving boundary layer equations we use a
fluid of grade three as a non-Newtonian fluidmodel. It is shown byRivlin and
Ericksen (1955) that the stress tensor is given by following relation

T=−pI+µA1+α1A2+α2A21+β(trA21)A1 (1.1)

where p is the pressure, µ is the viscosity, α1 and α2 are the second grade
fluid terms, β is the third grade fluid term, and A1, A2 are the first two
Rivlin-Ericksen tensors given by the relations

L= gradv A1 =L+L
⊤

(1.2)

A2 = Ȧ1+A1L+L
⊤
A1

where v is the velocity vector. Rivlin andEricksen (1955) showed thatmaking
equation (1.1) compatible with the thermodynamics and minimizing the free
energy when the fluid is at rest, the material constants should satisfy the
relations



Similarity solutions to boundary layer equations... 777

µ ­ 0 α1 ­ 0
(1.3)

β ­ 0 |α1+α2| ¬
√

24µβ

The dimensionless form of the equations of motion for a third grade fluid are
(Pakdermirli, 1992)

1

2
grad|q|2+ω×q=−gradp+ε∇2q+ε1(∇2ω×q)+ε1grad(q ·∇2q)+

+
1

4
(2ε1+ε2)grad |A1|2+(ε1+ε2)

[

A1 ·∇2q+2div
(

gradq(gradq)⊤
)]

+

+ε3A1 · grad|A1|2+ε3|A1|2∇2q
(1.4)

divq=0

where q is the dimensionless velocity vector, ∇denotes Laplacian, ω=curlq
and the dimensionless coefficients are defined as follows

ε =
µ

ρUL
=
1

Re
ε1 =

α1
ρL2

ε2 =
α2
ρL2

ε3 =
βU

ρL3

(1.5)

where L and U are some reference length and velocity, respectively, ρ is the
density, Re is the Reynolds number.

2. Coordinate system

Aspecial coordinate systemmaking the equations independentof thebody
shape is chosen (Fig.1). The φ coordinate is related to streamlines and the
ψ coordinate to velocity potential lines of the inviscid flow past an arbitrary
object.
If we defined a complex function

F(z)= φ+iψ (2.1)

we can easily write the well-known formulas

q0 = u0i+v0j u0− iv0 = F ′(z)
u0 = φx = ψy v0 = φy =−ψx

(2.2)



778 M.Yürüsoy

where q0 is the potential velocity field. The pressure follows fromBernoulli’s
equation as

p =−1
2
q20 +C (2.3)

where C is a constant. Themetric for the system can be then defined as

dz =
dF

F ′
(dx)2+(dy)2 =

(dφ)2+(dψ)2

|F ′(z)|2
=
(dφ)2+(dψ)2

q20
(2.4)

To simplify the equations of motion, we introduce new velocity components
as follows

Wφ =
qφ
q0

Wψ =
qψ
q0

(2.5)

The velocity and gradient operator in this coordinate system are given by

q= q0(Wφiφ +Wψiψ) ∇=
(

q0
∂

∂φ
,q0

∂

∂ψ

)

(2.6)

In our case the Christoffel symbols are

Γ
φ
φφ
= Γ

ψ
ψφ
=−∂Q

∂φ
Γ
φ
ψφ
= Γ

ψ
ψψ
=−∂Q

∂ψ

Γ
φ
ψψ
=
∂Q

∂φ
Γ
ψ
φφ
=
∂Q

∂ψ

(2.7)

where Q = logq0.

3. Boundary layer equations

We have now necessary tools to obtain the boundary layer equations for
a special third grade fluid. As the first approximation, the assume that the
second grade terms are negligible compared to the viscous and third grade
terms. The second grade terms spoil the scaling transformations which is the
only transformation leading to similarity solutions for third grade fluids (see
Pakdemirli, 1992). Equation (1.4)1 is reduced to that of a third grade fluid if
we take ε1 = ε2 =0.
We assume that ε3 is proportional to ε

2

ε3 = kε
2 (3.1)

Themethod ofmatched asymptotic expansions will be used in the derivation.
We have to construct an inner expansion inside the boundary layer and outer



Similarity solutions to boundary layer equations... 779

expansion outside out it. Letting the perturbation parameter ε → 0, we have
to obtain the limit flow which is inviscid and irrotational when

Wφ =1 Wψ =0 p =−
1

2
q20 +C (3.2)

The outer expansion will then consist of the first terms in (3.2) and of correc-
tions due to the boundary layer as follows

Wφ(φ,ψ;ε) = 1+β(ε)W
1
φ(φ,ψ)+ ...

Wψ(φ,ψ;ε) = β(ε)W
1
ψ(φ,ψ)+ ... (3.3)

p(φ,ψ;ε) =−1
2
q20 +C +β(ε)P

1(φ,ψ)+ ...

where β(ε) is as yet an unknown coefficient to be determined from matching
with the restriction that β(ε)→ 0 as ε → 0. The inner expansion variable is
defined by stretching the ψ coordinate

ψ∗ =
1

δ(ε)
ψ (3.4)

with δ(ε)→ 0 as ε → 0. Then the inner expansion will be
Wφ(φ,ψ;ε) = Wφ(φ,ψ

∗)+ δ(ε)W1φ(φ,ψ
∗)+ ...

Wψ(φ,ψ;ε) = δ(ε)Wψ(φ,ψ
∗)+ δ2(ε)W1ψ(φ,ψ

∗)+ ... (3.5)

p(φ,ψ;ε) = P(φ,ψ∗)+ δ(ε)P1(φ,ψ)+ ...

The form above leads to a nontrivial continuity equation, and the inviscid
velocity inside the boundary layer will approach the velocity on the surface as
follows

q0(φ,ψ) = qB(φ)+O(δ(ε)) (3.6)

If we substitute equations (3.4)-(3.6) into equations ofmotion (1.4) and retain
relatively larger terms in each group, we obtain the equations

Wφ
∂Wφ
∂φ
+Wψ

∂Wφ
∂ψ
+W2φ

dQB
dφ
=− 1

q2
B

∂P

∂φ
+

ε

δ2
∂2Wφ
∂ψ2

+

+6k
ε

δ4
q4B(φ)

∂2Wφ
∂ψ2

(∂Wφ
∂ψ

)2

0=−1
δ

1

q2
B

∂P

∂ψ
+ε
(

O
( 1

δ3

))

(3.7)

∂Wφ
∂φ
+
∂Wψ
∂ψ
=0



780 M.Yürüsoy

where δ, the boundary layer thickness, is a small parameter and ε is the
perturbation parameter.We eliminate the pressure in equation (3.7)1 by using
the equation P =−1

2
q2B+C.Nowwecan assume that ε is of order δ

2 and k of
order δ2. On these assumptions, we finallywrite the boundary layer equations
for a special third grade fluid and the boundary conditions as follows

∂Wφ
∂φ
+
∂Wψ
∂ψ
=0

(3.8)

Wφ
∂Wφ
∂φ
+Wψ

∂Wφ
∂ψ
+(W2φ −1)

dQB
dφ
=
∂2Wφ
∂ψ2

+6kq4B(φ)
∂2Wφ
∂ψ2

(∂Wφ
∂ψ

)2

Wφ(φ,0)= Wψ(φ,0)= 0 Wφ(φ,∞)= 1

where Q′B = q
′

B/qB, k are the third grade fluid coefficients. k =0corresponds
to the Newtonian flow. Note that the final equations are valid for arbitrary
profiles because the inviscid surface velocity distribution qB appears as an ar-
bitrary function φ in the equations. It is thereforemuchmore straightforward
to draw a general conclusion from equations. Lie’s group transformationsmay
be then useful in investigating particular forms of qB so that the partial dif-
ferential equations could be reduced to ordinary differential equations via the
similarity transformation.

Lie’s group theory is applied to the equations. The equations admit a
scaling symmetry. The scaling symmetry is used to transform the system of
partial differential equations into a system of ordinary differential ones. Nu-
merical solutions to the resulting nonlinear ordinary differential equations are
found by using a combination of the Runge-Kutta algorithm and a shooting
technique.

4. Equations determining infinitesimals generators

To find all possible exact solutions to equations (3.8)1,2, we prefer using
the general method of Lie’s group analysis rather than using special group
transformations. Details on the application of Lie’s groups to solutions to
differential equations can be found byFosdick amdRajagopal (1980), Bluman
and Kumei (1989).

A one parameter Lie’s group of transformations and the corresponding
generator X is defined as follows



Similarity solutions to boundary layer equations... 781

φ∗ =φ+εξ1(φ,ψ,Wφ,Wψ)

ψ∗ = ψ+εξ2(φ,ψ,Wφ,Wψ)
(4.1)

W∗φ = Wφ +εη1(φ,ψ,Wφ,Wψ)

W∗ψ = Wψ +εη2(φ,ψ,Wφ,Wψ)

X = ξ1
∂

∂φ
+ ξ2

∂

∂ψ
+η1

∂

∂Wφ
+η2

∂

∂Wψ
(4.2)

By carrying out a straightforward and tedious algebra, we obtained the follo-
wing infinitesimals and equations

ξ1 = ξ1(φ) ξ2 = c2(φ)ψ +α(φ)

η1 = c1(φ)Wφ η2 = Wφ(ψc
′
2+α

′)+Wψ(c1+ c2− ξ′1)
(4.3)

and

2c1
q′B
qB
+ c′1 =0 c

′
1+c

′
2 =0

c1+2c2− ξ′1 =0 4c2− c1− ξ′1−4ξ1
q′B
qB
=0

(4.4)

From the above equations, we conclude that either q′B =0or c1 =0.The two
cases will be considered separately.

i) qB = const

Solving equations (4.3) and (4.4), we finally obtain the form of the so-
called infinitesimals

ξ1 =3aφ+b ξ2 = aψ+α(φ)

η1 = aWφ η2 =−aWψ +Wφα′
(4.5)

These results agree with the ones by Yürüsoy and Pakdemirli (1999).

ii) c1 =0

Solving equations (4.3) and (4.4), we finally obtain the infinitesimals

ξ1 =2aφ+b ξ2 = aψ+α(φ)

η1 =0 η2 =−aWψ +Wφα′
(4.6)



782 M.Yürüsoy

Solving equation (4.4)4, we find that

qB = c
4
√

2aφ+ b (4.7)

If we consider case i, the problem can be transformed to the conventional
boundary layer problem, which was discussed by Yürüsoy and Pakdemirli
(1999). Therefore, it supplies no new information. Only case ii, which is a
scaling transformation, supplies useful information leading to the similarity
solutions.
Imposing the restrictions from the boundaries and from equation (3.8)3 on

the boundary conditions on the infinitesimals, one obtains the following form
of equations (4.6) and (4.7)

ξ1 =2aφ+ b ξ2 = aψ

η1 =0 η2 =−aWψ
qB = c

4
√
2aφ+ b

(4.8)

where c is anarbitrary constant.Only this infinitesimal generator andthe form
of qB, which is a scaling transformation, supplies useful information leading
to the similarity solutions. Note that qB is not a constant but a parabolic
function.

5. Similarity solution

In this section, we will derive the similarity transformations and solutions
using the infinitesimals given in (4.8). First we transform the equations into
a system of ordinary differential one, and solve this system numerically using
the Runge-Kutta method with shooting.
Leaving the details of the procedure, thoroughly described byFosdick amd

Rajagopal (1980),BlumanandKumei (1989),we choose only the scaling trans-
formation (a =1,b =0). The characteristic equations are

dφ

2φ
=
dψ

ψ
=
dWφ
0
=
dWψ
−Wψ

(5.1)

The similarity variable, similarity functions and qB are

ξ =
ψ√
φ

Wφ = f(ξ) Wψ =
g(ξ)√
φ

qB =
4

√

γ(φ) (5.2)



Similarity solutions to boundary layer equations... 783

where γ = c
4
√
2. Substituting equation (5.2) and their derivatives into boun-

dary layer equations (3.8)1,2, we finally obtain

1

2
(f2−1)+2gf ′− ξff ′−2f ′′−12κf ′′f ′2 =0

(5.3)

ξf ′−2g′ =0

and the boundary conditions take the form

f(0)= g(0)= 0 f(∞)= 1 (5.4)

where κ = k∗γ4.

Fig. 1. Orthogonal coordinate system (streamlines of inviscid flow are φ coordinates,
velocity potential lines are ψ coordinates)

Since the equations are highly nonlinear, a numerical approach towards
the solution would bemore appropriate. Although the problem is a boundary
value problem, it is converted to an initial value problem. We assign a trial
value to f ′(0), integrate the equations using the Runge-Kutta algorithm and
check whether the boundary condition is satisfied at infinity. We repeat the
procedure until we find an appropriate of value f ′(0). Numerical results for
variousnon-Newtonian coefficients κareplotted inFig.2-Fig.4.Thesefigures
present the functions f, g and f ′, respectively, for κ equal to 0, 10 and 30.
For κ = 0 the flow is Newtonian. An increase in κ yields an increase in the
non-Newtonian behaviour. From Fig.2 we conclude that the boundary layer
thickness grows as the non-Newtonian effects increase in magnitude. Figure 3
shows the vertical component of the velocity inside the boundary layer. g(ξ)
increases when the non-Newtonian effects get intensified.



784 M.Yürüsoy

Fig. 2. Function f for various values of κ (as indicated on the curves)

Fig. 3. Function g for various values of κ (as indicated on the curves)

The shear stress at the boundary is calculated from equation (1.1) using
the coordinate properties and neglecting the small term. The dimensionless
shear stress on the boundary comes out to be

tφψ =
1√
Re

[

q2B
∂Wφ
∂ψ
+2kq6B

(∂Wφ
∂ψ

)3]

ψ=0
(5.5)



Similarity solutions to boundary layer equations... 785

In terms of the similarity variables, the shear stress is

tφψ =
δ2√
Re

[

f ′(0)+2κ(f ′(0))3
]

(5.6)

Fig. 4. First derivative of f for various values of κ (as indicated on the curves)

In equation (5.5), f ′(0) is to be read from Fig.4, which gives

tφψ ∼=







































0.65√
Re

for κ =0 and δ =1

1.00√
Re

for κ =10 and δ =1

1.25√
Re

for κ =30 and δ =1

It is evident from the calculations that growing κ increases the shear stress
on the boundary.

6. Concluding remarks

A different approach to the boundary layer equations of third grade flu-
ids was presented. The geometry of the profile was included as an arbitrary
function in the boundary layer equations which allowed the general ideas to



786 M.Yürüsoy

be drawn more easily. The second grade effects were negligible compared to
the third grade and viscous effects. By using Lie’s group analysis, we first
found the general symmetries of the partial differential system. Thenwe redu-
ced the equations to a system of ordinary differential ones via the similarity
transformations. Finally, we solved numerically the resulting ordinary diffe-
rential equations. It occured that the boundary layer got thicker when the
non-Newtonian aspect of the fluid behaviour becamemore pronounced.

References

1. AcrivosA., ShahM.J.,PetersenE.E., 1960,Momentumandheat transfer
in laminar boundary layer flows of non-Newtonian fluids past external surface,
A. I. Ch. E. Jl., 6, 312-317

2. Astin J., Jones R.S., Lockyer P., 1973, Boundary layer in non-Newtonian
fluids, J. Mec., 12, 527-539

3. BeardD.W.,WaltersK., 1964,Elastico-viscousboundary layerflows,Proc.
Camb. Phil., 60, 667-674

4. Bluman G.W., Kumei S., 1989, Symmetries and Differential Equations,
Springer-Verlag, NewYork

5. Kaplun S., 1954, The role of coordinate systems in boundary layer theory,
ZAMP, 5, 111-135

6. Kevorkian J., Cole J.D., 1981,Perturbation Method in Applied Mathema-
tics, New York Springer

7. PakdemirliM., 1992,The boundary layer equations of third-grade fluids, Int.
J. Non-linear Mech., 27, 5, 785-793

8. PakdemirliM., 1993,Boundary layer flowof power-lawpast arbitraryprofile,
IMA Journal of Applied Mathematics, 50, 133-148

9. Pakdemirli M., 1994, Conventional andmultiple deck boundary layer appro-
ach to second and third grade fluids, Int. J. Engng. Sci., 32, 1, 141-154

10. Pakdemirli M., Suhubi E.S., 1992a, Boundary layer theory second order
fluids, Int. J. Engng. Sci., 30, 4, 523-532

11. Pakdemirli M., Suhubi E.S., 1992b, Similarity solutions of boundary layer
equations for second order fluids, Int. J. Engng. Sci., 30, 5, 611-629

12. Rivlin R.S., Ericksen J.L., 1955, Stress-Deformation relations for isotropic
materials, J. Ration. Mech. Analysis, 4, 323-425



Similarity solutions to boundary layer equations... 787

13. Stephani H., 1989,Differential Equations: Their Solution Using Symmetries,
Cambridge University Press

14. Yürüsoy M., Pakdemirli M., 1999, Exact solutions of boundary layer equ-
ations of a special non-Newtonian fluid over a stretching sheet,Mechanics Re-
search Communications, 26, 2, 171-175

Rozwiązania podobieństwa równań warstwy przyściennej cieczy
nieniutonowskiej trzeciego rzędu w specjalnym układzie współrzędnych

Streszczenie

W pracy przedstawiono dwuwymiarowe równania ruchu dla stacjonarnego prze-
pływucieczy trzeciego rzęduwspecjalnymukładziewspółrzędnych.Równaniawypro-
wadzono na bazie przepływu potencjalnego cieczy nielekkiej. Przy nielepkim opływie
dowolnego obiektu linie prądu tworzą współrzędną φ, a linie potencjału prędkości
współrzędną ψ. Obydwie generują ortogonalny układ współrzędnych krzywolinio-
wych. Przy takim opisie postać równań warstwy przyściennej nie zależy od kształ-
tu zanurzonego ciała poddanego opływowi. W pierwszym przybliżeniu założono, że
wyrażenia drugiego rzędu są pomijalne w stosunku do członówwiskotycznych i trze-
ciego rzędu. Człony drugiego rzędu uniemożliwiają transformację skalowania, będącą
jedynym przekształceniemprowadzącymdo rozwiązań podobieństwa cieczy trzeciego
rzędu. W pracy zastosowano metodę opartą na grupie Lie’a w generowaniu równań
warstwy przyściennej przy pomocywyrażeń infinitezymalnych. Równania przekształ-
conodoukładu równań różniczkowychzwyczajnych.Numeryczne rozwiązanie równań
nieliniowychuzyskanowdrodze kombinacji algorytmuRunge-Kutta i techniki trymo-
wania.

Manuscript received October 22, 2002; accepted for print February 27, 2003