Acta Mechanica
Acta Mechanica 181, 65–81 (2006) DOI 10.1007/s00707-005-0272-9
Printed in Austria
Group method analysis of mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder F. S. Ibrahim and M. A. A. Hamad, Assiut, Egypt Received January 10, 2005; revised May 24, 2005 Published online: December 19, 2005 Ó Springer-Verlag 2005
Summary. The transformation group theoretic approach is applied to the system of equations governing the unsteady mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder. The application of a two-parameter group reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature Tw, potential velocity U and sin / with position and time are derived in steady and unsteady cases. New formulae of dimensionless temperature are presented using the group method analysis. Hiemenz and Falkner-Skan equations are obtained as special cases. The new similarity representations and similarity transformations in steady/unsteady states are obtained. The family of ordinary differential equations has been solved numerically using a fourth-order Runge-Kutta algorithm with the shooting technique. The effect of varying parameters governing the problem is studied.
Nomenclature A, C’s, m, m1 ; T0 ; K 0 s; a0 s; b 0 s; c a1 ; a2 Pr Re D J x y t u v T U N F G qw
Real constants Essential parameters Prandtl number Reynolds number Diameter of the cylinder Microinertia per unit mass Distance along the surface Distance normal to the surface Time Velocity in x-direction Velocity in y-direction Temperature Potential velocity Angular velocity Dimensionless velocity Dimensionless microrotation Surface heat flux
66
F. S. Ibrahim and M. A. A. Hamad
Greek symbols g l q h / X m w r; d; D
Dimensionless coordinate Viscosity coefficient Density of the fluid Dimensionless temperature Angle Buoyancy parameter Kinematic viscosity Stream function Dimensionless material parameters
Subscripts w 1
Surface conditions Conditions far away from the surface
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. One of these models is the theory of micropolar fluids introduced by Eringen [1]–[3]. In this theory, the micropolar fluid exhibits the microrotational effects and micro-inertia. The difficulty of the study of such fluid problem is the paucity of boundary conditions, and the existence of deformable microelements as well as the time as the third independent variable. Many attempts were made to find analytical and numerical solutions, applying certain special conditions and using different mathematical approaches. Wilson [4] used the Karman-Polhausen approximate integral method to study the micropolar boundary-layer flow near a stagnation point. Peddieson and McNitt [5] used a finite difference scheme to solve the boundary-layer flow at a stagnation point under steady-state conditions. Ahmadi [6] provided a self-similar solution for the equations of micropolar boundary-layer flow over a semi-infinite plate, and he found the self-similar solutions when the micro-inertia is not constant. Takhar, Chamkha and Nath [7] used an implicit finite difference scheme to solve the unsteady laminar boundary layer. Gorla [8] analyzed the thermal boundary-layer fluid in the vicinity of a two-dimensional stagnation point. Hassanien and Gorla [9] presented the behavior of micropolar fluid flow near a stagnation point on a horizontal cylinder. The mathematical analysis used in the present analysis is the two-parameter group transformation, which leads to a similarity representation of the problem. A systematic formalism is presented for reducing the number of independent variables in systems which consist, in general, of a set of partial differential equations and auxiliary conditions (such as boundary and / or initial conditions). In engineering, such procedures are customarily termed similarity analysis. The formalism is a significant simplification of general group theory techniques developed by Moran and Gaggioli [10], based upon elementary group theory and earlier methods due to Birkhoff [11]. Moran and Gaggioli [12], [13] presented a general systematic group formalism for similarity analysis. Similarity analysis has been applied intensively by
Mixed convection boundary-layer flow
67
Gabbert [14]. Abd-el-Malek and Badran [15] studied fluid flow and heat transfer characteristics for steady laminar free convection on a vertical circular cylinder via group method analysis. The transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid studied by Abd-el-Malek et al. [16]. A group theoretic approach for solving the problem of diffusion of a drag through a thin membrane was discussed by Abd-el-Malek et al. [17]. Helal and Abd-el-Malek [18] applied the group theoretic method for solving the problem of the flow of an elastico-viscous liquid past an infinite flat plate in the presence of a magnetic field normal to the plate. Hassanien and Hamad [19] analyzed the problem of unsteady natural convection boundary-layer flow of a micropolar fluid along a vertical flat plate in a thermally stratified medium by using group method analysis. Hassanien et al. [20] presented group theoretic analyses of natural convection flow over an isothermal vertical wall immersed in a thermally stratified medium. The new systematic formalism introduced here is well studied for the similarity analysis of mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder. This new systematic formalism reduces the number of independent variables in systems which consist, in general, of a set of partial differential equations and auxiliary conditions (such as boundary and/or initial conditions) by two. These reduce the system of partial differential equations to a system of ordinary differential equations. The resultant system of ordinary differential equations with the appropriate boundary conditions is then solved numerically using a fourth-order Runge-Kutta scheme. The general analysis is developed in this study for the case of a surface temperature that varies exponentially with time and is uniform, 2 i.e., in the form Tw ¼ T1 þ T0 K4 em1 A t ; as well as the temperature variation with position as the power law variation, i.e., Tw ¼ T1 þ T0 K2 ðAx þ K1 Þm : Numerical results are obtained for the boundary layer characteristics. The effects of material parameter, buoyancy parameter and Prandtl number Pr are presented.
2 Mathematical analysis Consider an unsteady, laminar, incompressible, boundary layer flow of a micropolar fluid in the vicinity of the front stagnation point of a non-isothermal circular cylinder (see [9]). The free stream velocity, the wall temperature and the free stream temperature are taken as Uðx; tÞ, Tw ðx; tÞ and T1 , see Fig. 1. With the application of Boussinesq and boundary-layer approximations, the governing equations may be written in dimensionless form [9] as @u @v þ ¼ 0; @x @y
ð1Þ
@u @u @u @U @U @2u @N þu þv ¼ þU þ ð1 þ DÞ 2 þ XðT T1 Þ sin / þ D ; @t @x @y @t @x @y @y
ð2Þ
@T @T @T 1 @ 2T þu þv ¼ @t @x @y Pr @ y2
ð3Þ
and @N @N @N @u @ 2N þu þv ¼ r þ 2N þ k ; @t @x @y @y @ y2
ð4Þ
68
F. S. Ibrahim and M. A. A. Hamad
u•(x) q
x
D
Forward stagnation point
Wake
Separation point Boundary layer
Fig. 1. The physical model
where D ¼ K=l; r ¼
KD c Gr gbD3 ðTW0 T1 Þ ; k ¼ ; X ¼ 2 ; Gr ¼ ; qU1 j lj m2 Re
Re ¼
U1 D : m
ð5Þ
The specification of the initial and the boundary conditions is necessary to complete the statement of the problem. It is as follows: For t < 0, the flow is assumed to remain at rest. At t < 0, the flow starts to move impulsively with velocity Uðx; tÞ. In addition, the temperature of the cylinder is suddenly increased from that of the surrounding fluid at t = 0. Formally, these conditions may be stated as ) t < 0 : uðx; y; tÞ ¼ 0; Tðx; y; tÞ ¼ T1 every where, t ¼ 0 : u ¼ Uðx; tÞ; T ¼ T1 ð6Þ at y ¼ 0 t > 0 : u ¼ v ¼ 0; T ¼ Tw ðx; tÞ; N ¼ a @u @y u ¼ Uðx; tÞ; T ¼ T1 ; N ¼ 0
as y ! 1;
where u and v are the velocity components in the x- and y-directions, respectively. N is the angular velocity and T is the fluid temperature. From the continuity Eq. (1), there exists a nondimensional stream function wðx; y; tÞ such that u ¼ @w=@y and v ¼ @w=@x; which satisfies Eq. (1) identically. If we introduce the nondimensional temperature defined by: h¼
T T1 T T1 ¼ ; TW T1 T1
ð7Þ
Eqs. (2), (3) and (4) become @ 2 w @w @ 2 w @w @ 2 w @U @U @3w @N þ þU þ ð1 þ DÞ 3 þ X T1 h sin / þ D ; ¼ 2 @y@t @y @y@x @x @y @t @x @y @y T1
@h @T1 þh @t @t
þ
@w @h @T1 @w @h 1 @2h T1 þh ¼ T1 2 ; T1 @y @x @x @y Pr @y @x
@N @w @N @w @N þ ¼ r @ t @y @x @x @y
@2w @2N 2N þ 2 þ k ; @y @y2
ð8Þ
ð9Þ
ð10Þ
69
Mixed convection boundary-layer flow
with the boundary conditions t<0
@w ¼ 0; @y
9 > hðx; y; tÞ ¼ 0 > > =
@w t¼0 ¼ Uðx; tÞ h ¼ 0 @y
> > > ;
everywhere;
@w @w @2w t>0 ¼ ¼ 0 h ¼ 1; N ¼ a 2 @y @x @y @w ! Uðx; tÞ h ! 0 N ¼ 0 @y
ð11Þ at y ¼ 0; as
y ! 1:
3 The group of transformations In this section, a two-parameter transformation group is applied to the system of Eqs. (8) to (10) with the boundary conditions (11). The system of equations reduces to a system of ordinary differential equations in a single independent variable with the appropriate boundary conditions. The procedure is initiated with the group G, a class of two-parameter group of the form G : s ¼ CS ða1 ; a2 Þ s þ K S ða1 ; a2 Þ ;
ð12Þ 0
0
where s stands for x; y; w; N; U; T1 ; h and sin /, the C s and K s are real-valued and at least differentiable in their arguments ða1 ; a2 Þ, the parameters of the group. The class CG possesses complete sets of absolute invariants gðx; y; tÞ and gi ðx; y; t; w; N; h; T1 ; U; sin /Þ; i ¼ ð1; :::; 6Þ; where gi are the six absolute invariants corresponding to w; N; h; T1 ; U and sin /. If g ¼ gðx; y; tÞ is the absolute invariant of the independent variables, then gi ðx; y; t; w; N; h; T1 ; U; sin /Þ ¼ Fi ðgðx; y; tÞÞ
i ¼ ð1; :::; 6Þ:
4 The invariance analysis To transform the differential equations, transformations for the derivatives are obtained directly from G via chain rule operations: For example, ¼ ½CS =CX SX ; S X ¼ ½CS =CX CY SXY ; S XY
ð13Þ
¼ ½CS =CX CY CZ SXYZ : S XY Z Equation (8) is said to be invariantly transformed under (12) and (13) whenever t U U x ð1 þ DÞwyyy XhT 1 sin / DN y wyt þ wy wyx wx wyy U
ð14Þ
¼ H1 ða1 ; a2 Þ½wyt þ wy wyx wx wyy Ut UUx þ ð1 þ DÞwyyy XhT1 sin / DNy ; for some function H1 ða1 ; a2 Þ which may be a constant. Here the transformations in (13) are for the dependent and independent variables, and not for the derivatives. To transform the partial
70
F. S. Ibrahim and M. A. A. Hamad
differential equations, substitution from Eqs. (12) and (13) into the left-hand side of (14) and rearrangement yields ½Cw =C y Ct wyt þ ½ðCw Þ2 =ðC y Þ2 C x ½wy wyx wx wyy ½CU =Ct Ut ½ðCU Þ2 =C x UUx ð1 þ DÞ½Cw =ðC y Þ3 wyyy X½Ch CT1 Csin / hT1 sin / D½CN =C y Ny R1
ð15Þ
¼ H1 ða1 ; a2 Þ½wyt þ wy wyx wx wyy Ut UUx þ ð1 þ DÞwyyy XhT1 sin / DN; where R1 ¼ ½CU K U =C x Ux þ Xf½Ch CT1 K sin / hT1 þ ½Ch K T1 Csin / h sin / þ ½Ch K T1 K sin / h þ ½CT1 K h Csin / T1 sin / þ ½CT1 K h K sin / T1 þ ½K T1 K h Csin / sin /g:
ð16Þ
Thus, it follows that (16) is transformed invariantly whenever ½Cw =C y Ct ¼ ½ðCw Þ2 =ðC y Þ2 C x ¼ ½CU =Ct ¼ ½ðCU Þ2 =C x ¼ ½Cw =ðC y Þ3 ¼ ½Ch CT1 Csin / ¼ ½CN =C y ¼ H1 ða1 ; a2 Þ
ð17Þ
and R1 ¼ 0 :
K U ¼ K T1 ¼ K sin / ¼ K h ¼ 0:
ð18Þ
In a similar manner the invariance conditions of invariant transformation of (9) and (10) under (12) and (13) give the following: R2 ¼ 2r K N ;
ð19Þ
½CN =Ct ¼ ½Cw CN =C y C x ¼ ½ CN ¼ ½ Cw = ð C y Þ2 ¼ ½CN = ðC y Þ2 ¼ H2 ða1 ; a2 Þ;
ð20Þ
R2 ¼ 0 : K N ¼ 0;
ð21Þ
R3 ¼ ½K T1 Ch =Ct ht þ ½K h CT1 =Ct ðT1 Þt þ ½K T1 Ch Cw =C y C x wy hx þ ½K h CT1 Cw =C y C x wy ðT1 Þx ½K T1 Ch Cw =C y C x wx hx
1 T1 h ½K C =ðC y Þ2 hyy ; Pr
½CT1 Ch =Ct ¼ ½Cw CT1 Ch =C y C x ¼ ½CT1 Ch =ðC y Þ2 ¼ H3 ða1 ; a2 Þ;
ð22Þ ð23Þ
and R3 =0, which is satisfied in accordance with (18). Moreover, following Birkhoff [11], the boundary condition (11) is also invariant in form whenever the condition K y ¼ 0 is appended to (17), (18), (20), (21) and (23); that is ¼0: Y
wY ¼ wx ¼ 0;
! 1 : wY ¼ U; Y
¼ awyy ; N
h ¼ 1;
¼ 0; N
h ¼ 0:
ð24Þ
X; 0; tÞ ¼ 1, It is obvious that when K y ¼ 0, the transformation of h ðx; 0 ; tÞ ¼ 1 implies that hð which is only satisfied if Ch ¼ 1:
ð25Þ
Combining Eqs. (17), (20) and (23) and invoking the result (25), we get: C y ¼ 1; Ct ¼ 1 and
CU ¼ CN ¼ Cw ¼ C x ¼ CT1 Csin / :
ð26Þ
Thus, the foregoing restrictions indicate that groups which are of further interest are those in the class CG0 , with the form:
Mixed convection boundary-layer flow
0
G :
8 8 ¼ C x ða1 ; a2 Þ x þ K x ða1 ; a2 Þ x > > > > > > > > > < > 0 > > ¼y > S : y > > > > > > > > > > : > > t ¼ t þ K t ða1 ; a2 Þ > > > > > > > > > > ¼ C x ða1 ; a2 Þ w ¼ C x ða1 ; a2 Þ N > N > > > > > > > h¼h > > > > > > 1 ¼ CT1 ða1 ; a2 Þ T1 > T > > > > > > ¼ CxU > U > > > > : sin / ¼ Csin / ða1 ; a2 Þ sin /:
71
ð27Þ
Thus, as may be directly verified, any two-parameter group with the form (27) transforms (1) to (5) invariantly in the sense described.
5 Complete sets of absolute invariant The key feature of the systematic technique to be presented is the application of a basic theorem from group theory, see [10]. To emphasize the essential features of the theorem in a relatively uncomplicated form, it is now quoted for the case of two-parameter groups S, (27). At first, a function g(x, y, t) is an absolute invariant of a two-parameter group ¼ C y ða1 ; a2 Þ y þ K y ða1 ; a2 Þ; S : f x ¼ C x ða1 ; a2 Þx þ K x ða1 ; a2 Þ; y t ¼ Ct ða1 ; a2 Þ t þ K t ða1 ; a2 Þg;
ð28Þ
if and only if g satisfies the first-order linear partial differential equations ð a1 x þ a2 Þ
@g @g @g þ ð a3 y þ a4 Þ þ ð a5 t þ a6 Þ ¼ 0; @x @y @t
ð29:1Þ
ð b1 x þ b2 Þ
@g @g @g þ ð b3 y þ b4 Þ þ ð b5 t þ b6 Þ ¼ 0; @x @y @t
ð29:2Þ
where @ Cx a1 ða01 ; a02 Þ; @ a1 x @K a2 ða01 ; a02 Þ; @a1
@ Cx ða01 ; a02 Þ; @ a2 x @K b2 ða01 ; a02 Þ; ::: etc: @ a2
b1
ð30Þ
¼ x, y ¼ y and and wherein ð a01 ; a02 Þ denote the values of a1 and a2 which yield the identity: x t ¼ t, see [21]. 0 By definition, for each of the two-parameter groups S in the class CG0 there is one and only one functionally independent solution to (29) (the rank of the coefficient matrix for f @@ xg ; @@ yg ; @@ gt g is two, the matrix has rank two whenever at least one of its two-by-two sub matrices has a non-vanishing determinant). Furthermore, if g ðx; y; tÞ 6¼ const: is a solution to (29), for a group S, then every other solution to (29), for S, is given in the form Hðgðx; y; tÞÞ
72
F. S. Ibrahim and M. A. A. Hamad
where H is a differentiable function. It may be seen from (29) and the definitions of the constants ai ; bi that differences between the group S are reflected by the a0 s and b0 s: That is, in general, any particular group S possesses a characteristic set of a0 s and b0 s; and consequently a characteristic absolute invariant g is yielded by (29). Since K y ¼ 0, then a 4 ¼ b 4 ¼ 0:
ð31Þ
6 Derivation of distinct complete sets The similarity analysis of (8) to (11) now proceeds for the particular case of two-parameter groups of the form (27). According to the basic theorem from group theory, Eqs. (29) with Eq. (31) have one and only one solution, if at least one of the following conditions is satisfied: k 31 x þ k 32 6¼ 0;
k 35 t þ k 36 6¼ 0;
k 15 x t þ k16 x þ k 25 t þ k 26 6¼ 0;
ð32Þ
where k ij ai bj aj bi ; ði; j ¼ 1; 2; 3; 5; 6Þ: From Eqs. (31) and (32) and by using the transformations (27) and the definitions of the a’s, b’s and k’s, we have the result k31 ¼ k32 ¼ k35 ¼ k36 ¼ k15 ¼ k25 ¼ 0;
ð33Þ
which implies k31 x þ k32 ¼ 0;
k35 t þ k36 ¼ 0:
ð34Þ
Then the conditions (32) reduce to k16 x þ k26 6¼ 0:
ð35Þ
Applying Eqs. (34) and (35) to Eqs. (29) gives: (i) Equation (29.1) is identically satisfied. (ii) Equation (29.2) reduces to @g ¼ 0: @t
ð36Þ
For convenience, Eqs. (29) can be rewritten in the form ðk16 x þ k26 Þ
@g ¼ 0; @x
ð37Þ
and from (35), Eq. (37) gives @g ¼ 0: @x
ð38Þ
From Eqs. (36) and (38) we have g ¼ f ðyÞ:
ð39:1Þ
Without loss of generality for the independent absolute invariant gðyÞ in Eq. (39.1) we may assume the form: g ¼ Ay:
ð39:2Þ
For the absolute invariants corresponding to the dependent variables w; N; T1 ; U; sin / and h, we proceed as follows: from (25), h is itself an absolute invariant, and the functions wðx; y; tÞ; T1 ðx; tÞ; Nðx; y; tÞ; Uðx; tÞand sin / are expressed in terms of the absolute invariants FðgÞ; EðgÞ; GðgÞ; HðgÞ and IðgÞ in the formulae
73
Mixed convection boundary-layer flow
hðx; y; tÞ ¼ hðgÞ;
ð40Þ
wðx; y; tÞ ¼ C1 ðx; tÞFðgÞ;
ð41Þ
T1 ðx; tÞ ¼ C2 ðx; tÞEðgÞ;
ð42Þ
Nðx; y; tÞ ¼ C3 ðx; tÞGðgÞ;
ð43Þ
Uðx; tÞ ¼ C4 ðx; tÞHðgÞ;
ð44Þ
sin / ¼ C5 ðx; tÞIðgÞ:
ð45Þ
Since C2 ðx; tÞ and T1 ðx; tÞare independent of y, whereas g depends on y, it follows that E in (42) must be equal to a constant. Similarly H and I must be equal to constants. Then (42), (44) and (45) become T1 ðx; tÞ ¼ T0 C2 ðx; tÞ;
ð46Þ
Uðx; tÞ ¼ U0 C4 ðx; tÞ;
ð47Þ
sin / ¼ K0 C5 ðx; tÞ:
ð48Þ
The forms of the functions C1 ; C2 ; C3 ; C4 and C5 are those for which the governing Eqs. (8) to (10) reduce to ordinary differential equations.
7 The reduction to ordinary differential equations Making use of Eqs. (40), (41), (43), (46), (47) and (48) and invoking the relation (39.2), substitution for the functions and their partial derivatives into Eqs. (8) to (10), we have the following: C1 C2 0 1 C5 C6 00 02 0 ðFF F Þ 2 F þ 3 ðC3 þ C4 Þ þ 3 Xh þ 2 D G ¼ 0; A A A A A 1 00 C1 0 C9 0 C10 h þ Fh F þ 2 h ¼ 0; Pr A A A 000
ð1 þ DÞF þ
00
kG þ
C1 C7 0 C8 2 1 00 0 FG F G 2 G r 2 G þ F ¼ 0; A C6 A A A
ð49Þ ð50Þ
ð51Þ
with the boundary conditions g¼0 : g!1 :
0
00
G ¼ b F ;
F ¼ F ¼ 0; 0
F ! c ;
G ! 0;
h ¼ 1;
ð52Þ
h!0;
where the primes refer to differentiation with respect to g, and b ¼
A2 a ; C6
c ¼
U0 C4 ; A C1
ð53Þ U02 C4 @C4 C1 @x
1 C1 ¼ @C @x ;
1 C2 ¼ C11 @C @t ;
4 C3 ¼ UC10 @C @t ;
C4 ¼
C6 ¼ CC31 ;
3 C7 ¼ CC13 @C @x ;
3 C8 ¼ C13 @C @t ;
2 C9 ¼ CC12 @C @x ;
;
C5 ¼ T0 KC0 C1 2 C5 ; 2 C10 ¼ C12 @C @t ;
ð54Þ
where the C’s are constants to be determined for each individual case corresponding to each set of absolute invariants, and g is an arbitrary constant.
74
F. S. Ibrahim and M. A. A. Hamad
It remains to utilize g in turn with (54) to evaluate the C’s appearing in the ordinary differential Eqs. (49) to (51) and consequently to evaluate the corresponding expressions of the functions C1 ; C2 ; C3 ; C4 ; and C5 . We will consider the following three cases: Case (a): C1 ¼ C1 ðXÞ and C2 ¼ C2 ðXÞ In this case, we get from (54) the following: C1 ¼ C1 x þ K1 ;
C2 ¼ 0;
C2 ¼ K2 ðC1 x þ K1 Þm ;
C9 ¼ m C1 ;
C10 ¼ 0;
ð55Þ
where K1 and K2 are integral constants. By putting c = 1 in Eq. (53), we get U0 C4 ¼ AC1 ¼ AðC1 x þ K1 Þ;
C4 ¼ A2 C1 :
C3 ¼ 0;
ð56Þ
The constant C5 in (54) may be taken to be unity. This is achieved without restricting the expression of K0C5 as K0 C5 ¼
1 ðC1 x þ K1 Þ1m : T0 K2
ð57Þ
If we consider b ¼ a in (53), we get C6 ¼ A2 ;
C3 ¼ A2 C1 ¼ A2 ðC1 x þ K1 Þ;
C7 ¼ C1 ;
C8 ¼ 0:
ð58Þ
By substituting the above-obtained values of the constants into Eqs. (49) to (51), we get: 000
ð1 þ DÞF þ
C1 C1 X 00 02 0 ðFF F Þ þ þ h þ DG ¼ 0; A A A3
1 00 C1 0 C1 0 h þ Fh m F h ¼ 0; Pr A A 00
kG þ
ð59Þ ð60Þ
C1 C1 0 r 0 00 FG F G 2 ð2G þ F Þ ¼ 0; A A A
ð61Þ
with the boundary conditions 0
00
G ¼ b F ;
F ¼ F ¼ 0;
g¼0 : g!1 :
h ¼ 1; ð62Þ
0
F ! 1;
G ! 0;
h!0:
If we consider C1 = A, then the ordinary differential Eqs. (59), (60) and (61) become: 000
00
02
ð1 þ DÞF þ FF F þ 1 þ
X 0 h þ DG ¼ 0; A3
0 1 00 0 h þ Fh mF h ¼ 0; Pr 00
0
0
kG þ FG F G
r 00 ð2G þ F Þ ¼ 0; A2
ð63Þ ð64Þ ð65Þ
with the boundary conditions given by (62), and the forms of w; Tw ; N; U and sin / as follows: w ¼ ðAx þ K1 ÞFðgÞ; Tw ¼ T1 þ T0 K2 ðAx þ K1 Þm ; N ¼ A2 ðAx þ K1 ÞGðgÞ; U ¼ AðAx þ K1 Þ; sin / ¼ T01K2 ðAx þ K1 Þ1m :
ð66Þ
75
Mixed convection boundary-layer flow
For the above case, the boundary-layer characteristics are the velocity components u and v and the surface heat flux q, which are given by: 0
u ¼ AðA x þ K1 Þ F ðgÞ; ð67Þ
v ¼ A FðgÞ; m
0
q ¼ T0 K2 ðA x þ K1 Þ ½h ð0Þ: On the other hand if we consider b ¼ 2a in (53), the two ordinary differential Eqs. (63) and (65) will take the following forms: 000
00
02
ð1 þ DÞF þ FF F þ 1 þ 00
0
0
kG þ FG F G
X D 0 h þ G ¼ 0; A3 2
ð68Þ
2r 00 ðG þ F Þ ¼ 0; A2
ð69Þ
with the boundary conditions given by (62), and the forms of w; Tw ; N; U and sin / are as follows: w ¼ AxFðgÞ; Tw ¼ T1 þ ðAxÞm ; N ¼ A3 x GðgÞ;
ð70Þ
U ¼ A2 x; sin / ¼ ðAxÞ1m : As a special case, for the constant wall temperature T w, i.e., m=0, the ordinary differential Eqs. (64), (68) and (69) become 000
00
02
ð1 þ DÞF þ FF F þ 1 þ
X D 0 h þ G ¼ 0; 3 A 2
ð71Þ
0 1 00 h þ Fh ¼ 0; Pr 00
0
0
kG þ FG F G
ð72Þ 2r 00 ðG þ F Þ ¼ 0; A2
ð73Þ
with the boundary conditions given by (62), and sin / will take the form: sin / ¼ AX:
ð74Þ
Case (b): C1 ¼ C1 ðtÞ and C2 ¼ C2 ðtÞ In this case, we get: C1 ¼ K3 eC2 t ;
C1 ¼ 0;
C2 ¼ K4 eC10 t ;
C9 ¼ 0;
ð75Þ
where K3 and K4 are the integral constants. By putting c = 1 in Eq. (53), we get: U0 C4 ¼ AC1 ¼ AK3 eC2 t ;
C3 ¼ AC2 ;
C4 ¼ 0:
ð76Þ
The constant C5 in (54) may be taken to be unity. This is achieved without restricting the expression of K0 C5 as K0 C5 ¼
K3 ðC2 C10 Þ t e : T0 K4
ð77Þ
If we consider b ¼ a in (53), we get: C6 ¼ A2 ;
C3 ¼ A2 C1 ¼ A2 K3 eC2 t ;
C7 ¼ 0;
C8 ¼ C2 :
ð78Þ
76
F. S. Ibrahim and M. A. A. Hamad
By substituting the above-obtained values of the constants into Eqs. (49) to (51), we get: 000
ð1 þ DÞF
C2 0 C2 X 0 F þ 2 þ 3 h þ DG ¼ 0; A A2 A
1 00 C10 h 2 h ¼ 0; Pr A 00
kG
ð79Þ ð80Þ
C2 r 00 G 2 ð2G þ F Þ ¼ 0: 2 A A
ð81Þ
If we consider the constants C10 ¼ m1 C2 ¼ m1 A2 , then the ordinary differential Eqs. (79) to (81) become 000
0
ð1 þ DÞF F þ 1 þ
X 0 h þ DG ¼ 0; A3
ð82Þ
1 00 h m1 h ¼ 0; Pr 00
0
ð83Þ
0
kG þ FG F G
r 00 ð2G þ F Þ ¼ 0; A2
ð84Þ
with the boundary conditions given by (62) and 2
2
w ¼ K3 eA t FðgÞ;
Tw ¼ T1 þ T0 K4 em1 A t ;
2
2
N ¼ A2 K3 eA t GðgÞ; sin / ¼
K3 T0 K4
U ¼ AK3 eA t ;
ð85Þ
2
eð1m1 ÞA t :
For the above case, u; v and qw will take the following forms: 2
0
u ¼ AK3 eA t F ðgÞ; ð86Þ
v ¼ 0; 0
2
qw ¼ AT0 K4 em1 A t ½h ð0Þ: For the constant wall temperature (C10 ¼ 0) Eq. (80) becomes: 00
h ¼ 0:
ð87Þ
On the other hand if we consider b ¼ 2a, Eqs. (82) and (84) become: 000
0
0
0
ð1 þ DÞF F þ 1 þ 00
kG þ FG F G
X D 0 h þ G ¼ 0; A3 2
2r 00 ðG þ F Þ ¼ 0; A2
ð88Þ ð89Þ
with the boundary conditions given by (62). Case (C) C1 ¼ C1 ðXÞ and C2 ¼ C2 ðtÞ For this case, similarly by considering b ¼ a; c ¼ 1 C5 ¼ 1, C1 ¼ A and C10 ¼ A, we get the forms of C1 ; C2 ; C3 ; and C4 as in (55), (75), (58) and (56), respectively, and the ordinary differential equations will take the following forms: 000
00
02
ð1 þ DÞF þ FF F þ 1 þ 0 1 00 h þ Fh h ¼ 0; Pr
X 0 h þ DG ¼ 0; A3
ð90Þ ð91Þ
77
Mixed convection boundary-layer flow
1 0.9 0.8 F' 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
D =1.5
0
1
2
h
D =0.5
D =0.0
3
4
5
Fig. 2. Velocity distribution for Pr=10, k=1.5, r=0.5, X=1 and b=0
00
0
0
kG þ FG F G
r 00 ð2G þ F Þ ¼ 0; A2
ð92Þ
with the boundary conditions given by (62), sin / ¼
1 ðAx þ K1 ÞeAt : T0 K4
ð93Þ
8 Results and discussion We have presented a comprehensive account of the form similarity reduction that can arise from the two-dimensional unsteady mixed convection boundary-layer flow. Three cases are derived. Case (a): It was found that the similarity representations and similarity transformations described by Eqs. (63)–(65) and (64), (68), (69) in the steady case, and the wall temperature distribution taked the form Tw ¼ T1 þ T0 K2 ðAx þ K1 Þm : For m=0, the wall temperature is constant and the similarity solutions are given by Eqs. (71)–(73), when the main stream U ¼ A2 x and sin / ¼ Ax. Equation (71) is immediately recognizable as the Hiemenz flow [22], which is a special case of wedge-type flow for the case when X=0, and it gives Falkner-Skan flow when U ¼ 0 (see [22]). Equations (71)–(73) with boundary conditions (62) were solved numerically using a fourthorder Runge-Kutta method. Systematic ‘‘shooting’’ is required to satisfy their boundary conditions at infinity. For the purpose of numerical integration we have assumed A ¼ 2, b ¼ 0; k ¼ 0:5 and r = 0.5, see Hassanien and Gorla [9]. The buoyancy parameter X was varied from 0.1 to 10 and the Prandtl number Pr varied from 0.733 to 10. The double precision arithmetic was used in all the computations and the step size Dg=0.01 was selected. In order to assess the accuracy of the present results, the solutions have been compared with those of the corresponding results in Newtonian and non–Newtonian fluids. Our results from Eqs. (71) and 0 00 (72) are F ð0Þ ¼ 1:23259 and h ð0Þ ¼ 0.505 for D ¼ 0; b ¼ 0; X ¼ 0 and Pr ¼ 0:733, but for 0 00 D ¼0.5 we found F ð0Þ ¼ 0:99979 and h ð0Þ ¼ 0:4835: Agreement was found to be excellent with references [9] and [22], respectively. Figures 2–4 represent the distributions of velocity, microrotation and temperature within the boundary layer. Here we have chosen k ¼ 1:5 ; r ¼ 0:5; X ¼ 10 and Pr ¼ 10. From these
78
F. S. Ibrahim and M. A. A. Hamad
0.04 D = 1.5
0.035
D = 0.5
-G 0.03
D=0
0.025 0.02 0.015 0.01 0.005 0 0
1
2
3
h
4
5
Fig. 3. Angular velocity profiles for Pr ¼ 10, k=1.5, r=0.5, X=10, b=0
1 0.9 0.8 q 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
D = 0.0, 0.5,1.5
0
0.5
1
1.5
h
2
2.5
3
Fig. 4. Temperature profiles for Pr ¼ 10, k=1.5, r=0.5, X=10, b=0
1 0.9 0.8 F¢ 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
D=10, 5, 0.2
0
0.5
1
h
1.5
2
2.5
3
Fig. 5. Velocity profiles for Pr ¼ 10, k=0.5, r=0.5, D=0.5, b=0
figures we observed that, as the material parameter D increases, the boundary-layer thickness increases, the velocity distribution becomes more linear, the inflection point for the angular velocity distribution moves further away from the surface, and the temperature distribution becomes more uniform. The friction factor and Nusselt number decrease as D increases. We
79
Mixed convection boundary-layer flow
0.07 0.06 0.05 -G 0.04
W = 10,5,0.2
0.03 0.02 0.01 0 0
0.5
1
1.5
2
h
2.5
3
Fig. 6. Angular velocity profiles for Pr ¼ 10, k=0.5, r=0.5, D=0.5, b=0
1 0.9 0.8 0.7 0.6 q 0.5 0.4 0.3 0.2 0.1 0
W =10, 5, 0.2
0
0.5
1
1.5 h
2
2.5
3
Fig. 7. Temperature profiles for Pr ¼ 10, k=0.5, r=0.5, D=0.5, b=0
1.2 1 0.8 F' 0.6
W =20, 15, 10, 5, 0.5
0.4 0.2 0
0
1
2
h
3
4
5
Fig. 8. Velocity profiles for Pr ¼ 0:733, k=0.5, r=0.5, D=0.1, b=0
also observe that increasing values of D result in decreasing wall couple stress and increasing values of the magnitude of the maximum value of the angular velocity. Figures 5–8 display the effect of the buoyancy parameter X on the distributions of velocity, angular velocity and temperature within the boundary layer. It is found that, as the buoyancy force increases, the velocity distribution assumes a more uniform shape within the boundary layer. The angular velocity distribution is influenced by the buoyancy parameter. The friction factor and Nusselt number increase with the buoyancy parameter.
80
F. S. Ibrahim and M. A. A. Hamad
In many practical applications, it are usually the surface characteristics such as the local heat transfer rates that are of prime importance. The evaluation of such quantities requires only the information contained in Table 1. The present table illustrates the difference between the results of Newtonian ðD ¼ 0Þ and micropolar fluids (D 6¼ 0Þ. The results indicate that the micropolar fluid displays a reduction in the drag compared to the case of Newtonian fluids. The heat transfer rate is higher for a Newtonian fluid when compared to micropolar fluids. Further the friction factor and heat transfer rate get augmented with an increased value of the bouyancy parameter. It is found also that the flow and heat transfer are very sensitive to a change in the concentration parameter b: Case (b): In this case the similarity representations are found in the systems (82)–(84) and 2 (83), (88), (89). Here, the variable wall temperature takes the form Tw ¼ T1 þ T0 K4 eA t ; which indicates that the wall temperature varied exponentially with time t. In this case, it is interesting to note that (as observed in Eq. (86)) the flow velocity has only one component in the horizontal direction. Further from Eq. (83), the temperature distribution decays exponentially in the form pffiffiffiffiffiffiffiffiffi h ¼ e m1 Pr g . Furthermore, for constant wall temperature, i.e., Tw = const., with C2 ¼ A2 , we get the system (82), (84), (87), and the exact solution of (87) is of the form h ¼ K5 g þ K6 , where K5 and K6 are constants, which shows that the dimensionless temperature varied linearly with dimensionless coordinate g. Case (c): In this case the similarity representations are given by Eqs. (92)–(99), here we get: sin / ¼
1 ðAx þ K1 ÞeAt : T0 K4
Further, the relation between the ideal velocity distribution in the potential irrational flow past a circular cylinder UðxÞ and the free stream velocity U1 parallel to the x-axis is found identically with the expression UðxÞ ¼ 2U1 ðsin /Þ1=ð1mÞ , where U1 ¼ A=2 and T0 ¼ K2 ¼ 1. In this relation for m ¼ 0 we get UðxÞ ¼ 2U1 sin / which is of the same form as given by [22] in the Newtonian case.
9 Concluding remarks In this paper, we have presented new similarity solutions for mixed convection boundary layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder in three cases by using group method analysis. In our study, we have presented new formulae of the wall temperature Tw , the velocity far away from the cylinder UðxÞ and sin /. The exact solution of the dimensionless temperature is presented. Numerical results are presented for angular velocity and temperature profiles with the variation of material parameter and buoyancy.
References [1] [2] [3] [4]
Eringen, A.: Simple microfluids. Int. J. Eng. Sci. 2, 205–217 (1964). Eringen, A.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966). Eringen, A.: Theory of thermomicrofluids. J. Math. Anal. Appl. 9, 480–496 (1972). Willson, A. J.: Boundary layers in micropolar liquids. Proc. Cambridge Phil. Soc. 67, 46–57 (1970). [5] Peddieson, J., McNitt, R. P.: Boundary-layer theory for a micropolar fluid. Recent Adv. Eng. Sci. 5, 405–426 (1970).
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81
[6] Ahmadi, G.: Self-similar solution of incompressible micropolar boundary-layer flow over a semiinfinite plate. Int. J. Eng. Sci. 14, 639–651 (1976). [7] Takhar, H. S., Chamkha, A. J., Nath, G.: Unsteady three-dimensional MHD boundary-layer flow due to the impulsive motion of a stretching surface. Acta Mech. 146, 59–64 (2001). [8] Gorla, R. S. R.: Thermal boundary layer of a micropolar fluid at a stagnation point. Int. J. Eng. Sci. 18, 611–622 (1980). [9] Hassanien, I. A., Gorla, R. S. R.: Mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder. Int. J. Eng. Sci. 28, 153–165 (1990). [10] Moran, M. J., Gaggioli, R. A.: Reduction of the number of variables in systems of partial differential equations with auxiliary conditions. SIAM J. Appl. Math. 16, 202–215 (1968). [11] Birkhoff, G.: Hydrodynamics. Princeton: Princeton University Press 1960. [12] Gaggioli, R. A., Moran, M. J.: Group theoretic technique for the similarity solution of the system of partial differential equations with auxiliary conditions. U.S. Army Math. Research Center, Univ. of Wisconsin, Tech. Summary Report. No. 693 (1966). [13] Moran, M. J., Gaggioli, R. A.: A new systematic formalism for similarity analysis with application to boundary-layer flows. U. S. Army Math. Research Center, Tech. Summary Report No. 918 (1968). [14] Gabbert, C. H.: Similarity for unsteady compressible boundary layer. AIAA. J. 5, 1198–1200 (1967). [15] Abd-el-Malek, M. B., Badran, N. A.: Group method analysis of steady free-convective laminar boundary-layer flow on a nonisothermal vertical circular cylinder. J. Comp. Appl. Math. 36, 227–238 (1991). [16] Abd-el-Malek, M. B., Kassem, M. M., Mekky, M. L.: Similarity solutions for unsteady freeconvection flow from a continuous moving vertical surface. J. Comp. Appl. Math. 165, 11–24 (2004). [17] Abd-el-Malek, M. B., Kassem, M., Mekky, M. L.: Group theoretic approach for solving the problem of diffusion of a drug through a thin membrane. J. Comp. Appl. Math. 140, 1–11 (2002). [18] Helala, M. M., Abd-el-Malek, M. B.: Group method analysis of magneto-elastico-viscous flow along a semi-infinite flat plate with heat transfer. J. Comp. Appl. Math. 173, 199–210 (2005). [19] Hassanien, I. A., Hamad, M. A. A.: Group method analysis of unsteady free convection boundary-layer flow of a micropolar fluid along a vertical plate in a thermally stratified medium (submitted). [20] Hassanien, I. A., Ibrahim, F. S., Hamad, M. A. A.: Group theoretic analysis of natural convection flow over an isothermal vertical wall immersed in a thermally stratified medium (submitted). [21] Abd-el-Malek, M. B.: Application of the group-theoretical method to physical problems. J. Nonlinear Math. Phys. 5, 314 (1998). [22] Schlichting, H.: Boundary-layer theory. New York: Springer 2000. Authors’ address: F. S. Ibrahim and M. A. A. Hamad, Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt
