Moving wedge and flat plate in a micropolar fluid

International Journal of Engineering Science 44 (2006) 1225–1236

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Moving wedge and flat plate in a micropolar fluid

Anuar Ishak a, Roslinda Nazar a,*, Ioan Pop b

a School of Mathematical Sciences, National University of Malaysia, 43600 UKM Bangi, Selangor, Malaysiab Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

Received 20 June 2006; received in revised form 7 August 2006; accepted 13 August 2006Available online 30 October 2006

Abstract

Similarity solutions for a moving wedge and flat plate in a micropolar fluid may be obtained when the fluid and bound-ary velocities are proportional to the same power-law of the downstream coordinate. The governing partial differentialequations are transformed to the ordinary differential equations using similarity variables, and then solve numericallyusing a finite-difference scheme known as the Keller-box method. Numerical results are given for the dimensionless velocityand microrotation profiles, as well as the skin friction coefficient for several values of the Falkner–Skan power-law param-eter (m), the ratio of the boundary velocity to the free stream velocity parameter (k) and the material parameter (K). Impor-tant features of these flow characteristics are plotted and discussed. It is found that multiple solutions exist when theboundary is moving in the opposite direction to the free stream, and the micropolar fluids display a drag reduction com-pared to Newtonian fluids.� 2006 Elsevier Ltd. All rights reserved.

PACS: 47.15.Cb

Keywords: Forced convection; Micropolar fluid; Moving wedge; Multiple solutions; Similarity solutions

1. Introduction

The problem of laminar boundary-layer flows resulting from the flow of an incompressible fluid past asemi-infinite wedge is of considerable practical and theoretical interest. These types of boundary-layer prob-lems are expressed in the form of nonlinear third-order partial differential equations which cannot be solveddirectly in closed form. Accordingly, it is necessary to develop numerical methods capable of providing accu-rate solutions for problems of this type. The pioneering work by Falkner–Skan [1] used the similarity trans-formation method to reduce the partial differential equation associated with the steady incompressible laminarboundary-layer flow past a wedge to a third-order ordinary differential equation which is solved numerically.Hartree [2] carried out a detailed study of the same problem and the problem has been further studied by many

0020-7225/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

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* Corresponding author. Tel.: +603 89213371; fax: +603 89254519.E-mail address: [email protected] (R. Nazar).

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1226 A. Ishak et al. / International Journal of Engineering Science 44 (2006) 1225–1236

researchers [3–33]. Also there has been much work performed on the unsteady boundary-layer flow past awedge which is started impulsively from rest (see [34–38]).

Research interest in the flows of micropolar fluids has increased substantially over the past decades due tothe occurrence of these fluids in industrial processes. In the history of fluid mechanics, Eringen [39,40] is a pio-neering researcher who has formulated the theory of micropolar fluids. This theory displays the microscopiceffects arising from the local structure and micro-motions of the fluid elements. Eringen’s micropolar modelincludes the classical Navier–Stokes equations as a special case, but can cover, both in theory and applica-tions, many more phenomena than the classical model. Physically, the mathematical model underlying micro-polar fluids may represent the behaviour of polymeric additives, blood, lubricants, liquid crystals, dirty oilsand solutions of colloidal suspensions. In practice, the theory requires that one must solve an additional trans-port equation representing the principle of conservation of local angular momentum, as well as the usualtransport equations for the conservation of mass and momentum, and additional constitutive parametersare also introduced. Extensive reviews of the theory and applications can be found in the review articles byAriman et al. [41,42] and the recent books by Łukaszewicz [43] and Eringen [44]. Guram and Smith [45], Gorla[46], and Kumari and Nath [47] were the first to apply the micropolar boundary-layer theory to problems ofsteady and unsteady stagnation point flows and they claimed that the micropolar fluid model is capable ofpredicting results which exhibit turbulent flow characteristics, although it is difficult to see how a steady lam-inar boundary-layer flow could appear to be turbulent. Rees and Bassom [48] have studied quite recently themicropolar analogue of the Blasius boundary-layer flow. They derived nonsimilar boundary-layer equationsand solved them using the Keller-box method. They also performed an asymptotic analysis which is valid atlarge distances from the leading edge because the numerical results indicate that the boundary-layer develops atwo-layer structure. More recently, Kim [49] and Kim and Kim [50] have considered the steady boundary-layer flow of a micropolar fluid past a fixed wedge with both a constant surface temperature and constantsurface heat flux. The similarity variables found by Falkner–Skan [1] were employed to reduce the governingpartial differential equations to ordinary differential equations. Unfortunately the angular momentum equa-tion is not correctly derived so that the results of these papers are inaccurate.

In this paper, we analyse the steady forced convection boundary-layer flow over a wedge and past a flatplate moving in a micropolar fluid. In particular, we investigate the behaviour of the micropolar fluid inthe boundary-layer and also the skin friction at the walls of the wedge and at the flat plate for various valuesof the micropolar parameter. The objective of this paper is also to show that a similarity solution of the gov-erning equations can be obtained and this represents the classical Blasius and Falkner–Skan problems. From amathematical point of view, this problem is of interest because it represents one of the relatively few instancesin fluid mechanics of micropolar fluids where a similarity solution can be obtained. The existence of the exactNewtonian flow solution greatly facilitates an analysis of the results from the study of the correspondingmicropolar flow problem, in so much as one is able to highlight more easily the differences between the micro-polar and classical (Newtonian) fluids. To the best of the knowledge of the authors, this problem has not beenstudied before and therefore the results obtained are novel.

2. Basic equations

Consider the problem of forced convection boundary-layer flow over a wedge which moves with a velocityuw(x) in a micropolar fluid. The positive x-coordinate is measured along the surface of the wedge with the apexas origin, and the positive y-coordinate is measured normal to the x-axis in the outward direction towards thefluid, see Fig. 1. We assume that all fluid properties except the micro-inertia density and the spin-gradient vis-cosity are constants. By applying the boundary-layer approximations, the governing equations for steady forcedconvection flow over a continuously moving wedge can be written as (see Rees and Bassom [48] or Kim [49]):

ouoxþ ov

oy¼ 0; ð1Þ

uouoxþ v

ouoy¼ ue

due

dxþ lþ j

q

� �o2uoy2þ j

qoNoy; ð2Þ

Fig. 1. Physical model and the coordinate system.

A. Ishak et al. / International Journal of Engineering Science 44 (2006) 1225–1236 1227

qj uoNoxþ v

oNoy

� �¼ o

oycoNoy

� �� j 2N þ ou

oy

� �; ð3Þ

uojoxþ v

ojoy¼ 0; ð4Þ

where u and v are the velocity components along the x- and y-axes, respectively, N is the component of themicrorotation vector normal to the x–y plane, q is the density, j is the micro-inertia density, l is the dynamicviscosity, j is the gyro-viscosity (or vortex viscosity), c is the spin-gradient viscosity and ue(x) is the velocityoutside the boundary-layer or the velocity of the inviscid (potential) fluid. In order that Eqs. (1)–(4) reduce toa similarity form, we assume that the boundary conditions for these equations are of the following form:

v ¼ 0; u ¼ uwðxÞ ¼ �U wxm; j ¼ 0; N ¼ � 1

2

ouoy

at y ¼ 0;

u! ueðxÞ ¼ U1xm; N ! 0 as y !1;ð5Þ

where m = b/(2 � b), and b is the Hartree pressure gradient parameter which corresponds to b = X/p for atotal wedge angle X, and Uw and U1 are constants. We notice that 0 6 m 6 1, with m = 0 for the flow overa moving flat plate and m = 1 for the flow near the stagnation point on a moving wall. As it was shown byAhmadi [51], Kline [52] and Kline and Allen [53] the spin-gradient viscosity c can be defined as

cðx; yÞ ¼ ðlþ j=2Þjðx; yÞ ¼ lð1þ K=2Þjðx; yÞ; ð6Þ

where K = j/l denotes the dimensionless viscosity ratio and is called the material parameter. Eq. (6) has alsobeen very successfully used by Gorla [54], Yucel [55], and Ishak et al. [56] to study different problems of con-vective flow of micropolar fluids. It has also been stated by Ahmadi [51] that for nonconstant micro-inertia it ispossible using Eq. (6) to find similar and self-similar solutions for a large number of problems of micropolarfluids. It is also worth mentioning that the case K = 0 describes the classical Navier–Stokes equations for aviscous and incompressible fluid.

Now we introduce the following similarity variables:

wðx; yÞ ¼ 2mxueðxÞmþ 1

� �1=2

f ðgÞ; Nðx; yÞ ¼ ueðxÞðmþ 1ÞueðxÞ

2mx

� �1=2

hðgÞ;

jðx; yÞ ¼ 2mxðmþ 1ÞueðxÞ

iðgÞ; g ¼ ðmþ 1ÞueðxÞ2mx

� �1=2

y;

ð7Þ

where m = l/q is kinematic viscosity and w is the stream function defined in the usual way as u = ow/oy andv = �ow/ox so as to identically satisfy Eq. (1). Substituting (7) into Eqs. (2)–(4), we get the following ordinarydifferential equations:


ð1þ KÞf 000 þ ff 00 þ 2mmþ 1

� �ð1� f 02Þ þ Kh0 ¼ 0; ð8Þ

ð1þ K=2Þðih0Þ0 þ i fh0 � 3m� 1

mþ 1

� �f 0h

� �� Kð2hþ f 00Þ ¼ 0; ð9Þ

ð1� mÞf 0i� mþ 1

2fi0 ¼ 0; ð10Þ

subject to the boundary conditions

f ð0Þ ¼ ið0Þ ¼ 0; f 0ð0Þ ¼ �k; hð0Þ ¼ � 1

2f 00ð0Þ;

f 0ð1Þ ! 1; hð1Þ ! 0;ð11Þ

where primes denote differentiation with respect to g and k = Uw/U1 is the ratio of the wall velocity to the freestream fluid velocity, and k (>0) corresponds to the situation when the wall moves in the opposite direction tothe free stream and k(<0) when the wall moves in the same direction to the free stream, while k = 0 is for afixed flat wall. If we integrate Eq. (10), we get

i ¼ Af 2ð1�mÞ=ð1þmÞ; ð12Þ

where A is the non-dimensional constant of integration. We notice that Eqs. (8)–(10) were also derived by Kim[49] and Kim and Kim [50]. However, Eq. (9) was inaccurately derived in Kim [49] because in his paper Eq.(14) contains the extra term [m/(m + 1)]gf 0h 0.

The physical quantity of interest is the skin friction coefficient, which is defined as

Cf ¼sw

qu2w

; ð13Þ

where sw is the skin friction which is given by

sw ¼ ðlþ jÞ ouoyþ jN

� �y¼0

: ð14Þ

Using the variables (7), we get

Cf Re1=2x ¼ ð1þ K=2Þf 00ð0Þ; ð15Þ

where Rex = uw(x)x/m is the local Reynolds number.

2.1. Flat plate problem

In this case m = 0 and thus Eqs. (8) and (9) reduce to

ð1þ KÞf 000 þ ff 00 þ Kh0 ¼ 0; ð16Þð1þ K=2Þðih0Þ0 þ iðfhÞ0 � Kð2hþ f 00Þ ¼ 0; ð17Þ

subject to the boundary conditions (11). Also, (12) becomes

i ¼ Af 2: ð18Þ

The solution of Eqs. (16)–(18) subject to boundary conditions (11) for the case of a fixed plate can be found inAhmadi [51]. If K = 0 (Newtonian fluid), Eq. (16) reduces to the problem studied by Klemp and Acrivos [57].If K is not zero, but A = 0, i.e., i = 0, then from Eq. (17), we find

h ¼ � 1

2f 00; ð19Þ

i.e., the gyration is identical to the angular velocity. Then Eq. (16) becomes

ð1þ K=2Þf 000 þ ff 00 ¼ 0: ð20Þ


Following Rees and Bassom [48], we take

f ðgÞ ¼ ð1þ K=2Þ�1=2f ðgÞ; g ¼ ð1þ K=2Þ�1=2g; ð21Þ

and Eq. (20) reduces to the problem of boundary-layer flow over a semi-infinite flat plate moving in a viscous(Newtonian) fluid without mass transfer (impermeable plate) as studied by Fang [58], i.e.,

f 000 þ f f 00 ¼ 0; ð22Þ

with the boundary conditions

f ð0Þ ¼ 0; f 0ð0Þ ¼ �k; f 0ð1Þ ! 1; ð23Þ

–1.5 –1 –0.5 0 0.5 1 1.5–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

1.2

λ

f ’’ (

0)

m = 0, 0.03, 0.1, 0.3333, 1

K = 5

Fig. 2. Skin friction coefficient as a function of k for various values of m, when K = 5.

0 5 10 15 20 25 30 35 40 45 50–0.5

0

0.5

1

η

f ’(η

)

m = 0.03

K = 5

λ = 0.45

1st solution

2nd solution

3rd solution

Fig. 3. Velocity profiles for m = 0.03, k = 0.45 and K = 5.


where now primes denote differentiation with respect to g. The skin friction coefficient given by (15) becomes

Cf Re1=2x ¼ ð1þ K=2Þ�1=2f 00ð0Þ: ð24Þ

2.2. Wedge problem

Furthermore, if A = 0, i.e., i = 0, it can be easily shown, on using (21), that Eqs. (8) and (9) reduce to

f 000 þ f f 00 þ bð1� f 02Þ ¼ 0; ð25Þ

0 5 10 15 20 25 30 35 40 45 50–0.12

–0.1

–0.08

–0.06

–0.04

–0.02

0

η

h(η)

m = 0.03

K = 5

λ = 0.45

1st solution

2nd solution

3rd solution

Fig. 4. Microrotation profiles for m = 0.03, k = 0.45 and K = 5.

–1.4 –1.2 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

λ

f ’’ (

0)

K = 0, 1, 5

m = 0

Fig. 5. Skin friction coefficient as a function of k for various values of K when m = 0.


where b = 2m/(m + 1), subject to the boundary conditions (23). The solution to this problem can be found inRiley and Weidman [21].

3. Results and discussion

Eqs. (8), (9) and (12) subject to boundary conditions (11) were solved numerically using a finite-differencescheme known as the Keller-box method, as described by Cebeci and Bradshaw [59], for several values of m, kand K. Following Kim [49] and Ahmadi [51], we considered only the value of A being unity. In order to verifythe accuracy of the present method used on the simulation model, we have compared our results with thosecases reported by Ahmadi [51] and Riley and Weidman [21], and show good agreement.

0 2 4 6 8 10 12 14–0.5

0

0.5

1

η

f ’(η

)

m = 0

λ = 0.35

1st solution

2nd solution

K = 1

5 1

5

Fig. 6. Velocity profiles for various values of K when m = 0 and k = 0.35.

0 2 4 6 8 10 12 14–0.18

–0.16

–0.14

–0.12

–0.1

–0.08

–0.06

–0.04

–0.02

0

η

h(η)

m = 0 1

λ = 0.35

1st solution

2nd solution

5

1

K = 5

Fig. 7. Microrotation profiles for various values of K when m = 0 and k = 0.35.


Variations of the skin friction coefficient f 00(0) with k are shown in Fig. 2 for some values of m when K = 5.This figure shows that multiple solutions are possible for some values of m when k > 0, i.e., when the wall ismoving in the opposite direction to the free stream, while the solution is unique when k 6 0 for all m. Forexample, there are three solutions when m = 0.03 for the values of k within the range 0.41 6 k 6 0.46, and dualsolutions are obtained for m > 0.3333, when k > 1. The velocity and microrotation profiles that show the exis-tence of the three solutions when m = 0.03 and k = 0.45 are presented in Figs. 3 and 4, respectively. Thesefigures show that the boundary conditions (11) are satisfied. It is shown in Fig. 2 that for m = 1(X = 180�), there is no solution when k > 1.2467 = kc, while for m = 0 (flat plate), the solution no longer existswhen k > 0.3645 = kc. At these critical values of k (say kc), both solution branches are connected and thus a

–1.5 –1 –0.5 0 0.5 1–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

1.2

λ

f ’’ (

0)

K = 0, 5, 10

m = 0.3333

Fig. 8. Skin friction coefficient as a function of k for various values of K when m = 0.3333.

0 1 2 3 4 5 6 7–0.5

0

0.5

1

η

f ’(η

)

m = 0.3333

K = 0, 1, 5

λ = 0

Fig. 9. Velocity profiles for various values of K when m = 0.3333 and k = 0.


unique solution is obtained. Beyond these critical values, the boundary-layer separates from the surface andthus the solutions based upon the boundary-layer approximations are not valid. Further, it is seen that allcurves in Fig. 2 intersect at the point (�1,0). This is not surprising since there is no shear stress at the surfacewhen the wall and the fluid move with the same velocity. Moreover, all the solution curves for m > 0 (wedge)have (1,0) as the limit point, whereas for m = 0 (flat plate), the curve terminates at the origin. The detail resultswhen m = 0 and K = 0 can be found in Klemp and Acrivos [57], while for the case of the permeable plate, theyare given in Fang [58].

Fig. 5 shows the variation of the skin friction coefficient f 00(0) with k for the moving flat plate (m = 0) forvarious values of K. It is seen that the value of jf 00(0)j decreases with increasing K. This indicates that micro-polar fluids display drag reduction compared to Newtonian fluids (K = 0). Moreover, the value of f 00(0) ispositive when k > �1 and this means that the fluid exerts a drag force on the plate. This is not surprising sincethe fluid is moving with a higher velocity than the plate. The opposite is observed when k < �1. When k = �1,both the fluid and the plate are moving with the same velocity and this implies zero skin friction at the bound-ary. We found that there is always a solution for negative values of k, for all values of K. Therefore, noseparation occurs when the wedge and the fluid moving in the same direction. For opposite motions of thewedge and the fluid at large distances, the solution is obtained up to certain values of k. It is seen fromFig. 5 that the boundary-layer separation can be delayed by increasing the value of K. However, the effectof K on the boundary-layer separation is not very pronounced compared to the effects of m. The velocityand microrotation profiles when k = 0.35 are presented in Figs. 6 and 7, respectively, which supports theexistence of dual solutions for this case.

The variations of f 00(0) with k and K for a moving wedge are shown in Figs. 8 and 11 for m = 0.3333 andm = 1, respectively, while the respective velocity and microrotation profiles for some cases are presented inFigs. 9, 10, 12 and 13. We notice that for the same value of the material parameter K, the thickness of themicrorotation boundary-layer is slightly larger than the thickness of the velocity boundary-layer. Whenm = 0.3333 (X = 90�), we found that the solution is unique for all values of K, and the solution exists withinthe range k 6 1. Dual solutions are possible for m > 0.3333 when k > 1. Fig. 11 shows the case when m = 1(X = 180�). The solution curves terminate at the point (1,0), and dual solutions are obtained when k > 1.The effect of K on the skin friction coefficient is similar to the case of the flat plate.

Figs. 6, 9 and 12 show the resulting dimensionless velocity profiles f 0(g) for various values of the materialparameter K. It is observed that the velocity boundary-layer thickness increases with increasing values of K

0 1 2 3 4 5 6 7 8–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

η

h(η)

m = 0.3333

K = 1, 5

λ = 0

Fig. 10. Microrotation profiles for various values of K when m = 0.3333 and k = 0.


associated with a decrease in the wall velocity gradient, and hence produces a decrease in the magnitude of thereduced skin friction f 00(0). Further, for all values of m and K, the velocity becomes unity at the edge of theboundary-layer, which satisfies the boundary condition. The effect of K on the microrotation or angular veloc-ity can be seen in Figs. 7, 10 and 13. It is observed that the microrotation continuously changes with g andbecomes zero far away from the plate. As expected, the microrotation effects are most dominant near the wall.Also, the absolute value of the microrotation h(g) decreases as K increases in the vicinity of the plate but thereverse happens as one moves away from it. It is evident from these figures that both the velocity and angularvelocity profiles satisfy the boundary conditions (11), and this support the validity of the present results.

0 2 4 6 8 10 12 14 16 18 20–1.5

–1

–0.5

0

0.5

1

1.5

η

f ’(η

)

m = 1

λ = 1.2

1st solution

2nd solution

K = 5

10

5

10

Fig. 12. Velocity profiles for various values of K when m = 1 and k = 1.2.

–1.5 –1 –0.5 0 0.5 1 1.5–1

–0.5

0

0.5

1

1.5

2

λ

f ’’ (

0)

K = 0, 5, 10

m = 1

Fig. 11. Skin friction coefficient as a function of k for various values of K when m = 1.

0 2 4 6 8 10 12 14 16 18 20–0.3

–0.25

–0.2

–0.15

–0.1

–0.05

0

0.05

η

h(η)

m = 1

10 λ = 1.2

1st solution

2nd solution

K = 10

5

5

Fig. 13. Microrotation profiles for various values of K when m = 1 and k = 1.2.


4. Conclusions

We have theoretically studied the problem of steady two-dimensional laminar fluid flow past a movingwedge. The governing partial differential equations were transformed using suitable variables to get ordinarydifferential equations, and these equations were solved numerically using an implicit finite-difference schemeknown as the Keller-box method. Numerical results for the velocity and microrotation profiles, as well asthe skin friction coefficient, for various values of the Falkner–Skan power-law parameter m, the velocity ratioparameter k and the material parameter K, have been illustrated in graphical form. Comparisons with theexisting results for certain cases and a discussion of the effects of the parameters involved have been done.For the flat plate, dual solutions exist when the plate is moving in the opposite direction to the mainstream,while for the wedge, the number of solutions depends on the angle of the wedge. Moreover, micropolar fluidsdisplay drag reduction compared to classical Newtonian fluids (K = 0), and boundary-layer separation can bedelayed by increasing the values of K.

Acknowledgements

The authors wish to express their very sincere thanks to the reviewer of this paper for his valuable com-ments. They also wish to thank Prof. Derek B. Ingham for his generous help in reading the paper carefullyand for his interesting suggestions. This work is supported by a research grant (SAGA fund: STGL-013-2006) from the Academy of Sciences Malaysia.

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Moving wedge and flat plate in a micropolar fluid