On numerical and approximate solutions for stagnation point flow involving third order fluid

M. Ahmad, M. Sajid, T. Hayat, and I. Ahmad

Citation: AIP Advances 5, 067138 (2015); doi: 10.1063/1.4922878 View online: http://dx.doi.org/10.1063/14922878

View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/5/6?ver=pdfcov Published by the AIP Publishing

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On numerical and approximate solutions for stagnation point flow involving third order fluid

M. Ahmad,1,3 M. Sajid,2 T. Hayat,3,4 and I. Ahmad1

1 Department of Mathematics, University ofAzad Jammu & Kashmir, Muzaffarabad 13100, Pakistan

2Theoretical Physics Division, PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan 3Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan 4Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

(Received 4 April 2015; accepted 11 June 2015; published online 19 June 2015)

This article addresses the two-dimensional boundary layer flow of third order fluid in the region of a stagnation point over a surface lubricated with a power law fluid. The lubricant is assumed to have a thin layer of variable thickness over the surface. The third order fluid experiences a partial slip due to this lubrication layer. Mathematical model of the flow problem is represented through a system of nonlinear partial differential equations with nonlinear boundary conditions. The non-similar numerical and analytic solutions of the transformed ordinary differential equation are obtained using hybrid homotopy analysis method based on the combination of homotopy analysis and shooting methods. It is observed that extra drag force is required in order to achieve no-slip regime from full slip and thus slip has suppressed the effects of free stream velocity. The results varying from no-slip to full slip case are discussed under the influence of pertinent parameters. © 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4922878]

I. INTRODUCTION

The key importance of non-Newtonian fluids in industry and technology has received the attention of many recent researchers. The constitutive relationships of non-Newtonian fluids are nonlinear and can predict different phenomena like shear thinning, shear thickening, normal stress effects, stress relaxation and retardation, fluid memory etc. Such features cannot be described by a single constitutive relationship between shear stress and rate of deformation. A glance at literature shows that several constitutive relationships have been proposed for the non-Newtonian fluids. Amongst these models the differential type fluids of second and third grades have acquired special attention of the researchers. For details about the constitutive equations we refer the readers to the books by Bird1 and Harris.2 Amongst different non-Newtonian fluids, the second and third grade fluids become much popular.3,4 Important contributions to the topic include the works of Bandelli,5 Hayat et al.,6,7 Sajid et al.8 and references therein. One of the key problems while obtaining the numerical solutions of the two-dimensional flows of viscoelastic fluids is the vanishing of coefficient of the leading order derivative term appearing in the governing equation with the viscoelastic parameter. The order of the governing equation is reduced at the starting point of the domain and in the limit when viscoelastic parameter tends to zero. This poses challenges to mathematicians, physicists and numerical analysts to develop methods to cope up with the singularity appearing in the governing equation.

Two-dimensional flow near a stagnation point is one of the classical flow problems in fluid mechanics. After the pioneering work of Hiemenz,9 this problem was extended by Homann10 for

Corresponding author Tel.: +92 3459613554 E-mail: manzoorajku@gmail.com (M. Ahmad)

2158-3226/2015/5(6)/067138/10 5,067138-1 ©Author(s) 2015 I^B

an axisymmetric flow. Mahapta and Gupta11 investigated steady two dimensional flow of an incompressible viscous fluid towards a stretching sheet. In another article Mahapta and Gupta12 discussed the same phenomenon for a viscoelastic Walter-B fluid. Nazar et al.13 investigated the unsteady two-dimensional stagnation point flow in a viscous fluid over a flat deformable sheet. The analysis for the steady two-dimensional stagnation point flow of an incompressible micropolar fluid over a stretching surface was carried out by Nazar et al.14 Hayat et al.15 computed the series solutions for magnetohydrodynamic flow of an upper-convected Maxwell fluid near a stagnation point. The stagnation point flow for an Oldroyd-B fluid has been discussed by Sajid et al.16 Ramesh et al.17 investigated the numerical solution for the steady two-dimensional MHD flow of a dusty fluid near the stagnation point with heat source/sink. The analysis for MHD stagnation point flow and heat transfer for a nanofluid has been carried out by Ibrahim et al.18 Turkyimazoglu and Pop19 examined the flow and heat transfer of a Jeffery fluid near the stagnation point on a stretching/shrinking sheet with parallel external flow and obtained exact analytic solutions. The effects of slip on an unsteady stagnation point flow of a nanofluid over a stretching surface were reported by Malvandi et al.20

In all the above mentioned articles the stagnation point flow is considered over a rigid plate. The stagnation point flow considered in Ref. 10 has been discussed against a thin lubrication layer by Yeckel et al.21 Blyth and Pozrikidis22 provided an analysis of the stagnation point flow of a viscous fluid flowing over another viscous fluid. A slip boundary condition for the flow over a lubricated rotating disk is deduced by Andersson and Rousselet.23 Axisymmetric stagnation point flow of viscous fluid has been considered by Santra et al.24 They provided a numerical solution of the problem for a similar flow. Sajid et al.25 extended the analysis of Ref. 24 by defining a general slip boundary condition. The literature regarding the stagnation point flow of non-Newtonian fluid over a lubricated surface is scarce. Very recently Sajid et al.26 analyzed the stagnation point flow of a viscoelastic Walters B' fluid with power law lubrication. They obtained a numerical solution using the hybrid method. The analysis of non-Newtonian fluid flows over a thin lubricated layer is an open area of research. This fact motivated us to discuss the boundary layer flow of a third order fluid over a lubricated layer of variable thickness. The transformed equations are solved using a hybrid homotopy analysis method27'28 which is merger of analytic and numerical methods namely the homotopy analysis method29-36 and shooting method.37 The detail of the adopted numerical scheme is presented in section III and obtained results are validated through residual errors.

II. MATHEMATICAL FORMULATION

Consider a steady, incompressible, laminar, two dimensional stagnation point flow of a third order fluid over a surface which is lubricated with a power law fluid. The lubricant spreads over the sheet in the form of a thin layer of variable thickness. For the development of mathematical model we use Cartesian coordinate system (x, y, z) with y - axis is normal to the sheet. The flow rate of the lubricant is given by

where h(x) is the variable thickness of the lubricant and U is the velocity component of power law fluid in x-direction. The boundary layer equation that governs the flow of a third order fluid are38

d x dy

du du 1 dp d2u a1 T d3u du d2u du d2v d3u

du d u d u d v d u

+ + ^--^—ô + V—-r

(3 u \ 2

d/)' (5)

where u and v are the velocity components of third order fluid in x and y directions respectively, p is the density, v is the kinematic viscosity, a1, a2, ¡32, ¡33 are the fluid parameters, p and p are respectively the pressure and modified pressure.

The domain is divided in two regions, a thin layer of power law lubricant i.e. 0 < y < h(x) and a third order fluid that occupies the space h(x) < y < 1. The boundary conditions relevant to the present flow situation are24-28

U (x, y) = 0, V (x, y) = 0, at y = 0, (6)

u = U, v = V at y = h (x), (7)

Idu) \ d2u d2u „du du 1 I du)3 /dU)n , . .

+ a1 [uddd-y + + dyj + 2(^2 + dy) = Hdy), at y = h(x), (8)

u = U0 (x), as y ^ to. (9)

in which ju is the dynamic viscosity, k is the consistency index and n is the power law index. Under the assumption that vertical component of lubricant does not change inside the thin layer we have

V (x,y) = 0, for y € [0, h (x)]. (10)

Using boundary condition (7) we get

v = 0 at y = h (x), (11)

Following Ref. 24, it is assumed that U varies linearly inside the thin lubrication layer therefore

U (x,y) = °-Wr- «12»

in which U (x) is the interfacial component of velocity for both the fluids at the interface y = h (x). Hence by using Eq. (1) one obtains the thickness of the lubrication layer

h (x) = , (13)

and hence the boundary condition (8) becomes

( du )

"d* + ai

d2u + d 2u + ^ d u du d x dy dy 2 d x dy

■( I )3

+ 2 (H2 + = (2Q2 u2n. (14)

For the flow near a stagnation point the free stream velocity takes the form

U0 (x) = cx, (15)

In which c is constant having dimensions of t-1. Which gives

dp = c2x. (16)

Defining the dimensionless variables

, u = cxf' (n), v = -V cvf (n), (17)

n = y^[V' u = cxf' (n ) ' v = -Vcvf (n ),

We have the following boundary value problem for the stagnation point flow of a third order fluid in the region of stagnation point

f''' - f'2 + ff'' + 1 + ^ (2f'f''' - ffv - f''2) + 6<p<pif'''f''2 = 0, (18)

f (0) = 0, f " (0) + 3ef ' (0) f " (0) + 2001 f ''3 (0) - A{f ' (0)}2n = 0, f ' («,) = 1, (19)

kVv x2n-1c2n"3 aiC c2 (fo + fo) cx2

--2— , e = -, 0 =-, 0i = -

J (2g)2 j j v

Eq. (18) is solved subject to boundary conditions (19) using a hybrid homotopy analysis method given in detail in next section.

III. SOLUTION BY HYBRID HOMOTOPY ANALYSIS METHOD

In the first step we convert the boundary value problem given in Eqs. (18) and (19) into initial value problem by using shooting method37 for this we assume

f' (0) = s. (21)

The nonlinear boundary condition given by Eq. (19) is cubic in f' ' (0) and has the following real root

f'' (0) - 21/3(1+3cs) + A (22)

f (0) = A + 62^' (22)

( I-\ 1/3

A - 1108A02021s2n + ^ 11664220404s4n + 8640301(1 + 3es)3l .

Differentiating Eqs. (18), (19), (21) and (22) with respect to s we get

g ''' + g f'' + fg '' - 2 f 'g' + e (2 f 'g''' + 2g 'f''' - fgv - g fv - 2 f ''g '')

+6001 (g'''f' '2 + 2 f'' f' g'') - 0, (23)

g (0) - (0) - 1^(0) - -^ + (^ - ^) dA, <24)

The governing initial value problems (18) and (23) are higher order than the available initial conditions. Further the coefficient of the leading order derivative terms vanishes at q - 0 and when e ^ 0. Therefore, one cannot integrate them by a standard integration scheme like Runge-Kutta method. The relevant first order initial value problems are

f' - F, (25)

F' - G, (26)

G' - - fG + F2 - 1 - e (2FG' - fG'' - G2) - 600G'G2, (27)

g' - Y, (28)

Y' - Z, (29)

Z' - -gG - fZ + 2FY - e (2FZ' + 2YG' - fZ'' - gG'' - 2GZ) - 6001 (Z'G2 + 2G'GZ), (30)

21/3 1 + 3es A

/(0) - 0, f(0) -g(0) - --i^-i + ^, m>

g (0) - 0, Y (0) - 1 Z (0) - + - dA. (32)

For numerical computations we replaced ^ by a value q^ and divided the domain 0 < q < n<x> into subintervals [(i - 1) H, H], i - 1,2,3 ... having length H. The initial value problems in each subintervals takes the form

-T- - F1, (33)

d^ = GI., dn

dG' = fG 1 + (F*? « (2FidG' P d'2°l _ =-fG-1+(F )- « ^ — -f —

dg' = yi

"(G')j - 6001 (dG(G')j , (35)

dYl = Zi,

f = 'G' - f'Zi + 2F'Y' - « (< + 2Y'f - f 'f - g'f -6001 ( f (G')2 + 2 f G' Z ' ) '

f1 (0) = 0' F1 (0) = s, G1 (Q) = + 627500

(2GiZi)

g1 (0) = 0' y1 (0) = 1' Z1 (0) = + - ^^^ £ (40)

A \ 621/3002 A2 ) ds

The numerical values computed at the end points of the it h subinterval becomes the initial conditions for the (i + 1)th subinterval.

A. Zeroth order deformation problem

The zero order deformation equations are given by

(1 - p) L [f (n, p) - f0 (n)] = -p

(1 - p) L [F' (n,p) - F0 (n)] = -p

df (n, P)

dn dFl (n, p) dn

- F' (n, p)

- Gl (n, p)

(1 - p) L [Gl (n, p) - GQ (n)] dG' (n, p)

+ f (n, p) Gl (n, p) + 1 - (F ' (n, p))

, ) O 17li \ dG' p) ri , X d2G' (n'p) (7i, ,y

+«1 2F (n,p)-^--f (n,p)-ZZ2--\G (n,p))

+6001( (Gi (n' p))2)

(1 - p) L [gi (n,p) - gQ (n)] = -p

dgi(n, p)

(1 - p) L [Yi (n,p) - YQ (n)] = -p

dYi (n, p)

- yi (n,p)

- Zi (n,p)

(1 - p) L [Zi (n, p) - ZQ (n)] = dZi (n, p)

+ gi (n,p) G (n,p) + fi (n,p) Z (n,p) - 2Fi (n,p) Y (n,p)

2Fi n p) + 2Yi (n,p) dG'(n p)

f np) ^Znr1 - (2Gi np) Zi np)) - gi np) ^

+6001 (dn(Gi (n,p))2 + 2Gi (n,p) Zi (n,p)

where p and L are respectively the embedding parameter and auxiliary linear operator and f (n), F0 (n), G0 (n), (n), Y0 (n) and Z0 (n) are initial guesses. In the present case we have chosen the numerical values at the starting point of each subinterval as initial guesses. In the present case we set auxiliary parameter to be -1 and auxiliary function to be 1. The convergence in the proposed hybrid homotopy analysis method is controlled through the length of the subinterval i.e. H and the order of approximation.

B. mth order deformation problem

The mth order deformation problem in each subinterval are given by

L [f'm - Xmf'-l] = -— + F

L [Flm - XmFm-l] = -

L [G'm - XmGm-l] = + (Xm - l) + £

-fl Gl + f' f'

f kGm-1 + r' r-

k m-l-k

dn J k dn2

- (GkG'm-l-k)

k dG i

Gk-lGl 1

L [g'm - Xmg'm-l] = -

L [y' - XmYlm-l] = -

L [z'm - XmZ'm-l]

gkGm-l-k '

k m- 1-

2F lk~

+ 2Y V

k m- 1- k

i d2 z'

2G kZ 1

k m-1-

2T=0 (

1 m-l-k^i ^i r, m-l-k^i

Gk-l Gl + 2-;-Gu ,Z,

fl (0) = 0, Fo1 (0) = s, GO (0) = -

2l/3(l + 36 s)

gO (0) = 0, Yo1 (0) = l, ZO (0) = -

2l/3(36 )

62l/300f

62l/3002 2l/3(l + 36 s)

m < l m > l

The final solutions in each subinterval are thus given by

[f,F,G,g,Y,Z]' = £m=0 [f,F,G,g,Y,Z]' .

The solution of the governing problem is obtained in the following steps. First an approximate value of s is chosen and the system of initial value problems is solved for i = l. Then the initial condition at the second subinterval is evaluated from this solution and a solution is found in the second subinterval. This procedure continues and an analytic solution is evaluated in each subinterval.

TABLE I. Numerical values of f ''(0) when n = 1, e = 0.1, tp1 = 1.0.

À 0 = 0.1 0 = 0.2 0 = 0.3 0 = 0.4 0 = 0.5 0 = 0.6 0 = 0.7

0.1 0.073740 0.073681 0.073620 0.073559 0.073498 0.073438 0.073378

0.5 0.308774 0.304627 0.300693 0.297001 0.293504 0.290166 0.286958

1.0 0.497937 0.482828 0.467114 0.453128 0.439128 0.424481 0.408554

5.0 0.862098 0.765715 0.671058 0.607705 0.616997 0.554423 0.453273

10.0 0.923735 0.797164 0.683032 0.656209 0.631329 0.610487 0.506951

50.0 0.971369 0.817961 0.798342 0.795003 0.692006 0.653159 0.626298

FIG. 1. Residual error of obtained solutions.

FIG. 2. Influence of slip parameter A on the velocity profile f'.

A zero finding algorithm is chosen to evaluate the correct value of s which leads to F0N (q^) = 1. We choose the order of approximation and size of interval such that the obtained residual error lies within an accuracy of 10-5.

IV. NUMERICAL RESULTS AND DISCUSSION

The hybrid homotopy analysis method explained in detail in previous section is employed through mathematica for finding the solution of stagnation point flow of a third grade fluid over

FIG. 3. Influence of second grade fluid parameter e on the velocity profile f' .

FIG. 4. Influence of third grade parameter 0 on the velocity profile f'.

a lubricated surface. The analytic solutions of required accuracy are evaluated in each subinterval together with the respective numerical values at each node point of the subinterval. In table I the numerical values of /"(0) are given for different values of the slip parameter and third order fluid parameter. These numerical values provide the missing conditions for obtaining the velocity profiles. To ensure the accuracy and correctness of the obtained hybrid homotopy analysis method solutions we have plotted residual error curves in each case. Here we are shown one of these error curves for a set of parameters in Fig. 1. It is evident from this figure that the obtained solutions have an accuracy of 10-5. The influence of pertinent parameters appearing in the problem on the velocity profiles are shown in Figs. 2-5.

Fig. 2 is shown to present the effects of slip parameter on the velocity field /'. The velocity profiles depict that velocity decreases by increasing slip parameter A. The full slip case is represented through the profile when A ^ 0 and the no-slip is represented when A ^ ж. It is clear from this figure that in the full slip case the slip on the surface dominates the stagnation point effects and almost no change is observed in the velocity throughout the semi-infinite domain. Influence of the second grade fluid parameter on the velocity profile /' is observed through Fig. 3. This figure

FIG. 5. Influence of power law index n on the velocity profile f'.

elucidate that the velocity oscillates inside the boundary layer when we vary the second grade fluid parameter. The velocity decreases and boundary layer thickness increases by increasing the third grade parameter as one can see from Fig. 4. Figure 5 is made to see the effect of power law index n on the velocity profile f'. The results show that for a shear thinning lubricant the boundary layer thickness is more when compared with shear thickening fluid. Furthermore, the velocity increases by an increase in the power law index.

V. CONCLUDING REMARKS

The stagnation point flow of a third grade fluid over a flat surface lubricated with a power law fluid is analyzed. The flow problem is governed by nonlinear partial differential equation and a nonlinear condition at the interface. The governing equations are transformed to a non-similar ordinary differential equation subject to nonlinear boundary conditions. A hybrid homotopy solution of the governing problem is evaluated and results are discussed under the influence of fluid parameters appearing in the problem. The main features of the present study are listed below:

• Slip effects dominates the effects of free stream velocity.

• An increase in the slip enhances the fluid velocity and suppresses the boundary layer thickness.

• Velocity oscillates inside the boundary layer when we increase the second grade fluid parameter.

• Fluid velocity is an increasing function of power law index n.

• The fluid velocity has a higher value when third grade fluid is flowing over a shear thickening lubricant.

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