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Symmetry groups of boundary layer equations of a class of non-Newtonian fluids
Pergamon
Im J Non-Linear Mechanrcs, Vol. 31, No. 3, pp. 267-216, 1996 Copyright Q 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved OOZO-7462/96 $15.00 + 0.00
0020-7462(95)00071-2
SYMMETRY GROUPS OF BOUNDARY LAYER EQUATIONS OF A CLASS OF NON-NEWTONIAN FLUIDS Mehmet Departments
Pakdemirli,? of Mechanical
Muhammet Engineering
Yiiriisoy
and Physics, Turkey
and Atalay
Celal Bayar
KiiCiikbursa
University,
45040 Manisa,
1995; in revised form 21 September 1995)
(Receioed 6 January
Abstract-A non-Newtonian fluid model in which the stress is an arbitrary function of the symmetric part of the velocity gradient is considered. Symmetry groups of the two-dimensional boundary layer equations of the mode1 are found by using exterior calculus. The complete isovector field corresponding to some cases, such as arbitrary shear function, Newtonian fluids, and powerlaw fluids, are found. Similarly, solutions for some special transformations are presented. Results obtained in a previous paper [M. Pakdemirli, Int. J. Non-Linear Mech. 29, 187 (1994)] using special groups of transformations (scaling, spiral) are verified in this study using a general approach. Copyright % 1996 Elsevier Science Ltd. Keywords: boundary
layer, non-Newtonian
fluids
1. INTRODUCTION Boundary layers of non-Newtonian fluids have received considerable attention in the last few decades. Boundary layer theory has been applied successfully to various non-Newtonian fluid models. For power-law fluids, some of the previous work is due to Acrivos et al. [2], Schowalter [3], Na and Hansen [4] and Pakdemirli [S]. Mansutti and Rajagopal [6] investigated the boundary layer flow of a shear thinning power-law fluid. The boundary layers they studied are conceptually different from the usual boundary layers. Despite the flow being non-inertial, boundary layers develop due to non-linearities in the equation. For second grade fluids, the early work includes Srivastava [7] and Rajeswari and Rathna [S], while more recent work includes Rajagopal et al. [9, lo], Garg and Rajagopal [11] and Pakdemirli and $uhubi [12, 131. Boundary layer equations for third grade fluids are presented by Pakdemirli [ 141 using a convenient coordinate system. Multiple deck boundary layers as well as new conventional boundary layers have been proposed for third grade fluids by Pakdemirli [ 1.51. For the rate type of fluids, see Beard and Walters [ 161 and Astin et al. [17] for example. In this study, a non-Newtonian fluid model in which the shear stress is related to the symmetric part of the velocity gradient by an arbitrary function is considered. This model includes Prandtl, Powell-Eyring, Williamson and power-law fluids as special cases. Hansen and Na [18] applied scaling transformation to the two-dimensional boundary layer equations of this model. Timol and Kalthia [19] applied scaling and spiral group transformations to the three-dimensional equations of the same model. Pakdemirli [1] extended the analysis of previous work [18,19] by finding the special forms of the shear stress where richer similarities can be retrieved. There is also some theoretical work concerning the non-Newtonian fluid models considered here. A very recent work is due to Malek et al. [20], where existence and stability of solutions for general power-law fluids are investigated. That work [20] is an extension of previous work by Malek et al. [21]. Other related early work includes refs [22] and [23]. The work of ref. [l] is extended here by employing a general symmetry group approach rather than using special group transformations. Exterior calculus is used to find the isogroups of the partial differential equation. For details of the method, see Harrison and Estabrook [24] and Edelen [25]. For an application to general balance equations, an
t Author to whom correspondence Contributed by K. R. Rajagopal.
should
be addressed.
261
M. Pakdemirli et al.
268
algorithm has been developed by Suhubi [26]. We first derive the determining equations for the isovector field. Then, we investigated the cases of arbitrary shear, Newtonian flow and power-law flow. From the general isovector fields we show how special transformations of ref. [l] can be retrieved. We derive the similarity equations for some of those transformations.
2. EQUATIONS
We consider follows:
the two-dimensional
OF
laminar,
MOTION
incompressible
boundary
layer equations
as
.T
U+o”=O
(1)
ax ay
au au azxy u~+v-=-+o.g ay ay where u and v are the velocity U(x) is the velocity distribution shear stress to be an arbitrary we write
(2)
components in the x and y directions, respectively, and in the x direction outside the boundary layer. Assuming function of the symmetric part of the velocity gradient,
TX),=
This non-Newtonian fluid model includes law fluids as special cases. The boundary
au zxy ay
(-)
Prandtl, Powell-Eyring, Williamson and powerconditions associated with the problem are u(x, co) = U(x).
u(x, 0) = v(x, 0) = 0,
3. DETERMINING
EQUATIONS
BY EXTERIOR
(4)
CALCULUS
To find the symmetry groups of equations (l)-(4), we apply the formalism given by Harrison and Estabrook [24], which is based on Cartan’s geometric formulation of partial differential equations in terms of exterior differential forms [27]. We first reduce the order of equations to first order by defining new variables as follows: Xl
x2 = Y,
=x,
Ul
=
u,
al2
8th
au1
v,
=
2111 =g,
2312=3g>
au2 VZl
=ax,
vz* = 2.
(5) 2
In terms of these new variables,
fh)
we define new functions
=
ug,
7-(42)
=
(6)
~‘(Vl2).
1
Equations
(1) and (2) are written
Tav12 ax2
We define a seven-dimensional (x1 >x2, follows:
ul,
u2,
vll,
v12,
~2~).
UlVll
now in terms of (5) and (6) as follows: -
u2v12
+f=
0,
2122 =
-v11.
(7)
cover differentiable manifold M = R1 with a coordinate Exterior differential forms associated with (5) and (6) are as
(~1 = dul - vlldxl
- v12dx2
(8)
02 = du2 - v2idx1 - vlldxz w = TdvlzAdxl
+ (uiv1r + u2v12 -f
(9) )dxlAdx2
(10)
da, = - dvl lAdx,
- dv12Adx2
(11)
da2 = -d~lAdxi
+ dvl,Adx2.
(12)
Symmetry
groups
of boundary
269
layer equations
The ideal formed by l-forms ol, crz and 2-forms o, dai, do2 is closed. The symbol d denotes exterior differentiation and A exterior product. The associated isovector field is
where oi, 02, Qi, Q2, Vil, VI2 and vzl all depend on x1, x2, ul, u2, ull, u12, vzl. The vector field V is an isovector field of the ideal if and only if there exists appropriate 0- and l-forms such that the following relations hold: Y”ai = i,cr, + A.,02
(14)
zY”a2 = 2301 + 1402
(15)
~2’~c.o= &co - &do1 - &da2 + plAal
+ p2Ac2
(16)
where _.YVdenotes the Lie derivative with respect to vector I/, Aiare arbitrary O-forms and pj are arbitrary l-forms. The left-hand side of equation (14) can be calculated by employing the well-known relation
where ] represents the interior product operator between a differential form and a vector field. Defining F=
V]Oi
(18)
makes simplifications in the analysis. Inserting (18) into (17) and calculating the Lie derivative, we finally insert the result and (8) and (9) into (14) and obtain aF
aF w1=
O2=
_dv,l’
aF v12
Q
=
1
8x2
aF +
UlZI&
llaull
0
aF -
l;_-2,
=F-v
-= aF avzl
-ii&’
ullg&
z
(19)
12av12’
Defining G=VJa2
(20)
and proceeding in the same way, we finally obtain the following results from (15): 01=--,
v,,=
dG
aG
-= aG av12
cc)2 =G’
aVZl
0
+u,,g+qg 2
1
2
(21)
R2=G-~21~-~ll~. 21
11
M. Pakdemirli et nl.
210
Comparing now (19) with (21) we write F=cI(xI,xz,u~,u~)u~~+B(x~,xz,u~,uZ)U~~
+Y(x~,x~,u~,uz)
(22)
G = B(xr>xz, ~1, &U
+ w(xr, ~2, ~1, ~2)
(23)
- x(x1, ~2, ~lruz)rr1
(24)
02 = - CI, Rl = y,
ml= -P,
aa
Q2= w
(25)
afl
+ -v12u21 + 7$w21 au2
(26)
+(g-g)u1A2
(27)
+(-&311u21
v,, =-g+(g+g)u21 +(~--g)ull a&
ab u:,+--2,. au, au2
(28)
We now treat (16). The left-hand side is calculated using relation (17). Inserting forms to the right-hand side of (16), taking the limit as g1 and rs2 approaches zero and annihilating the coefficients of all forms and eliminating the arbitrary O-forms and l-forms, we finally have C&u11+ Vllul
+ Rzu12 + V12u2- qf’
- T
ah,
(2 dx
w2
+
u12au
1
v
-
ah2
11- au2>
am2 am2 au, av,, --f) = 0 + G + z”l2 -auZ’ll - au12 v12F > wll + u2u12 awl
am1 awl av12 l2~-ull-&=j--Q’ dx+O 2
(29)
(30)
Inserting equations (25) and (27) into (30), we obtain a polynomial equation in terms of the variables ul 1 and u12. Since arbitrary values can be assigned to the variables, the coefficients should vanish. Solving the three outcoming equations together with (24)4,5, we remove the dependence of some of the variables from the following functions:
+1, x23%I, ay am jjy+x=o, 1 2
P(Xl)Y Yh, X2?41, g+g=o, 1 1
4x19x2,Ul,u2)
!g+3L!!F&-“=o. 1
au,
2
au2
Equations (24)l_3 are unused at this time and are re-written in (32) for convenience.
(31) (32)
Symmetry groups of boundary layer equations
271
Inserting (25H27) into (29), using the new forms of functions (31), collecting similar terms, we finally obtain
Inspecting (33), we see that there are three groups of terms (coefficient of T, coefficient of T I/T and remaining terms). Each group is a polynomial in itself of the variables vl 1 and o12. However, T is a function of v12 and depending on the specific form of T, each group may interact with each other. To proceed further, we first assume that T is completely arbitrary and then investigate other special cases.
4. ARBITRARY
SHEAR
STRESS
In this section, we consider the shear stress to be completely arbitrary. In this case, the three groups of terms in (33) seperate into three equations. Each equation now is a polynomial in terms of the variables vl 1 and v12. Equating the coefficients of each polynomial to zero, we have the following set of equations: (34)
(35)
~Ul+W+Uu,+3f&=o 1
2
1
acr
-_=
au, a3
0
(37)
o
--
(3%
ax; -
a3 $+2-E ax2aul
o
2
2
a%
axy
zz++,u,=” 1 2 a2ci au:-
-
0
1
(39)
212
M. Pakdemirli
et al.
(44)
ay
-=
ax2
0
(45)
(46) Solving (34)-(46) together with (32), we find CI,/I, y and w. Using components of the isovector field
then (25) we write the
w1 = 3axl + b
(47)
w2 = ax2 + h(x,)
(48)
01 = au,
(49)
C& = - au2 + h’(xr)ur.
(50)
Parameter a corresponds to scaling transformation and parameter b corresponds to translation in the x1 coordinate. The scaling transformation corresponding to parameter a is exactly the same transformation given in (17) of ref. [l]. Note that spiral group transformation cannot be retrieved from this isovector field in agreement with ref. [l]. We see that symmetry corresponding to h(x,) is new and cannot be found with the transformations used in ref. [l]. The remaining equation which seperated from (34) is
af = 0.
(3uxr + b)f’ + Remembering that f = UU’ and solving c1 = b/3u. Setting cl = 0 we obtain
for
U, we obtain
(51) U = k(xI + c~)“~
U = kx;‘3
where
(52)
in agreement with the results of ref. [l]. This l/3 factor represents a wedge flow of 90”. This means that only wedge flow of 90” is possible for arbitrary shear stress if the invariant solution corresponding to parameter a is chosen. However, for specific forms of the stress (Newtonian, power-law), we show in the next sections that arbitrary wedge angles are possible. Note also that if a = b = 0, then an invariant solution corresponding to h(x,) does not put any restriction on the form of for U, which means that a solution corresponding to that symmetry accepts arbitrary profiles. However, imposing further the boundary conditions reduces this solution to a trivial solution.
5. NEWTONIAN
FLUIDS
For this case, we take T = v and substitute u12. Equating
(v a constant)
this into (33) which now becomes a polynomial the coefficients of each term to zero we obtain
(53) equation
in terms of v1 1 and
(54)
(55)
~u,+U-&U2+3f*-V 1
2
au,
(56)
273
Symmetry groups of boundary layer equations
ac!0
-_=
(57)
au,
(58)
(59) Solving (54)-(59) together with (32), we find CI,p, y and o. Using components of the isovector field
then (25), we write the
o1 = ax, + b
(60)
02 = cx2 + h(x,)
(61)
Sz, = (a - 2c)uI
(62)
Q2 = - cu2 + h’(Xl)Ul.
(63)
The above results are in full agreement with those presented by Suhubi that reference]. The remaining equation which separated from (54) is (ax1 + b)f’
+ (4c - a)f=
[26] [i.e. (6.16) of
0.
(64)
U = k(xI + cl)“’ where c1 = b/a.
Remembering thatf= UU’ and solving for U, we obtain Without loss of generality, we set c1 = 0 and therefore U = kx;
(65)
where m = 2 - 4c/a. This arbitrary m parameter implies that for Newtonian fluids similarity solutions are possible for arbitrary wedge angles. We now retrieve the spiral group transformation given in ref. [l] from the isovector field sim(60)-(63). Selecting h(x,) = 0, a = 0 and b = - (2/ m) c, we write the corresponding ilarity equations dxl
dx2
-2/m=F from which we define the similarity 5 = X2e(m/2)x,, Solving
dur =_=-2ur
variable
du2
(66)
-2l2
and functions
u1 = e”“lF(<),
u2 = e(mi2)x1G(~).
(64) for this special choice of parameters,
we obtain
U = kemxl. Substituting (67), (68) and (53) into the original differential equations: m dF
(67)
equations,
dG
(68) we obtain
the following
ordinary
(69)
(70) Results
are identical
with those of Section
4.3 of ref. Cl].
6. POWER-LAW
For power-law
FLUIDS
fluids, we take T = v(u12)B
(v, n constants)
(71)
274
M. Pakdemirli
et al.
where n # - 1. Substituting equation (71) into (33), we have
aa v12- 2~u20:2 + ;xU,+W-&U2+3f& 1 2 1 Ul2-2J%V au, 1 11 ( )
We assume that n # -2. Separating (72), solving the resulting equations together with (32) and using further (25), we finally obtain o1 = ((n - 1)a + (n + 2)b)xr + c
(73)
02 = bxz + h(x,)
(74)
RI = --au,
(75)
Cl, = -(na + (1 + n)b)u2 + h’(xl)ul.
(76)
By redefining appropriate constants, it can be shown that the above results reduce to those of the Newtonian case for n = 0. The isovector fields (73)-(76) include scaling as well as a spiral group of transformations in agreement with ref. [l]. The last equation to be satisfied is (((n - 1)~ + (n + 2)b)xl + c)f’ + ((n + 1)~ + (n + 2)b)f= 0. Remembering that f= UU’ and solving for U, we obtain U = k(xI + cJ” c1 = c/[(n - 1)~ + (n + 2)b]. Without loss of generality, we set c1 = 0 and obtain U = kxy
(77) where
(78)
where k is an arbitrary constant and m = -u/[(n - 1)~ + (n + 2)b]. This arbitrary m parameter implies that for power-law fluids, similarity solutions are possible for arbitrary wedge angles in agreement with refs [ 1,4,5]. We now present the derivation of similarity equations for the power-law case corresponding to scaling transformation. Remembering that m = -u/[(n - 1)~ + (n + 2)b], we solve a in terms of b m(n + 2) b (79) a = m(l -n) - 1 ’ Substituting this result into isovector components (73)-(76), writing the base vector corresponding to parameter b, we obtain the system of differential equations
dxl
-zr
Xl
dxz
dul
m(n - 1) + 1 = G x2 n+2
duz = m(2n + 1) - (n + 1) u2 n+2
(80)
where all denominators are multiplied by (m(n - 1) + l)/(n + 2). Equation (80) is identical with (3.46) of ref. [l] if n is replaced by 2n. This difference stems from different definitions of the shear stress. Note also that our problem is a contraction to two-dimensional flow.
Symmetry
groups
of boundary
275
layer equations
The similarity variable and functions are defined by solving equation (80) m(1 -IIq
=
1
1)-(n+
m(Zn+
x2x1-,
a1 =
xTF(r),
u2
1)
n+2
Xl
=
WI).
(81)
Substituting (81) into original equations (7), using (71) and (78), we finally obtain the corresponding ordinary differential equations mF + mF2 +
m(1 - n) - 1 dF
dG
q-+-=0 dv dvl
n+2
(82)
m(1 - n) - 1
(83)
n+2
where 5 = v/(n + 1). Boundary conditions now read f(0) = g(0) = 0,
f(a)
= k.
(84)
Similarity solutions corresponding to other parameters and other cases may be produced in a similar way. Many of those equations, which are obtained by using special transformations, are given in ref. [l].
7. OTHER
NON-NEWTONIAN
MODELS
Shear stresses are given below for some non-Newtonian
fluid models:
z = A sin-l (u12/c) (Prandtl) 1 z = vu12 + Bsinh-l
z
=
WI12
+
B
Au12 +
(85)
(Powell-Eyring)
(Williamson). 012
Remembering that T = r’, we take derivatives and substitute results into (33). The equations seperate exactly the same way as that of arbitrary shear stress case. Therefore results of Section 4 are valid also for the above fluid models. The geometry of the flow for useful solutions is therefore restricted to wedge flow of 90”.
8. CONCLUDING
REMARKS
A general symmetry group approach is applied to boundary layer equations of a class of non-Newtonian fluids in which the stress is an arbitrary function of the symmetric part of the velocity gradient, Exterior calculus is used in the analysis and the equations determining the isovector fields are derived. Solution of the equations yield the isovector fields. Special cases such as arbitrary stress, Newtonian fluids and power-law fluids are treated. Similarity solutions corresponding to some base vectors are presented. The related previous work includes constructing special group transformations and generating the corresponding similarity solutions. However, the general isovector field is given in this work for the first time. Results of scaling and spiral group transformations existing in the literature are retrieved using this general approach. It is also shown that some non-Newtonian fluid models such as Prandtl, Powel-Eyring, Williamson, etc., possess the same isovector field with the arbitrary shear stress case. Acknowledgement-Special thanks are due to Professor E. S. Suhubi whose calculations for Newtonian guided us greatly in our analysis. We also thank M. G. Timol for his private communication.
fluids
REFERENCES 1. M. Pakdemirli, Similarity analysis of boundary layer equations of a class of non-Newtonian fluids. Int. J. Non-Linear Mech. 29, 187 (1994). 2. A. Acrivos, M. J. Shah and E. E. Petersen, Momentum and heat transfer in laminar boundary-layer flows of non-Newtonian fluids past external surfaces. A.I.Ch.E. J. 6, 312 (1960).
216
M. Pakdemirli
et al.
The application of boundary-layer theory to power-law pseudoplastic fluids: similar 3. W. R. Schowalter, solutions. A.1.Ch.E. J. 6, 24 (1960). 4. T. Y. Na and A. G. Hansen, Similarity solutions of a class of laminar three dimensional boundary layer equations of power-law fluids. Int. J. Non-Linear Mech. 2, 373 (1967). 5. M. Pakdemirli, Boundary layer flow of power-law fluids past arbitrary profiles. IMA J. Appl. Math. 50, 133 (1993). and K. R. Rajagopal, Flow of a shear thinning fluid between intersecting planes. lnt. J. 6. D. Mansutti Non-Linear Mech. 26, 169 (1991). 7. A. C. Srivastava, The flow of non-Newtonian liquid near a stagnation point. ZAMP 9, 80 (1958). visco-elastic and visco-inelastic 8. G. K. Rajeswari and S. L. Rathna, Flow of a particular class of non-Newtonian fluids near a stagnation point. ZAMP 13, 43 (1962). fluids. Lett. 9. K. R. Rajagopal, A. S. Gupta and A. S. Wineman, On a boundary layer theory for non-Newtonian Appl. Engng Sci. 18, 875 (1980). fluid past an 10. K. R. Rajagopal, A. Z. Szeri and W. Troy, An existence theorem for the flow of a non-Newtonian infinite porous plate. Int. J. Non-Linear Mech. 21, 279 (1986). 11. V. K. Garg and K. R. Rajagopal, Flow of a non-Newtonian fluid past a wedge. Acta Mech. 88, 113 (1991). and E. S. $uhubi, Boundary layer theory for second order fluids. Int. J. Engng Sci. 30, 523 12. M. Pakdemirli (1992). 13. M. Pakdemirli and E. S. $uhubi, Similarity solutions of boundary layer equations for second order fluids. Int. J. Engng Sci. 30, 611 (1992). 14. M. Pakdemirli, The boundary layer equations of third grade fluids. Int. J. Non-Linear Mech. 27, 785 (1992). and multiple deck boundary layer approach to second and third grade fluids. 15. M. Pakdemirli, Conventional Int. J. Engng Sci. 32, 141 (1994). 16. D. W. Beard and K. Walters, Elastico-viscous boundary layer flows. Proc. Camb. Phil. Sot. 60, 667 (1964). 17. J. Astin, R. S. Jones and P. Lockyer, Boundary layers in non-Newtonian fluids. J. Mecanique 12, 527 (1973). boundary layer equations of 18. A. G. Hansen and T. Y. Na, Similarity solutions of laminar, incompressible, non-Newtonian fluids. ASME J. Basic Engng 71 (1968). 19. M. G. Timol and N. L. Kalthia, Similarity solutions of three dimensional boundary layer equations of non-Newtonian fluids. Int. J. Non-Linear Mech. 21, 475 (1986). 20. J. Malek, K. R. Rajagopal and M. Ruzicka, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Meth. Appl. Sci. 5, 789 (1995). 21. J. Malek, J. Necas and M. Ruzicka, On the non-Newtonian incompressible fluids. Math. Models Meth. Appl. sci. 3, 35 (1993). 22. 0. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969). 23. J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris (1969). 24. B. K. Harrison and F. B. Estabrook, Geometric approach to invariance groups and solution of partial differential systems. J. Math. Phys. 12, 653 (1971). 25. D. G. B. Edelen, Applied Exterior Calculus. Wiley, New York (1985). 26. E. S. Suhubi, Isovector fields and similarity solutions for general balance equations. Int. J. Engng Sci. 29, 133 (1991). 27. E. Cartan, Les Systemes DifSerentiels Exterieurs et Leurs Applications Geometriques. Hermann, Paris (1945).
Im J Non-Linear Mechanrcs, Vol. 31, No. 3, pp. 267-216, 1996 Copyright Q 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved OOZO-7462/96 $15.00 + 0.00
0020-7462(95)00071-2
SYMMETRY GROUPS OF BOUNDARY LAYER EQUATIONS OF A CLASS OF NON-NEWTONIAN FLUIDS Mehmet Departments
Pakdemirli,? of Mechanical
Muhammet Engineering
Yiiriisoy
and Physics, Turkey
and Atalay
Celal Bayar
KiiCiikbursa
University,
45040 Manisa,
1995; in revised form 21 September 1995)
(Receioed 6 January
Abstract-A non-Newtonian fluid model in which the stress is an arbitrary function of the symmetric part of the velocity gradient is considered. Symmetry groups of the two-dimensional boundary layer equations of the mode1 are found by using exterior calculus. The complete isovector field corresponding to some cases, such as arbitrary shear function, Newtonian fluids, and powerlaw fluids, are found. Similarly, solutions for some special transformations are presented. Results obtained in a previous paper [M. Pakdemirli, Int. J. Non-Linear Mech. 29, 187 (1994)] using special groups of transformations (scaling, spiral) are verified in this study using a general approach. Copyright % 1996 Elsevier Science Ltd. Keywords: boundary
layer, non-Newtonian
fluids
1. INTRODUCTION Boundary layers of non-Newtonian fluids have received considerable attention in the last few decades. Boundary layer theory has been applied successfully to various non-Newtonian fluid models. For power-law fluids, some of the previous work is due to Acrivos et al. [2], Schowalter [3], Na and Hansen [4] and Pakdemirli [S]. Mansutti and Rajagopal [6] investigated the boundary layer flow of a shear thinning power-law fluid. The boundary layers they studied are conceptually different from the usual boundary layers. Despite the flow being non-inertial, boundary layers develop due to non-linearities in the equation. For second grade fluids, the early work includes Srivastava [7] and Rajeswari and Rathna [S], while more recent work includes Rajagopal et al. [9, lo], Garg and Rajagopal [11] and Pakdemirli and $uhubi [12, 131. Boundary layer equations for third grade fluids are presented by Pakdemirli [ 141 using a convenient coordinate system. Multiple deck boundary layers as well as new conventional boundary layers have been proposed for third grade fluids by Pakdemirli [ 1.51. For the rate type of fluids, see Beard and Walters [ 161 and Astin et al. [17] for example. In this study, a non-Newtonian fluid model in which the shear stress is related to the symmetric part of the velocity gradient by an arbitrary function is considered. This model includes Prandtl, Powell-Eyring, Williamson and power-law fluids as special cases. Hansen and Na [18] applied scaling transformation to the two-dimensional boundary layer equations of this model. Timol and Kalthia [19] applied scaling and spiral group transformations to the three-dimensional equations of the same model. Pakdemirli [1] extended the analysis of previous work [18,19] by finding the special forms of the shear stress where richer similarities can be retrieved. There is also some theoretical work concerning the non-Newtonian fluid models considered here. A very recent work is due to Malek et al. [20], where existence and stability of solutions for general power-law fluids are investigated. That work [20] is an extension of previous work by Malek et al. [21]. Other related early work includes refs [22] and [23]. The work of ref. [l] is extended here by employing a general symmetry group approach rather than using special group transformations. Exterior calculus is used to find the isogroups of the partial differential equation. For details of the method, see Harrison and Estabrook [24] and Edelen [25]. For an application to general balance equations, an
t Author to whom correspondence Contributed by K. R. Rajagopal.
should
be addressed.
261
M. Pakdemirli et al.
268
algorithm has been developed by Suhubi [26]. We first derive the determining equations for the isovector field. Then, we investigated the cases of arbitrary shear, Newtonian flow and power-law flow. From the general isovector fields we show how special transformations of ref. [l] can be retrieved. We derive the similarity equations for some of those transformations.
2. EQUATIONS
We consider follows:
the two-dimensional
OF
laminar,
MOTION
incompressible
boundary
layer equations
as
.T
U+o”=O
(1)
ax ay
au au azxy u~+v-=-+o.g ay ay where u and v are the velocity U(x) is the velocity distribution shear stress to be an arbitrary we write
(2)
components in the x and y directions, respectively, and in the x direction outside the boundary layer. Assuming function of the symmetric part of the velocity gradient,
TX),=
This non-Newtonian fluid model includes law fluids as special cases. The boundary
au zxy ay
(-)
Prandtl, Powell-Eyring, Williamson and powerconditions associated with the problem are u(x, co) = U(x).
u(x, 0) = v(x, 0) = 0,
3. DETERMINING
EQUATIONS
BY EXTERIOR
(4)
CALCULUS
To find the symmetry groups of equations (l)-(4), we apply the formalism given by Harrison and Estabrook [24], which is based on Cartan’s geometric formulation of partial differential equations in terms of exterior differential forms [27]. We first reduce the order of equations to first order by defining new variables as follows: Xl
x2 = Y,
=x,
Ul
=
u,
al2
8th
au1
v,
=
2111 =g,
2312=3g>
au2 VZl
=ax,
vz* = 2.
(5) 2
In terms of these new variables,
fh)
we define new functions
=
ug,
7-(42)
=
(6)
~‘(Vl2).
1
Equations
(1) and (2) are written
Tav12 ax2
We define a seven-dimensional (x1 >x2, follows:
ul,
u2,
vll,
v12,
~2~).
UlVll
now in terms of (5) and (6) as follows: -
u2v12
+f=
0,
2122 =
-v11.
(7)
cover differentiable manifold M = R1 with a coordinate Exterior differential forms associated with (5) and (6) are as
(~1 = dul - vlldxl
- v12dx2
(8)
02 = du2 - v2idx1 - vlldxz w = TdvlzAdxl
+ (uiv1r + u2v12 -f
(9) )dxlAdx2
(10)
da, = - dvl lAdx,
- dv12Adx2
(11)
da2 = -d~lAdxi
+ dvl,Adx2.
(12)
Symmetry
groups
of boundary
269
layer equations
The ideal formed by l-forms ol, crz and 2-forms o, dai, do2 is closed. The symbol d denotes exterior differentiation and A exterior product. The associated isovector field is
where oi, 02, Qi, Q2, Vil, VI2 and vzl all depend on x1, x2, ul, u2, ull, u12, vzl. The vector field V is an isovector field of the ideal if and only if there exists appropriate 0- and l-forms such that the following relations hold: Y”ai = i,cr, + A.,02
(14)
zY”a2 = 2301 + 1402
(15)
~2’~c.o= &co - &do1 - &da2 + plAal
+ p2Ac2
(16)
where _.YVdenotes the Lie derivative with respect to vector I/, Aiare arbitrary O-forms and pj are arbitrary l-forms. The left-hand side of equation (14) can be calculated by employing the well-known relation
where ] represents the interior product operator between a differential form and a vector field. Defining F=
V]Oi
(18)
makes simplifications in the analysis. Inserting (18) into (17) and calculating the Lie derivative, we finally insert the result and (8) and (9) into (14) and obtain aF
aF w1=
O2=
_dv,l’
aF v12
Q
=
1
8x2
aF +
UlZI&
llaull
0
aF -
l;_-2,
=F-v
-= aF avzl
-ii&’
ullg&
z
(19)
12av12’
Defining G=VJa2
(20)
and proceeding in the same way, we finally obtain the following results from (15): 01=--,
v,,=
dG
aG
-= aG av12
cc)2 =G’
aVZl
0
+u,,g+qg 2
1
2
(21)
R2=G-~21~-~ll~. 21
11
M. Pakdemirli et nl.
210
Comparing now (19) with (21) we write F=cI(xI,xz,u~,u~)u~~+B(x~,xz,u~,uZ)U~~
+Y(x~,x~,u~,uz)
(22)
G = B(xr>xz, ~1, &U
+ w(xr, ~2, ~1, ~2)
(23)
- x(x1, ~2, ~lruz)rr1
(24)
02 = - CI, Rl = y,
ml= -P,
aa
Q2= w
(25)
afl
+ -v12u21 + 7$w21 au2
(26)
+(g-g)u1A2
(27)
+(-&311u21
v,, =-g+(g+g)u21 +(~--g)ull a&
ab u:,+--2,. au, au2
(28)
We now treat (16). The left-hand side is calculated using relation (17). Inserting forms to the right-hand side of (16), taking the limit as g1 and rs2 approaches zero and annihilating the coefficients of all forms and eliminating the arbitrary O-forms and l-forms, we finally have C&u11+ Vllul
+ Rzu12 + V12u2- qf’
- T
ah,
(2 dx
w2
+
u12au
1
v
-
ah2
11- au2>
am2 am2 au, av,, --f) = 0 + G + z”l2 -auZ’ll - au12 v12F > wll + u2u12 awl
am1 awl av12 l2~-ull-&=j--Q’ dx+O 2
(29)
(30)
Inserting equations (25) and (27) into (30), we obtain a polynomial equation in terms of the variables ul 1 and u12. Since arbitrary values can be assigned to the variables, the coefficients should vanish. Solving the three outcoming equations together with (24)4,5, we remove the dependence of some of the variables from the following functions:
+1, x23%I, ay am jjy+x=o, 1 2
P(Xl)Y Yh, X2?41, g+g=o, 1 1
4x19x2,Ul,u2)
!g+3L!!F&-“=o. 1
au,
2
au2
Equations (24)l_3 are unused at this time and are re-written in (32) for convenience.
(31) (32)
Symmetry groups of boundary layer equations
271
Inserting (25H27) into (29), using the new forms of functions (31), collecting similar terms, we finally obtain
Inspecting (33), we see that there are three groups of terms (coefficient of T, coefficient of T I/T and remaining terms). Each group is a polynomial in itself of the variables vl 1 and o12. However, T is a function of v12 and depending on the specific form of T, each group may interact with each other. To proceed further, we first assume that T is completely arbitrary and then investigate other special cases.
4. ARBITRARY
SHEAR
STRESS
In this section, we consider the shear stress to be completely arbitrary. In this case, the three groups of terms in (33) seperate into three equations. Each equation now is a polynomial in terms of the variables vl 1 and v12. Equating the coefficients of each polynomial to zero, we have the following set of equations: (34)
(35)
~Ul+W+Uu,+3f&=o 1
2
1
acr
-_=
au, a3
0
(37)
o
--
(3%
ax; -
a3 $+2-E ax2aul
o
2
2
a%
axy
zz++,u,=” 1 2 a2ci au:-
-
0
1
(39)
212
M. Pakdemirli
et al.
(44)
ay
-=
ax2
0
(45)
(46) Solving (34)-(46) together with (32), we find CI,/I, y and w. Using components of the isovector field
then (25) we write the
w1 = 3axl + b
(47)
w2 = ax2 + h(x,)
(48)
01 = au,
(49)
C& = - au2 + h’(xr)ur.
(50)
Parameter a corresponds to scaling transformation and parameter b corresponds to translation in the x1 coordinate. The scaling transformation corresponding to parameter a is exactly the same transformation given in (17) of ref. [l]. Note that spiral group transformation cannot be retrieved from this isovector field in agreement with ref. [l]. We see that symmetry corresponding to h(x,) is new and cannot be found with the transformations used in ref. [l]. The remaining equation which seperated from (34) is
af = 0.
(3uxr + b)f’ + Remembering that f = UU’ and solving c1 = b/3u. Setting cl = 0 we obtain
for
U, we obtain
(51) U = k(xI + c~)“~
U = kx;‘3
where
(52)
in agreement with the results of ref. [l]. This l/3 factor represents a wedge flow of 90”. This means that only wedge flow of 90” is possible for arbitrary shear stress if the invariant solution corresponding to parameter a is chosen. However, for specific forms of the stress (Newtonian, power-law), we show in the next sections that arbitrary wedge angles are possible. Note also that if a = b = 0, then an invariant solution corresponding to h(x,) does not put any restriction on the form of for U, which means that a solution corresponding to that symmetry accepts arbitrary profiles. However, imposing further the boundary conditions reduces this solution to a trivial solution.
5. NEWTONIAN
FLUIDS
For this case, we take T = v and substitute u12. Equating
(v a constant)
this into (33) which now becomes a polynomial the coefficients of each term to zero we obtain
(53) equation
in terms of v1 1 and
(54)
(55)
~u,+U-&U2+3f*-V 1
2
au,
(56)
273
Symmetry groups of boundary layer equations
ac!0
-_=
(57)
au,
(58)
(59) Solving (54)-(59) together with (32), we find CI,p, y and o. Using components of the isovector field
then (25), we write the
o1 = ax, + b
(60)
02 = cx2 + h(x,)
(61)
Sz, = (a - 2c)uI
(62)
Q2 = - cu2 + h’(Xl)Ul.
(63)
The above results are in full agreement with those presented by Suhubi that reference]. The remaining equation which separated from (54) is (ax1 + b)f’
+ (4c - a)f=
[26] [i.e. (6.16) of
0.
(64)
U = k(xI + cl)“’ where c1 = b/a.
Remembering thatf= UU’ and solving for U, we obtain Without loss of generality, we set c1 = 0 and therefore U = kx;
(65)
where m = 2 - 4c/a. This arbitrary m parameter implies that for Newtonian fluids similarity solutions are possible for arbitrary wedge angles. We now retrieve the spiral group transformation given in ref. [l] from the isovector field sim(60)-(63). Selecting h(x,) = 0, a = 0 and b = - (2/ m) c, we write the corresponding ilarity equations dxl
dx2
-2/m=F from which we define the similarity 5 = X2e(m/2)x,, Solving
dur =_=-2ur
variable
du2
(66)
-2l2
and functions
u1 = e”“lF(<),
u2 = e(mi2)x1G(~).
(64) for this special choice of parameters,
we obtain
U = kemxl. Substituting (67), (68) and (53) into the original differential equations: m dF
(67)
equations,
dG
(68) we obtain
the following
ordinary
(69)
(70) Results
are identical
with those of Section
4.3 of ref. Cl].
6. POWER-LAW
For power-law
FLUIDS
fluids, we take T = v(u12)B
(v, n constants)
(71)
274
M. Pakdemirli
et al.
where n # - 1. Substituting equation (71) into (33), we have
aa v12- 2~u20:2 + ;xU,+W-&U2+3f& 1 2 1 Ul2-2J%V au, 1 11 ( )
We assume that n # -2. Separating (72), solving the resulting equations together with (32) and using further (25), we finally obtain o1 = ((n - 1)a + (n + 2)b)xr + c
(73)
02 = bxz + h(x,)
(74)
RI = --au,
(75)
Cl, = -(na + (1 + n)b)u2 + h’(xl)ul.
(76)
By redefining appropriate constants, it can be shown that the above results reduce to those of the Newtonian case for n = 0. The isovector fields (73)-(76) include scaling as well as a spiral group of transformations in agreement with ref. [l]. The last equation to be satisfied is (((n - 1)~ + (n + 2)b)xl + c)f’ + ((n + 1)~ + (n + 2)b)f= 0. Remembering that f= UU’ and solving for U, we obtain U = k(xI + cJ” c1 = c/[(n - 1)~ + (n + 2)b]. Without loss of generality, we set c1 = 0 and obtain U = kxy
(77) where
(78)
where k is an arbitrary constant and m = -u/[(n - 1)~ + (n + 2)b]. This arbitrary m parameter implies that for power-law fluids, similarity solutions are possible for arbitrary wedge angles in agreement with refs [ 1,4,5]. We now present the derivation of similarity equations for the power-law case corresponding to scaling transformation. Remembering that m = -u/[(n - 1)~ + (n + 2)b], we solve a in terms of b m(n + 2) b (79) a = m(l -n) - 1 ’ Substituting this result into isovector components (73)-(76), writing the base vector corresponding to parameter b, we obtain the system of differential equations
dxl
-zr
Xl
dxz
dul
m(n - 1) + 1 = G x2 n+2
duz = m(2n + 1) - (n + 1) u2 n+2
(80)
where all denominators are multiplied by (m(n - 1) + l)/(n + 2). Equation (80) is identical with (3.46) of ref. [l] if n is replaced by 2n. This difference stems from different definitions of the shear stress. Note also that our problem is a contraction to two-dimensional flow.
Symmetry
groups
of boundary
275
layer equations
The similarity variable and functions are defined by solving equation (80) m(1 -IIq
=
1
1)-(n+
m(Zn+
x2x1-,
a1 =
xTF(r),
u2
1)
n+2
Xl
=
WI).
(81)
Substituting (81) into original equations (7), using (71) and (78), we finally obtain the corresponding ordinary differential equations mF + mF2 +
m(1 - n) - 1 dF
dG
q-+-=0 dv dvl
n+2
(82)
m(1 - n) - 1
(83)
n+2
where 5 = v/(n + 1). Boundary conditions now read f(0) = g(0) = 0,
f(a)
= k.
(84)
Similarity solutions corresponding to other parameters and other cases may be produced in a similar way. Many of those equations, which are obtained by using special transformations, are given in ref. [l].
7. OTHER
NON-NEWTONIAN
MODELS
Shear stresses are given below for some non-Newtonian
fluid models:
z = A sin-l (u12/c) (Prandtl) 1 z = vu12 + Bsinh-l
z
=
WI12
+
B
Au12 +
(85)
(Powell-Eyring)
(Williamson). 012
Remembering that T = r’, we take derivatives and substitute results into (33). The equations seperate exactly the same way as that of arbitrary shear stress case. Therefore results of Section 4 are valid also for the above fluid models. The geometry of the flow for useful solutions is therefore restricted to wedge flow of 90”.
8. CONCLUDING
REMARKS
A general symmetry group approach is applied to boundary layer equations of a class of non-Newtonian fluids in which the stress is an arbitrary function of the symmetric part of the velocity gradient, Exterior calculus is used in the analysis and the equations determining the isovector fields are derived. Solution of the equations yield the isovector fields. Special cases such as arbitrary stress, Newtonian fluids and power-law fluids are treated. Similarity solutions corresponding to some base vectors are presented. The related previous work includes constructing special group transformations and generating the corresponding similarity solutions. However, the general isovector field is given in this work for the first time. Results of scaling and spiral group transformations existing in the literature are retrieved using this general approach. It is also shown that some non-Newtonian fluid models such as Prandtl, Powel-Eyring, Williamson, etc., possess the same isovector field with the arbitrary shear stress case. Acknowledgement-Special thanks are due to Professor E. S. Suhubi whose calculations for Newtonian guided us greatly in our analysis. We also thank M. G. Timol for his private communication.
fluids
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