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Stagnation point flow of Maxwell fluid towards a permeable surface in the presence of nanoparticles

G.K. Ramesh a,b*, B.J. Gireeshab, T. Hayatc,d, A. Alsaedid

a Department of Mathematics, SEA College of Engineering and Technology, K.R. Puram, Bengaluru 560049, Karnataka, India b Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta 577 451, Shimoga, Karnataka, India c Department of Mathematics, Quaid-T-Azam University, Islamabad 44000, Pakistan

d Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 10 June 2014; revised 23 January 2016; accepted 20 February 2016

KEYWORDS

Stagnation point flow; Maxwell fluid; Permeable surface; Nanoparticles; Numerical solution

Abstract Analysis has been carried out to study the stagnation point flow of Maxwell fluid towards a permeable stretching sheet in the presence of nanoparticles. Using suitable transformations, the governing partial differential equations are first converted to ordinary one and then solved numerically by fourth-fifth order Runge-Kutta-Fehlberg method with MAPLE. The flow and heat transfer characteristics are analyzed and discussed for different values of the parameters. Present work reveals that the velocity increases whereas the temperature and concentration decrease with the increase of Maxwell parameter. The thermal and concentration boundary layer thickness decreases with velocity ratio, Lewis number, Prandtl number suction, Brownian motion and ther-mophoresis parameters. Comparison with known results for Newtonian fluid flow is found an excellent agreement.

© 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

There is no doubt that human society development greatly depends upon energy. However the rapid development of human society during the past few years leads to the shortage of global energy and the serious environmental protection. Sustainable energy generation in recent time is thus a challeng-

* Corresponding author at: Department of Studies and Research in

Mathematics, Kuvempu University, Shankaraghatta 577 451,

Shimoga, Karnataka, India. Tel.: +91 9900981204.

E-mail address: gkrmaths@gmail.com (G.K. Ramesh).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

ing issue globally. Solar energy in which circumstances has been regarded one of the best sources of renewable energy via least environmental impact. Solar power in fact is a natural way of obtaining water, heat and electricity. Power tower solar collectors could benefit from the potential efficiency improvements that arise from using a nanofluid as a working fluid. Particle size of nanomaterial is similar or smaller than the wavelength of de Broglie and coherent waves. It is now recognized that solar thermal system with nanofluids becomes the new study hotspot.

On the other hand several industrial fluids are non-Newtonian in their flow characteristics. In a Newtonian fluid, the shear stress is directly proportional to the rate of shear strain, whereas in a non-Newtonian fluid, the relationship

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1110-0168 © 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

A velocity ratio Uw stretching sheet velocity

c stretching rate U free stream velocity

Cf skin friction x coordinate along the stretching sheet

cP specific heat y distance normal to the stretching sheet

C nanoparticle volume fraction

Db Brownian diffusion coefficient Greek symbols

DT thermophoresis diffusion coefficient V kinematic viscosity

Nb Brownian motion parameter / rescaled nanoparticle volume fraction

Nt thermophoresis parameter Pf density of the base fluid

Nux local Nusselt number Pp density of the particles

Le Lewis number k suction parameter

Pr Prandtl number b Maxwell parameter

Shx local Sherwood number h dimensionless temperature

T temperature of the fluid g similarity variable

Tw temperature at the wall a thermal diffusivity

T1 ambient fluid temperature sw wall shearing stress

u, v velocity components along x and y directions

between the shear stress and the rate of shear strain is nonlinear. Most of the particulate slurries such as china clay and coal in water, multiphase mixtures such as oil-water emulsions, paints, synthetic lubricants, and biological fluids including blood at low shear rate, synovial fluid, and saliva and foodstuffs such as jams, jellies, soups, and marmalades are examples of non-Newtonian fluids. Because of the large variety of the non-Newtonian fluids, many models of non-Newtonian fluids exist. Maxwell model is one subclass of rate type fluids. This fluid model predicts the relaxation time effects. Such effects cannot be predicted by differential-type fluids. This fluid model is especially useful for polymers of low molecular weight. A review of non-Newtonian fluid flow problems may be found in [1-3].

Initially, Sakiadis [4] introduced the concept of boundary layer flow over a moving surface. Crane [5] modified the idea introduced by Sakiadis and extended this concept linear stretching sheet. Flow in the neighborhood of stagnation point in a plane was first studied by Hiemenz [6]. Mahapatra and Gupta [7-9] investigated the magnetohydrodynamic stagnation point flow towards a stretching sheet. They shown that the velocity at a point decreases/increases with increase in the magnetic field when the free stream velocity is less/greater than the stretching velocity. Also they have studied the temperature distribution when the surface has constant temperature and constant heat flux. Further they have extended their work on power law fluid and discussed the uniqueness of solutions of stagnation-point flow towards a stretching surface. Accordingly, researchers in the [10-13] studied the stagnation point flow over a surface.

Aforementioned studies were primarily concerned with the laminar flow of a clear fluid. Nanotechnology is an emerging research topic having extensive use in industry due to the unique chemical and physical properties which the nano-sized materials possess. These fluids are colloidal suspensions, typically metals, oxides, carbides or carbon nanotubes in a base fluid. The term nanofluid was coined by Choi [14] in his seminal paper presented in 1995 at the ASME Winter Annual

Meeting. It refers to fluids containing a dispersion of submicronic solid particles (nanoparticles) with typical length of the order of 1-50 nm. Kuznetsov and Nield [15] analytically studied the natural convective boundary layer flow of nano-fluid past a vertical plate. Khan and Pop [16] first time studied the problem of laminar fluid flow resulting from the stretching of a flat surface in a nanofluid. Mustafa et al. [17] investigated the stagnation point flow of viscous nanofluid towards a stretching surface using homotopy analysis method. Alsaedi et al. [18] examined the influence of heat generation/absorption on the stagnation point flow of nanofluid towards a linear stretching surface. Rahman et al. [19] examined the dynamics of natural convection boundary layer flow of water based nanofluids over a wedge. They discussed the analysis in the presence of a transverse magnetic field with internal heat generation or absorption. Nandy and Mahapatra [20] analyzed the effects of velocity slip and heat generation/absorption on magnetohydrodynamic stagnation-point flow and heat transfer over a stretching/shrinking surface and then obtained the solution numerically using fourth order Runga-Kutta method with the help of shooting technique. Different from a stretching sheet, it was found that the solutions for a shrinking sheet are non-unique. Makinde et al. [21] studied the combined effects of buoyancy force, convective heating, Brownian motion and thermophoresis on the stagnation point flow and heat transfer of an electrically conducting nanofluid towards a stretching sheet. Effect of magnetic field on stagnation point flow and heat transfer due to nanofluid towards a stretching sheet has been investigated by Ibrahim et al. [22]. Nadeem et al. [23,24] reported the numerical solutions of non-Newtonian nanofluid flow over a stretching sheet using the Maxwell fluid model. Further they obtained the analytic solution for non-orthogonal stagnation point flow of a nanosecond grade fluid toward a stretching surface with heat transfer. Hady et al. [25] studied the natural convection boundary-layer flow over a downward-pointing vertical cone in a porous medium saturated with a power-law nanofluid in the presence of heat generation or absorption. Unsteady boundary layer

flow of viscous nanofluid with thermal radiation has been discussed by Khan et al. [26]. Sheikholeslami et al. [27] examined the natural convection flow of nanofluid in the presence of magnetic field. MHD flow of viscous nanofluid due to rotating disk is addressed by Rashidi et al. [28]. Turkyilmazoglu and Pop [29] discussed thermal radiation effect in unsteady natural convection flow of nanofluids past a vertical infinite plate. Mohamad et al. [30] examined Hiemenz flow of nanofluid due to porous wedge. Turkyilmazoglu [31,32] explores slip and convection effects in the flow of nanofluids. Nadeem et al. [33] analyzed the flow of three-dimensional water-based nanofluid over an exponentially stretching sheet. Very recently Ramesh and Gireesha [34,35] studied the heat source/sink effects on Maxwell fluid over a stretching surface with convec-tive boundary condition in the presence of nanoparticles, and also obtained the numerical solution of the influence of heat source on stagnation point flow towards a stretching surface of a Jeffrey nanoliquid.

In this paper, we study the behavior of the stagnation point flow of Maxwell fluid towards a stretching sheet in the presence of nanoparticles. The sheet is taken permeable. Similarity transforms are used for this problem, and non-dimensionalized equations are solved numerically. Graphical results for various values of the parameters are presented to gain thorough insight towards the physics of the problem. To the best of our knowledge, this problem has not been studied before.

2. Mathematical analysis

Consider the flow of an incompressible non-Newtonian Maxwell fluid in the region y > 0 driven by a stretching surface located at y — 0 with a fixed stagnation point at x — 0. The x and y axes are chosen along and perpendicular to the sheet. The stretching velocity Uw(x) and the ambient fluid velocity U(x) are assumed to vary linearly from the stagnation point, i.e., Uw(x) — cx and U(x) — bx where c and b are rate constant.

We assume that flow is laminar, steady and two-dimensional. The sheet is flat and permeable, and the temperature T and the nanoparticle fraction C take constant values Tw and Cw respectively. The ambient values attained as y tend to infinity of T and C denoted by Tx and CM respectively. All the thermo-physical properties are taken constant.

The flow problem under consideration is governed by the following boundary layer equations:

du dv o

dx dy '

du du du 1 dp I 2u u 2 „ „ dx dy dy2 p dx \ dx2 dy2

dT dT _ _ dx dy dy2 dy dy

d2u dxdy

u@£+v@£_d - -dx dy B dy2 V T

where u and v are the velocity components along the x and y axes, respectively. Further, a, pp, pp, v, T and T1 are respectively the thermal diffusivity, density of the base fluid, density of the particles, kinematic viscosity of the fluid, fluid temperature and ambient fluid temperature. k0 is the relaxation time of the UCM fluid, DB is the Brownian diffusion coefficient, DT is the thermophoresis diffusion coefficient and cp is the specific heat at constant pressure. Here s is the ratio of the effective heat capacity of the nanoparticle material and the heat capacity of the ordinary fluid and C is the nanoparticle volume fraction.

The associated boundary conditions for the present problem are

u — Uw(x), v — — Vw(x), T — Tw; C — Cw at y — 0, u ! U(x), v ! 0, T ! C ! CM as y !i

where Uw{x) — cx is the stretching sheet velocity, c > 0 this is known as stretching rate. Tw and Cw are the temperature of fluid and nanoparticles fraction at wall, and CM is ambient nanoparticle volume fraction.

To employing the generalized Bernoulli's equation, in the free stream U{x) — bx Eq. (2) reduces to

UjdU 1 dp

dx p dx

Using (6) into (2) one can obtain

du du d2u ydU ^ ( 2 @2u dx dy dy2 dx 0 \ dx2

, d u dy2'

The specific forms of the stretching velocity and the surface temperature and concentration are chosen to allow the coupled nonlinear partial differential Eqs. (1)-(4) to be converted to a set of coupled, nonlinear ordinary differential equations by the similarity transformation

y; f(g) —

(xvUw)

u(g) —

CC C — C

where g is the similarity variable, f h and / are the dimension-less stream function, temperature and concentration respectively. The velocity components u and v in Eq. (8) automatically satisfy the continuity Eq. (1). In terms of f(g); h(g) and /(g) the momentum Eq. (2), energy Eq. (3) and concentration Eq. (4) can be written as

f2 — ff — f ' + A2 — bff' ' + 2-ßfff"

—h' + fh + Nbe's' + Nte02 — 0

/'' + LePrf /' + N-b h' — 0

(9) (10)

Here f represents an ordinary derivative with respect to g and the corresponding boundary conditions in the non-dimensional form are

f — k, f — 1, h — 1, / — 1 at g — 0 f ! A, h ! 0, / ! 0 as g !i

The dimensionless parameters in Eqs. (9)—(11) are A — | is the ratio of rates of velocities, b — k0c is the Maxwell parameter, k — ^ is suction parameter, Nb — sPB(C»~Ci) is the

Brownian motion, Nt — s"Pr(TT~T'1'1 is the thermophoresis parameter, Le — pis the Lewis number and Pr — J is the Prandtl number.

For practical purposes, the functions /(g), h(g) and /(g) allow us to determine the skin friction coefficient, Nusselt number and Sherwood number respectively.

C =(1 +R/(0) ; ^ = -ReX/20'(0), Shx = -Re-/2/'(0)

where Re-,

_ xUw(x)

is the local Reynolds number.

3. Numerical solutions

Numerical solutions to the governing ordinary differential Eqs. (9)-(11) with the boundary conditions (12) are obtained using a fourth-fifth order Runge-Kutta-Fehlberg method with the help of MAPLE to obtain the missing values of f "(0), h'(0) and /'(0). The problem for a regular (non-Newtonian) fluid involves four parameters, namely Maxwell parameter, Prandtl number, suction parameter, and velocity ratio parameters. The present extension involves three more parameters Nb, Le and Nt. Therefore, we need to be very selective in the choice of the values of the parameters. Since most nanofluids examined to date have large values of the Lewis number Le, we are interested mainly in the case Le p 1. Because the physical domain in this problem is unbounded, whereas the computational domain has to be finite, we apply the far field boundary conditions for the similarity variable g at a finite value denoted here by gmax. We run our bulk computations with the value gmax = 5, which was sufficient to achieve the far field boundary conditions asymptotically for all values of the parameters considered. Researchers can solve the above nonlinear differential equations analytically see [36,37].

4. Result and discussion

In order to validate the method used in this study and to judge the accuracy of the present analysis, a comparison with available results corresponding to the skin-friction coefficient f00(0) and Nusselt number —h'(0) for Nt — Nb — Le — 0 (in the absence of nanoparticles), b = 0 (in the absence of Maxwell parameter) and k — 0 (i.e. for stretching impermeable plate) with the available published results of Mahapatra and Gupta [8], Ibrahim et al. [22], and Hayat et al. [40] for various values of A is made and presented in Tables 1 and 2. A comparison is also made with Nusselt number —h'(0) in the presence of nanoparticles (b — A — k — 0, the absence of Maxwell parameter, velocity rates ratio and suction) for various values of Nt, which can be seen in Table 3. These show a favorable agreement and thus give confidence that the numerical results obtained are accurate.

Fig. 1 exhibits the velocity profiles for several values of A. It is found that when the stretching velocity is less than the free stream velocity (A > 1), the flow has a boundary layer structure, physically saying that the straining motion near

Table 1 Comparison of the values of skin friction coefficient

/"(0) for various values of velocity ratio A.

A Mahapatra and Hayat et al. Ibrahim Present

Gupta [8] [40] et al. [22] study

0.01 - -0.9982 -0.9980 -0.9991

0.1 -0.9694 -0.9695 -0.9694 -0.9696

0.2 -0.9181 -0.9181 -0.9181 -0.9181

0.5 -0.6673 -0.6673 -0.6673 -0.6672

2.0 2.0175 2.0176 2.0175 2.0175

3.0 4.7293 4.7296 4.7292 4.7292

Table 2 Comparison of the values of Nusselt number - h (0)

for various values of velocity ratio A.

Pr A Mahapatra and Hayat Ibrahim Present

Gupta [8] et al. [40] et al. [22] study

1 0.1 0.603 0.6021 0.6022 0.6048

0.2 0.625 0.6244 0.6245 0.6256

0.5 0.692 0.6924 0.6924 0.6925

1.5 0.1 0.777 0.7768 0.7768 0.7769

0.2 0.797 0.7971 0.7971 0.7971

0.5 0.863 0.8647 0.8648 0.8647

Table 3 Comparison results for the local Nusselt number — h'(0) in the absence of Maxwell parameter, velocity ratio and suction parameter.

Nt Khan and Pop [12] Present study -h (0) Errors

Nb = 0.1 Nb = 0.1

0.1 0.9524 0.9523 0.0001

0.3 0.5201 0.5200 0.0001

0.5 0.3211 0.3210 0.0001

Figure 1 Velocity profile for different values of A.

the stagnation region increases so the acceleration of the external stream increases which leads to decrease in the thickness of the boundary layer with increase in A. When the stretching velocity cx of the surface exceeds the free stream velocity bx (A < 1) inverted boundary layer structure is formed and for

Figure 2 Temperature profile for different values of A.

Figure 4 Velocity profile for different values of b.

Figure 3 Concentration profile for different values of A.

Figure 5 Temperature profile for different values of b.

A — 1 there is no boundary layer formation because the stretching velocity is equal to the free stream velocity. The temperature and concentration profiles for different values of A with other fixed parameter are presented in Figs. 2 and 3 respectively. It can be seen from these figures that both 0(g) and /(g) decrease with increase in A.

Fig. 4, shows plots for velocity distribution for different values of b. When A < 1 the velocity increases with the increase of b, and hence the boundary layer thickness decreases. Similar effect can be found when A > 1. Converse part can be seen when increase of b, shows decrease in temperature 0 and concentration / profiles see the Figs. 5 and 6. This is exactly opposite effects for regular Maxwell fluid in the absence of nanoparticles.

The graph of velocity profile f versus g for different values of k is plotted in Fig. 7. It is found that for a fixed value of A < 1 the velocity decreases with the increase of k. The velocity profiles tends asymptotically to the horizontal axis, and the non-dimensional velocities absorbs maximum at the wall. It is fact that suction stabilizes the boundary layer growth. At A > 1 the velocity increases with the increase of k. It is also interesting to note that there is a significant enhancement of temperature and concentration at the wall, when it is porous.

Figure 6 Concentration profile for different values of b.

The temperature and concentration profiles start to decrease monotonically from the very beginning which can be seen from Figs. 8 and 9.

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

A = 2.0

/ X = 0.1,0.5, 1.0

/ Le= 1.0, Nb = 0.1, Nt = 0.1, Pr = 3.0t p = 0.2 1

X = 0.1, 0.5, 1.0

A = 0.2

Figure 7 Velocity profile for different values of k.

Figure 10 Temperature profile for different values of Nb.

Figure 8 Temperature profile for different values of k.

Figure 11 Concentration profile for different values of Nb.

Figure 9 Concentration profile for different values of k.

Figs. 10 and 11, illustrate the variations of 0 and / with g for various values of Nb. It is found that the increase in the value of Nb is to increase 0(g) in the boundary layer, whereas concentration boundary layer reduces as Nb increases which

Figure 12 Temperature profile for different values of Nt.

thereby enhances the nanoparticles concentration at the sheet. The graph of Nt on the 0 and / profiles is depicted in Figs. 12 and 13. From these plots, it is observed that the effect of

Figure 13 Concentration profile for different values of Nt .

Figure 16 Concentration profile for different values of Pr.

Figure 17 Variation of —h'(0) with Nt for different values of Nb.

Figure 14 Concentration profile for different values of Le.

Figure 15 Temperature profile for different values of Pr.

increasing values of Nt is to increase the temperature and concentration profiles. Fig. 14, displays the effect of Le on concentration profiles. It is noted that the concentration of fluid

Figure 18 Variation of —/' (0) with Nt for different values of Nb.

decreases with increase of Le. Physically this is due to the fact that mass transfer rate increases as Le increases. It also reveals that the concentration gradient at surface of the plate increases.

Table 4 Values of skin friction coefficient —/"(0), Nusselt number — h'(0) and Sherwood number —/'(0) for different values of the physical parameters with A = 0.2.

b k Le Nb Nt Pr -f '(0) -/' '(0) -h '(0)

0 0. 2 1 0.1 0.1 3 1.1465 0.9255 1.8027

0.1 1.1605 0.9264 1.8025

0.2 1.1745 0.9274 1.8024

0.2 0 1 0.1 0.1 3 0.9333 0.7313 0.9265

0.1 0.9814 0.7730 1.0848

0.5 1.1745 0.9274 1.8024

0.2 0.5 1 0.1 0.1 3 1.1745 0.9274 1.8024

2 1.1745 2.8591 1.6933

3 1.1745 4.6009 1.6406

0.2 0.5 1 0.1 0.1 3 1.1745 0.9274 1.8024

0.3 1.1745 2.0012 1.2903

1.0 1.1745 2.2960 0.3260

0.2 0.5 1 0.1 0.01 3 1.1745 2.1467 1.9526

0.1 1.1745 0.9274 1.8024

0.3 1.1745 -1.0223 1.5126

0.2 0.5 1 0.1 0.1 2 1.1745 0.5903 1.4193

3 1.1745 2.1467 1.9526

7 1.1745 2.6723 2.5741

Table 5 Values of skin friction coefficient /''(0), Nusselt number — h' (0) and Sherwood number —/'(0) for different values of the physical parameters with A = 2.0.

b k Le Nb Nt Pr f '(0) -/' '(0) -h '(0)

0 0.2 1 0.1 0.1 3 2.3101 1.3887 2.0733

0.1 2.4135 1.3946 2.0783

0.2 2.5208 1.4005 2.0833

0.2 0 1 0.1 0.1 3 2.1145 1.1966 1.2519

0.1 2.1961 1.2425 1.4047

0.5 2.5208 1.4005 2.0833

0.2 0.5 1 0.1 0.1 3 2.5208 1.4005 2.0833

2 2.5208 3.3331 1.9733

3 2.5208 5.0605 1.9174

0.2 0.5 1 0.1 0.1 3 2.5208 1.4005 2.0833

0.3 2.5208 2.4373 1.4901

1.0 2.5208 2.7057 0.3756

0.2 0.5 1 0.1 0.01 3 2.5208 2.5340 2.2752

0.1 2.5208 1.4005 2.0833

0.3 2.5208 -0.1647 1.7155

0.2 0.5 1 0.1 0.1 2 2.5208 1.0058 1.7290

3 2.5208 1.4005 2.0833

7 2.5208 3.2781 2.7331

Temperature and concentration profiles for the selected values of Pr are plotted in Figs. 15 and 16. The graph depicts that the temperature decreases when the values of Pr increase. This is due to the fact that a higher Pr fluid has relatively low thermal conductivity, which reduces conduction and thereby the thermal boundary layer thickness, and as a result, temperature decreases. We note that the Nb, Nt, Le and Pr have no influence on the flow field, which is clear from Eq. (9). Figs. 17

and 18 show that the variations —0'(0) and —/'(0). From these graph we observe that —0' (0) decreases with the increase of Nt —/' (0) increases with the increase of Nt. It is also noted that in the absence of b, A and k our results are similar to those by Khan and Pop (Newtonian fluid). From Figs. 17 and 18 one can see Nusselt and Sherwood number for that non-Newtonian fluid is higher than the Newtonian fluid.

From Table 4 one can see that the skin friction coefficient /''(0) is negative at A < 1. Physically, negative value of/''(0) means the surface exerts a drag force on the fluid, and positive value means the opposite. This is not surprising since in the present problem, we consider the case of a stretching sheet, which induces the flow. Negative value of 0 (0) means that the heat flows from the fluid to the solid surface. This is not surprising since the fluid is hotter than the solid surface. In the future, the same problem can be extended to power law fluid see [38,39].

5. Conclusions

In the present investigation, the influence of different parameters on the velocity, temperature and concentration profiles is illustrated and discussed. The numerical results give a view toward understanding the response characteristics of the stagnation point flow of a Maxwell fluid in the presence of nanoparticles and suction. It is found that boundary layer is formed when A > 1 on the other hand inverted boundary layer is formed when A < 1. Some results of thermal characteristics at the wall are usually analyzed from the numerical results and the same are documented in Tables 4 and 5. Analyzing this table, it reveals that the effects of increasing the values of b are to increase —/''(0) and decrease the —/' (0), whereas for A < 1 —0(0) increases but decreases at A > 1. Also one can observe that there is no change in —/''(0), when Le, Nb, Nt and Pr vary.

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