Flow of viscous fluid along an exponentially stretching curved surface

Accepted Manuscript Flow of viscous fluid along an exponentially stretching curved surface N.F. Okechi, M. Jalil, S. Asghar PII: DOI: Reference:

S2211-3797(17)31109-9 http://dx.doi.org/10.1016/j.rinp.2017.07.059 RINP 832

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Results in Physics

Received Date: Revised Date: Accepted Date:

26 June 2017 24 July 2017 24 July 2017

Please cite this article as: Okechi, N.F., Jalil, M., Asghar, S., Flow of viscous fluid along an exponentially stretching curved surface, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.07.059

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Flow of viscous fluid along an exponentially stretching curved surface N.F. Okechi ab*, M. Jalilb, S. Asghar ac a

Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, 44000, Pakistan

b

Department of Mathematics Programme National mathematical Centre , Sheda-Kwali, P.M.B 118, Gwagwalada, Abuja, Nigeria c

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia

Abstract In this paper, we present the boundary layer analysis of flow induced by rapidly stretching curved surface with exponential velocity. The governing boundary value problem is reduced into self- similar form using a new similarity transformation. The resulting equations are solved numerically using shooting and Runge-Kutta methods. The numerical results depicts that the fluid velocity as well as the skin friction coefficient increases with the surface curvature, similar trend is also observed for the pressure. The dimensionless wall shear stress defined for this problem is greater than that of a linearly stretching curved surface, but becomes comparably less for a surface stretching with a power-law velocity. In addition, the result for the plane surface is a special case of this study when the radius of curvature of the surface is sufficiently large. The numerical investigations presented in terms of the graphs are interpreted with the help of underlying physics of the fluid flow and the consequences arising from the curved geometry.

Keywords Boundary layer flow, Curved surface, Exponential stretching, Curvature. Introduction Sakiadis [1], pioneered the study of boundary layer flow over a moving solid with constant speed. Crane [2] investigated the flow of Newtonian fluid over a linearly stretching sheet, giving exact solution of the two-dimensional boundary layer equations. The effects of various physical phenomenon like heat and mass transfer as well the influence of suction, injection, magnetic field, among others on such flow due to the stretching of surface have been investigated by different authors in literature. Gupta and Gupta [3], examined the stretching flow subject to

suction or injection. Mgyari and Keller [4] investigated the flow behavior and heat transfer due to exponentially stretching of surface with an exponential temperature distribution. Elbashbeshy [5], extended the work of [4] by subjecting the stretching surface to suction, and showed that the boundary layer thickness decreases with increasing parameter of suction. Ahmad and Asghar [6] found the analytical and numerical solution for the flow and heat transfer over hyperbolic stretching surface. Moreover because of the increasing industrial significance of non-Newtonian fluid several authors have been motivated to study the flow of non-Newtonian fluid over a stretching surface under the influence of different physical phenomena. Rajagopal and Gupta [7] gave the exact solution for a boundary layer flow of non-Newtonian fluid flow past an infinite plate. Anderson and Kumaran [8] obtained analytical and numerical solution for non-Newtonian power-law fluid over power-law stretching sheet. Analytical solutions for the flow of power-law fluid over a power law stretching of flat surface was given by Jalil et al. [9]. Jalil and Asghar [10] also presented analytical and numerical solution for flow of power-law fluid over exponentially stretching surface. Hussain et al [16] recently analysed MHD Prandtl-Eyring fluid flow over a stretching sheet numerically. All the preceding papers address the flow of Newtonian or non-Newtonian fluids over linearly or non-linearly stretching flat surface. However, the flow of viscous fluid past a curved surface has not been given its due importance; especially non-linearly stretching of the curved surface. Sajid et al. [11] presented linear stretching on a curved surface by a numerical approach, showing the boundary layer thickness is greater for a curved surface as compared to flat surface. They further indicated that the drag force is reduced on a curved surface as compared with the flat surface and the pressure variation is not negligible. The viscous flow over a power law stretching curved surface was investigated by Sanni et al [12] numerically, in this paper the authors showed that a slight variation in the curvature causes a significant increase in the velocity and skin friction coefficient, and the non-linearity of the stretching velocity also increases the skin friction. Moreover some numerical investigations on flow over a rather stretching cylinder have been also given (see [15-16]). In this paper, we study the flow of a viscous fluid over a rapidly stretching curved surface by redefining the stretching velocity in terms of an exponential function. Curvilinear coordinates are employed as the best fit for the boundary of the surface. The nonlinearity in the system is due to the curvilinear nature of the surface and the non-linear convective part of the governing equations. The numerical solution of this highly non-linear system is perhaps the only way forward. We define appropriate similarity variables to transform the partial differential equation and the boundary conditions into a self-similar boundary form; which is then solved numerically.

Statement of the problem We consider the flow of an incompressible viscous fluid passing over a stretching curved surface. The surface is stretched with an exponential velocity (
() =  ).) along the sdirection with the fluid forming a boundary layer in the r-direction. The distance of surface from

the origin determines the shape of the curved surface, i.e., the surface tends to flatness for large value of .

Fig1. Flow geometry for an exponentially stretching curved surface. For a steady incompressible flow, the governing equations consisting of the continuity and the Navier Stokes equations respectively are the written the in curvilinear coordinate system as:
[( + )] +

= 0, 










 + − +  +

1   1  

   2
=− +  + − + − ,    ( + )  ( + )  

+ ( + )







 + + +  +

1  
1



 
2  =− +  + − + + ,    ( + )  ( + )   + 

+ ( + )

Applying boundary layer theory on (1)-(3), the governing equations for the flow reduces to the following boundary layer equations:

(1)

(2)

(3)

∂ ∂u [(r + R)v] + R = 0, ∂r ∂s

(4)

u2 1 ∂p , = r + R ρ ∂r

u

(5)

 ∂ 2u ∂v R ∂u uv 1 R ∂p 1 ∂u u  + u + =− +ν  2 + − , ∂r r + R ∂s r + R ρ r + R ∂s r + R ∂r ( r + R ) 2   ∂r

(6)

The appropriate boundary conditions for the problem are:

u = ae s , v = 0 at r = 0 u → 0,

(7)

∂u → 0 as r → ∞ ∂r

Where
and  are the and  components of velocity respectively,  is the pressure,  is the kinematic viscosity of the fluid and  is its density. To transform (4)-(7) into ordinary differential equations we introduce the flowing similarity variable:

η=

ae s

ν

r, v =

− R aν e s r+R 2

{ f (η ) + η f ′(η )} ,

u = ae f ′(η ), p = ρ a 2 e 2 s H (η ), λ = s

ae s

ν

(8)

R

The radial variable is non-dimensionalized by the boundary layer thickness and the velocity by the stretching velocity. Using the above similarity transformations; Eqs. (4) - (7) are transformed into self-similar nonlinear ordinary differential equations given by: H′ =

f ′2 η+λ

(9)

ηλ η + 2λ λ λ 4λ f ′′ f′ H′ + H = f ′′′ + ff ′′ + ff ′ − − λ f ′2 + 2 2 2 (η + λ ) (η + λ ) η +λ η + λ (η + λ ) (η + λ ) (η + λ )

(10)

f ′(0) = 1, f (0) = 0, f ′( ∞) = 0, f ′′( ∞) = 0.

(11)

Where Eq. (5) is a balance between centrifugal force and radial pressure gradient, and has been has been transformed to (9). Eqs. (9) and (11) together gives an additional boundary equation  (0) = 1/λ, and using Eqs. (9) in (10) yields the following equation:

f iv + −

λ λ λ 2 f ′′′ f ′′ f′ ff ′′′ + ff ′′ − ff ′ − + + 2 3 2 3 η + λ (η + λ ) (η + λ ) (η + λ ) (η + λ ) (η + λ ) 3

(η + λ )

2

λ f ′2 −

3 λ f f′ ′′ = 0 (η + λ )

(12)

with the boundary conditions given by Eq. (11). The skin friction coefficient for this problem is given by:

! = " #$ − %& + =0 '( = ⟹ +

,' (

!  1⁄2 


1 = .  (0) − .  (0)

λ

(13) where the second term in the above expression is the contribution due to the curvature of the surface. Numerical results and Discussion

The solution of the transforned ODEs have been obtained numerically using shooting method with the Runge-Kutta alogorithm. The curvature effect on the velocity and pressure for an exponentially stretching curved surface are illustrated graphically in Figures [a &b]. These graphs show that the horizontal velocity increases slightly whereas the vertical velocity increases significantly. This in turn translates into a growing viscous bounday layer thickness. The pressure also shows an increasing trend for increasing curvature. The observations made through the numerical computations does indeed need the support of the physical reasoning; without which the numerical results are neither reliable nor valuable. The physical explanation comes through the presence of centifugal force experienced by the fluid as it traverses circular path over the curved stretching surface. The curvature of the surface gives rise to the centrifugal force, in response to the centripetal acceleration, which is directed away from the center of curvature of the surface ensuring that the flow remains circular. The effect of curvature on the flow properties is directed towards the normal to the curved surface such that the magnitude of this effect increases with increasing centrifugal force (increasing curvature of the surface or decreasing radius of curvature). The centrifugal force gives rise to an additional disturbance called the secondary wave that is superimposed on the primary wave generated by the stretching of the surface. Since, the centrifugal force is directed in the normal direction, it influences the vertical velocity the most. Now, it could be possible that the centrifugal force when acting on the small perturbations (secondary wave) give rise to a weak horizontal velocity. Therefore, although the centrifugal force is balanced by the vertical pressure gradient [ Eq. 5 ] ; nevertheless there

remains the tendency of the centrifugal force to generate a small amount of horizontal velocity disturbing the balance slightly. In the limiting case of large radius of curvature, the surface takes the form of a flat surface and the centrifugal force vanishes. For this situation the secondary wave becomes nonexistent and the flow becomes a dominant primary flow generated by the stretching of the surface alone. Another important observation comes through the wall shear stress. The dimensionless wall ,

shear stress − + - '( for the present problem increases with increasing curvature as shown in Table1. For large radius of curvature, the shear stress for the present problem matches with that of exponentially stretching surface given in [4] and [5]. The comparison between a non-linearly (power-law) stretching [12] and an exponentially stretching curved surfaces show that the dimensionless wall shear stress for a power-law stretching curved surface is much greater than an exponentially stretching curved surface. This explains the reduction in the velocities as the resistance to flow is much higher for the power-law stretching curved surface than the latter. Furthermore the results also show greater dimensionless wall shear stress for this problem than a linearly stretching curved surface when the power-law index becomes unity.

(a)

(b)

(c) Fig. 2. Effects of dimensionless radius of curvature λ on horizontal and vertical components of velocity profiles (a) f′(η) (b) f(η) and (c) dimensionless pressure P(η). Table 1

Effect of dimensionless radius of curvature λ on skin friction-Res1/2Cf

λ 5 10 20 30 40 50 100 200 1000 ∞

-Res1/2Cf 1.4196 1.3467 1.3135 1.3028 1.2975 1.2944 1.2881 1.2850 1.2826 1.2818

[4]

1.28180

[5]

1.20181

Conclusion

This study is carried with an objective of investigating the viscous fluid flow generated by exponential stretching of the curved surface. In line with our interest we have defined an exponential velocity for the surface keeping in mind the desired characteristics of an exponential function. The boundary layer equations are transformed into ODEs by defining suitable similarity variable and the solution is obtained numerically. The observations of this study are validated by comparing with the existing literature and the arguments entailing to physics of fluid flow It is observed that the curvature and thus the centrifugal force have profound effect on the field quantities due to an additional force experienced by the fluid when it is set into a curvilinear path

due to stretching and the no slip condition. In addition to the primary wave generated by the stretching of the sheet, the curvature gives rise to the secondary wave. The total effect on the velocity field is the superposition of the two effects. The vertical velocity increases significantly and the horizontal velocity undergoes a small increase. The reason for such behavior has been discussed in the previous section. The effect of increasing curvature shows an increasing value of the shear stress. A comparison with the power law stretching of the curved surface is also made. These results may have applications in scenarios that require the transport of fluid over a rapidly stretching curved surface to achieve a desired outcome. In particular, it has a great practical relevance in chemical and polymer industry where the sheets are rapidly stretched to enhance the quality of the polymer sheets. Further to that, the present problem gives an overview of the stretching problems for linear, non-linear and exponential stretching over the plane surfaces and the curved surfaces. The investigations will hopefully be useful in recent researches being carried out for the blood flow in the arteries being taken as straight tubes and curved tubes. The aspect of considering these arteries as flexible and being stretchable can be a field of future investigations.

Acknowledgements

This work is jointly supported by The World Academy of Science (TWAS) and Comsats Institute of Information Technology (CIIT) fellowship programme.

References

[1] B.C. Sakiadis. Boundary layer behaviour on continuous solid surfaces: Boundary layer equations for two-dimensional and axisymmetric flow. A.I.Ch. E.J.,1961;7, 26-28. [2] Crane LJ. Flow past a stretching plate. Z Angew Math Phys 1970;21:645–7. [3] Gupta P, Gupta A. Heat and mass transfer on a stretching sheet with suction or blowing. Canad J Chem Eng 1977;55:744–6. [4] Magyari E, Keller B. Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J Phys D Appl Phys 1999;32: 577–85. [5] Elbashbeshy E.M.A. Heat transfer over an exponentially stretching continuous surface with suction. Archives of mechanics.2001;53:643-51. [6] Ahmad A, Asghar S. Flow and heat transfer over hyperbolic stretching sheets. Appl Math Mech 2012;33:445–54. [7] Rajagopal KR, Gupta AS. An exact solution for the flow of a non-Newtonian fluid past an infinite plate. Meccanica 1984;19:158–60. [8] Andersson H, Kumaran V. On sheet-driven motion of power-law fluids. Int J 253 Non-Linear Mech 2006;41:1228–34. [9] Jalil M, Asghar S, Mushtaq M. Analytical solutions of the boundary layer flow of power-law fluid over a power-law stretching surface. Commun Nonlinear Sci Numer Simulat 2013;18:1143–50. [10] Jalil M, Asghar S. Flow of power-law fluid over a stretching surface: a Lie group analysis. Int J Non-Linear Mech 2012;48:65–71.

[11] Sajid M, Ali N, Javed T, Abbas Z. Stretching a curved surface in a viscous fluid.Chin Phys Lett 2010;27:024703. [12] Sanni K.M, Asghar S, Jalil M, Okechi N.F. Flow of viscous fluid along a nonlinearly stretching curved surface. Results in Physics.2017:7:1-4. [13] Vajravelu K. Viscous flow over a nonlinearly stretching sheet. Applied Mathematics and Computation.2001;124:281-88. [14] Schlichting H., Gersten K..Boundary layer theory, 8th revised and enlarge edition. Springer 2001. [15] Malik, M Y, and Salahuddin T. Numerical Solution of MHD Stagnation Point Flow of Williamson Fluid Model over a Stretching Cylinder. International Journal of Nonlinear Sciences and Numerical Simulation 2015;16(3-4):161-164. [16] Malik M Y, Salahuddin T, Hussain A, Bilal S. MHD flow of tangent hyperbolic fluid over a stretching cylinder: using Keller box method. J Magn Magn Mater 2015; 395:271–276. [17] Hussain A, Malik M , Awais M. Computational and physical aspects of MHD PrandtlEyring fluid flow analysis over a stretching sheet. Neural Comput & Applic 2017. DOI 10.1007/s00521-017-3017-5.