Viscous dissipation effects on unsteady mixed convective stagnation point flow using Tiwari-Das nanofluid model

Results in Physics 7 (2017) 280–287

Contents lists available at ScienceDirect

Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics

Viscous dissipation effects on unsteady mixed convective stagnation point flow using Tiwari-Das nanofluid model F. Mabood a,⇑, S.M. Ibrahim b, P.V. Kumar b, W.A. Khan c a

Department of Mathematics, University of Peshawar, 25120, Pakistan Department of Mathematics, GITAM University, Vishakhapatnam, Andhra Pradesh 530045, India c Department of Mechanical and Industrial Engineering, Majmaah University, Majmaah 11952, Saudi Arabia b

a r t i c l e

i n f o

Article history: Received 17 November 2016 Accepted 24 December 2016 Available online 29 December 2016 Keywords: Nanofluid Stagnation-point flow Shrinking/stretching sheet Viscous dissipation

a b s t r a c t A mathematical model has been developed using Tiwari-Das model to study the MHD stagnation-point flow and heat transfer characteristics of an electrically conducting nanofluid over a vertical permeable shrinking/stretching sheet in the presence of viscous dissipation. Formulated partial differential equations are converted into a set of ordinary differential equations using suitable similarity transformation. Runge-Kutta-Fehlberg method with shooting technique is applied to solve the resulting coupled ordinary differential equations. The profiles for velocity, temperature, skin friction coefficient and local Nusselt number for various parameters are displayed through graphs and tabular forms. In this problem, we considered two types of nanoparticles, namely, copper (Cu) and Alumina (Al2O3) with water as base fluid. Ó 2016 Published 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/).

Introduction Due to the nature of stability and free from additional problems like sedimentation, erosion, additional pressure drop and nonNewtonian behaviour, nanofluids have been widely used in many scientific and industrial applications. They also play a vital role in the enhancement of thermal conductivity of conventional fluids like water, oil, toluene, etc. Choi [1] was the first researcher who used the term ‘nanofluid’ to describe a new class of fluid. Masuda et al. [2] discussed the phenomena of enhancement of thermal conductivity by nanofluids. Later many researchers [3–7] focused on this problem to study the characteristics of heat transfer in nanofluids under various physical situations. Chamka and Aly [8] presented the effects of magnetic field, heat generation or absorption and suction or injection on boundary layer flow of nanofluid over a permeable vertical plate. Ahmad and Pop [9] studied the characteristics of mixed convection flow of nanofluids over a vertical flat plate embedded in a porous medium. Rana and Bhargava [10] investigated the effect of temperature dependent heat source/sink on mixed convection flow of nanofluid along the vertical plate. Tiwari and Das [11] proposed a new mathematical model for nanofluids by taking solid volume fraction into account. This

⇑ Corresponding author. E-mail addresses: [email protected] (F. Mabood), ibrahimsvu@gmail. com (S.M. Ibrahim), [email protected] (P.V. Kumar), wkhan1956@ gmail.com (W.A. Khan).

model was successfully adopted by Pop et al. [12,13], Abu-Nada [14] and Lee et al. [15] in their papers. Stagnation-point flow has been studied extensively due to its industrial applications such as cooling of electronic devices by fans, cooling of nuclear reactors, etc. Hiemenz [16] initiated the problem of steady two dimensional stagnation-point flow. Later this problem has been extended by Mahapatra and Gupta [17], Bhattacharya et al. [18] and Mabbod and Khan [19]. Bachok et al. [20,21] discussed the steady two dimensional stagnation point flow of a nanofluid over a permeable shrinking/stretching sheet. In recent years, many researchers focused on the problem of viscous dissipation effect on free, forced and mixed convective flows of nanofluids. Motsumi and Makinde [22] studied the radiation and viscous dissipation effects on boundary layer flow of nanofluids over a permeable moving flat plate. The effects of viscous dissipation and variable magnetic field on mixed convection MHD flow of nanofluid over a non-linear stretching sheet have been studied by Habibi Matin et al. [23]. Motivated by the investigation mentioned above, the aim of the present work is to investigate the effects of viscous dissipation on mixed convection stagnation point nanofluid flow past a stretching/shrinking sheet. The nonlinearity of basic equations associated with their inherent mathematical difficulties has led us to use numerical method. Thus the transformed dimensionless governing equations are solved numerically by using the Runge–Kutta–Fehl berg method (RKF) method along with shooting technique. To

http://dx.doi.org/10.1016/j.rinp.2016.12.037 2211-3797/Ó 2016 Published 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/).

281

F. Mabood et al. / Results in Physics 7 (2017) 280–287

Nomenclature a; b; c A B Cf g k M Nux Gr x Rex Pr Ec qw T T w ðx; tÞ T1 u; v V1 x; y U 1 ðx; tÞ uw ðx; tÞ f ðgÞ

Greek symbols thermal diffusivity b thermal expansion coefficient r electrical conductivity e velocity ratio parameter / nanoparticle volume fraction g similarity variable hðgÞ dimensionless temperature k buoyancy or mixed convection parameter l dynamic viscosity m kinemaic viscosity q fluid density sw wall sher stress w stream function

constants unsteadiness parameter magnetic field skin friction coefficient acceleration due to gravity thermal conductivity magnetic parameter local Nusselt number local Grashof number local Reynolds number Prandtl number Eckert number Surface heat flux fluid temperature surface temperature ambient temperature velocity components wall transient parameter Cartesian coordinates free stream velocity surface velocity dimensionless stream function

a

Subscripts w condition at the surface of the plate 1 ambient condition f base fluid nf nanofluid s soild

the best of our best knowledge such a study is not investigated in the scientific literature. Mathematical formulation Consider an unsteady two dimensional mixed convection flow of viscous incompressible and electrically conducting nanofluid near the stagnation point on a vertical permeable shrinking/ stretching flat plate. Rectangular co-ordinate system ðx; yÞ is considered for the present problem. The flow is subjected to a transverse magnetic felid of strength B ¼ B0 1 is applied along the

The thermophysical properties of the base fluid and the nanoparticles Oztop and Abu-Nada [4] are shown in Table 1. Under these assumptions and by taking Tiwari-Das model into consideration, the governing equations are

@u @ v þ ¼ 0; @x @y

ð1Þ

@u @u @u lnf @ 2 u 1 dp rB2 þu þv ¼   u @t @x @y qnf @y2 qnf dx qnf þ

ð1c tÞ2

normal direction to the plate. In this problem we assumed that: ax The velocity of the ambient fluid is U 1 ðx; tÞ ¼ 1c where a > 0 t is the strength of the stagnation point flow and c is a constant with ct < 1. Both a and c have dimension ‘‘time1”. The shrinking/ bx stretching velocity uw ðx; tÞ ¼ 1ct where b < 0 and b > 0 corresponds to shrinking and stretching case respectively. a; b and c have dimension ‘‘time1”. T0x 1. The surface temperature T w ðx; tÞ ¼ T 1 þ ð1ctÞ 2 where T 1 is the

ambient fluid temperature and T 0 is the characteristic temperature. 2. Nanoparticles and the base fluid are in thermal equilibrium and no slip occurs between them. 3. V w is the uniform surface mass flux with V w < 0 for suction and V w > 0 for injection.

/qs bs þ ð1  /Þqf bf

qnf

lnf @T @T @T @2T þu þv ¼ anf 2 þ @t @x @y @y ðqC p Þnf




gðT  T 1 Þ;

ð2Þ

2 @u ; @y

ð3Þ

subject to the boundary conditions

t < 0 : u ¼ v ¼ 0;

T ¼ T1

for any x; y;

t P 0 : u ¼ uw ðx; tÞ;

v ¼ V w ðtÞ

u ! U 1 ðx; tÞ;

T ! T1

as y ! 1:

T ¼ T w ðx; tÞ at y ¼ 0;

ð4Þ

Following the generalized Bernoulli’s equation, in free-stream, Eq. (2) written as

dU 1 dU 1 1 dp rB2  U : þ U1 ¼ dt dx qnf dx qnf 1

ð5Þ

Table 1 Thermophysical properties of the base fluid and the nanoparticles. Base fluid and nanoparticles

Molecular formula

C p (J/kg K)

q (kg/m3)

k (W/mK)

a  107 (m2/s)

b  105 (l/K)

Water Aluminum oxide Copper

H2 O Al2 O3 Cu

4179 765 385

997.1 3970 8933

0.613 40 400

1.47 131.7 1163.1

21 0.85 1.67

282

F. Mabood et al. / Results in Physics 7 (2017) 280–287

subject to the boundary conditions

Substituting (5) in (2), it becomes

@u @u @u lnf @ u dU 1 dU 1 rB þu þv ¼ þ ðU  uÞ þ U1 þ qnf 1 @t @x @y qnf @y2 dt dx 2

2

þ

/qs bs þ ð1  /Þqf bf

qnf

gðT  T 1 Þ:

ð6Þ

Here u and v are the velocity components along the x and y directions, respectively. T is the temperature of the nanofluid, r is electrical conductivity, bf and bs are the thermal expansion coefficients of the base fluid and nanofluid, g is the acceleration due to gravity, / is the solid volume fraction of the nanofluid, qf is the density of the base fluid, qs is the density of the solid particle. In the above expression, lnf is the viscosity of the nanofluid, anf is the thermal diffusivity of the nanofluid, qnf is the density of the nanofluid which are given by lf lnf ¼ ð1/Þ 2:5 ; qnf ¼ ð1  /Þqf þ /qs ; ðqC p Þnf ¼ ð1  /ÞðqC p Þf þ /ðqC p Þs ;

where lf is the viscosity of the base fluid, kf and ks are the thermal

a g¼ v f ð1  ctÞ

12

 av 12 f y; w ¼ x f ðgÞ; 1  ct

T  T1 hðgÞ ¼ Tw  T1

ð8Þ

ax 0 0 f ðgÞ ¼ U 1 ðx; tÞf ðgÞ; v 1  ct

  1  f 000 þ ff 00  f 02 þ 1 þ A 1  f 0  gf 00 2 ð1  /Þ   1 0 1  / þ / qqs bbs M  ð1  f 0 Þ þ @ f  f Akh ¼ 0 þ 1  / þ / qqs 1  / þ / qqs 

1

knf kf


 f 00  0 h þ Pr f ðqcp Þs

ð1  /Þ þ / ðqcp Þ

ð9Þ

f


 h  1 0  PrA 2h þ g h 2 h0 

f

þPr

f

number and k is the buoyancy or mixed convection parameter, which is defined as:

Grx Re2x

¼ gbf

1 2   Ecðf 00 Þ ¼ 0 ðqc Þ ð1  /Þ2:5 ð1  /Þ þ / ðqcpp Þs f

b ; a2

Grx ¼ gbf ðT w  T 1 Þ

x3

t2f

;

Rex ¼

U1 x

tf

;

where Grx and Rex are respectively the local Grashof number and Reynolds number. k is a constant with k > 0 corresponding to assisting flow, k ¼ 0 represents the case when the buoyancy force is absent. The physical quantities of interest are the skin friction coefficient C f and the local Nusselt number Nux are defined as

Cf ¼

sw qf U 2 =2

Nux ¼

xqw ; kf ðT w  T 1 Þ

sw


 @u ¼ lnf ; @y y¼0

sw and the wall heat flux qw are given by:

qw ¼ knf


 @T : @y y¼0

Using similarity variables (8), we get

1 ð1  /Þ2:5

0

f ; ð0Þ;

1

½Rex 2 Nux ¼ 

knf 0 h ð0Þ: kf

The nonlinear and coupled Eqs. (9) and (10) with boundary conditions (11) are solved numerically using Runge-Kutta-Fehlberg method with shooting technique for different values of parameters. The effects of the emerging parameters on the dimensionless velocity, temperature, skin friction coefficient, the rate of heat transfer are investigated. The step size and convergence criteria were chosen to be 0.001 and 106 respectively. The asymptotic boundary conditions in Eq. (6) were approximated by using a value of 10 for gmax as follows:

gmax ¼ 10; f 0 ð10Þ ¼ 1; hð10Þ ¼ 0; :

1  / þ / qqs f

f

is the Eckert number. The velocity ratio parame-

Solution method

 v a 12 ¼ f ðgÞ; 1  ct

where prime denotes differentiation with respect to g.  v a 12 Here, we take V w ðtÞ ¼ 1ct V w ,where dimensionless constant V w denotes the transpiration rate with V w > 0 for suction, V w < 0 for injection and V w ¼ 0 for an impermeable surface. Making use of Eq. (8), Eqs. (2) to (4) can be written as

2:5

is the magnetic parameter, A ¼ ac unsteadies parame-

u2w ðC p Þf ðT w T 1 Þ

C f ½Rex 2 ¼

Using Eq. (8), the velocity components can be written as

u¼

rB20 qf a

ð11Þ

ter e is defined as the ratio of stretching rate of the sheet and strength of the stagnation point flow with e > 0; e < 0 and e ¼ 1 corresponds stretching, shrinking sheets and the flow with no boundary layer ðuw ¼ U 1 Þ, respectively, while e ¼ 0 is the planar v stagnation flow towards a stationary sheet. Pr ¼ af is the Prandtl

1

@w @w and v ¼  : @y @x

hð0Þ ¼ 1 at g ¼ 0

hð1Þ ¼ 0 as g ! 1

where M ¼ ter, Ec ¼

b ¼ e; a

where the wall stress

here, g is the similarity variable, w is the stream function, f and h are dimensionless quantities. The continuity equation (1) is being satisfied by the stream function w as

u¼

0

ð7Þ

conductivity of the base fluid and nanoparticle, knf is the effective thermal conductivity of the nanofluid approximated by the Maxwell-Garnett model. To transform the momentum and energy equations into a set of ordinary differential equations, we define the following transformation as

f ð0Þ ¼

f ð1Þ ¼ 1;

k¼

f Þ2/ðkf ks Þ anf ¼ ðqkCnfp Þnf ; kknff ¼ ðkðkssþ2k þ2kf Þþ/ðkf ks Þ




0

f ð0Þ ¼ V w ;

ð10Þ

ð12Þ

This ensures that all numerical solutions approached the asymptotic values correctly. To validate the mathematical model, comparisons have been made with previously published data for skin friction coefficient and Nusselt number in Tables 2 and 3, and favourable agreement is observed. Results and discussion In order to get a clear insight of the problem, numerical computations have been carried out for various parameters such as

283

F. Mabood et al. / Results in Physics 7 (2017) 280–287 Table 2 Comparison of skin friction coefficient and Nusselt number for different values of Pr when k ¼ 1 and A ¼ M ¼ e ¼ / ¼ V w ¼ 0:. Pr

0.7 1 7 10 20 40 50

00

h0 ð0Þ

f ð0Þ Ishak et al. [24]

Dinarvand et al. [25]

Present

Ishak et al. [24]

Dinarvand et al. [25]

Present

1.7063 1.6754 1.5179 1.4928 1.4485 1.4101 1.3989

1.7063 1.6754 1.5178 1.4927 1.4482 1.4104 1.3986

1.70632 1.67543 1.51791 1.49283 1.44848 1.41005 1.39893

0.7641 0.8708 1.7224 1.9446 2.4576 3.1011 3.3415

0.7641 0.8708 1.7225 1.9444 2.4573 3.1014 3.3418

0.76406 0.87077 1.72238 1.94461 2.45758 3.10109 3.34145

Table 3 Comparison of skin friction coefficient and Nusselt number for different values of /; k and Quantity

/

e when A ¼ V w ¼ 0:5; M ¼ 1; Ec ¼ 0: (In case of Copper-Water).

Dinarvand et al. [25] k¼1

Present k ¼ 1

k¼1

k ¼ 1

e ¼ 0:5

e ¼ 0:5

e ¼ 0:5

e ¼ 0:5

e ¼ 0:5

e ¼ 0:5

e ¼ 0:5

e ¼ 0:5

Cf 1=2 Rex

0.0 0.1 0.2

2.9297 4.1872 5.6224

1.2002 1.7240 2.3181

2.5561 3.7751 5.1679

0.9207 1.4124 1.9723

2.9249 4.1843 5.6188

1.1993 1.7235 2.3176

2.5505 3.7715 5.1634

0.9197 1.4119 1.9718

Nux 1=2 Rex

0.0 0.1 0.2

4.0445 4.4249 4.8141

5.2995 5.7506 6.2389

3.9922 4.3732 4.7628

5.2774 5.7267 6.2138

4.0431 4.4241 4.8130

5.2994 5.7505 6.2388

3.9907 4.3723 4.7614

5.2773 5.7266 6.2137

nanoparticle volume fraction ð/Þ, magnetic parameter ðMÞ, wall transpiration parameter ðV w Þ, mixed convection parameter ðkÞ, unsteadiness parameter ðAÞ, velocity ratio parameter ðeÞ and Eckert number ðEcÞ. In this discussion all the graphs and Tabular forms are taken for two nanofluids, namely Copper and Alumina. The Prandtl number corresponding to the base fluid is kept constant at Pr ¼ 6:2. Fig. 1(a) and (b) depict the effect of nanoparticle volume fraction / on velocity distribution in the presence and absence of magnetic parameter M. It is observed that there is a sharp rise in the velocity within the layer g < 1:8 and then it becomes uniform as g ! 1. Further, we observed that Al2 O3 water nanofluid exhibits relatively less velocity than that of Cu water nanofluid. Figs. 2(a) and (b) show the influence of suction/injection parameter V w for zero and non zero mixed convection parameter k. The effect of suction or injection is to increase the velocity up to certain level of g ðg < 1:8Þ and then it becomes uniform as g ! 1. Figs. 3(a) and (b) illustrate the behavior of velocity profile under the influence of velocity ratio parameter e for steady ðA ¼ 0:0Þ and unsteady ðA ¼ 5:0Þ cases. When e < 1 then there is a rapid increase in the velocity up to certain level of g and then it becomes uniform

as g increases. Similarly if e > 1 there is a rapid decrease in the velocity and then it becomes uniform as g increases. The advancement of velocity with nanoparticle volume fraction / in the presence and absence of viscous dissipation ðEcÞ is demonstrated in Fig. 4(a) and (b). Temperature increases marginally with the increases of unsteady parameter A and magnetic parameter M as shown in the Figs. 5(a) and (b). The effect of mixed convection parameter k and velocity ration parameter e on skin friction coefficient with the presence and absence of magnetic parameter M is shown in Fig. 6(a) and (b). We have seen a fall in the skin friction coefficient as k increases along with M. In Figs. 7(a) and 7(b) the combined effect of nanoparticle volume fraction / and unsteadiness parameter A for different values of wall transpiration parameter V w on skinfriction factor is explored. Here we noticed that for Nusselt number increases about nanoparticle volume fraction / corresponds to both parameters unsteadiness parameter A and wall transpiration parameter V w . Nusselt number decreases as magnetic field parameter M and Eckert number Ec increasing in presence of velocity ration parameter e as shown in Fig. 8(a) and (b). The effect of Eckert number Ec

0

Fig. 1. Effect of / and M on f ðgÞ.

284

F. Mabood et al. / Results in Physics 7 (2017) 280–287

0

Fig. 2. Effect of V w and k on f ðgÞ.

Fig. 3. Effect of A and

e on f 0 ðgÞ.

Fig. 4. Effect of / and Ec on hðgÞ.

F. Mabood et al. / Results in Physics 7 (2017) 280–287

285

Fig. 5. Effect of A and M on hðgÞ.

Fig. 6. Effects of e, M and k on skin friction coefficient.

Fig. 7. Effects of /, A and V w on skin friction coefficient.

along with velocity ratio parameter e on Nusselt number is given in Fig. 9(a) and (b). It is clear that Nusselt number declines as Ec increases along with e.

To assess the present method, comparison is made with the results of Ishak et al. [24] and Dinarvand et al. [25] as shown in Tables 2 and 3. Table 4 presents the values of skin friction coeffi-

286

F. Mabood et al. / Results in Physics 7 (2017) 280–287

Fig. 8. Effects of e, M and Ec on local Nusselt number.

Fig. 9. Effects of /, V w and A on local Nusselt number.

Table 4 Values of skin friction coefficient and Nusselt number for different values of /; Ec and V w when A ¼ M ¼ e ¼ k ¼ 0:5:. Quantity

/

Copper-Water

Alumina-Water

Ec ¼ 0:1

Ec ¼ 1

Ec ¼ 0:1

Ec ¼ 1

Vw ¼ 0

Vw ¼ 1

Vw ¼ 0

Vw ¼ 1

Vw ¼ 0

Vw ¼ 1

Vw ¼ 0

Vw ¼ 1

Cf 1=2 Rex

0.0 0.1 0.2

0.9452 1.3468 1.8132

1.2050 1.8664 2.5893

0.9573 1.3606 1.8291

1.2161 1.8803 2.6062

0.9452 1.1799 1.4719

1.2050 1.5356 1.9208

0.9573 1.1929 1.4861

1.2161 1.5482 1.9351

Nux 1=2 Rex

0.0 0.1 0.2

3.2591 3.7274 4.2184

7.5791 7.8188 8.1008

2.5308 2.5564 2.5381

6.2824 5.6868 5.0743

3.2590 3.6896 4.1379

7.5791 7.7617 7.9814

2.5307 2.7263 2.8922

6.2824 6.0844 5.8763

cient and Nusselt number for different values of nanoparticle volume fraction ð/Þ, wall transpiration parameter ðV w Þ and Eckert number ðEcÞ. Skin friction increases with the increase of /, V w and Ec.

 Temperature decreases with unsteadiness parameter ðAÞ and magnetic parameter ðMÞ.  Skin friction coefficient increases with the increase of nanoparticle volume fraction ð/Þ, wall transpiration parameter ðV w Þ and Eckert number ðEcÞ.

Conclusions Some important observations of this study can be summarized as follows:  Velocity rises as nanoparticle volume fraction ð/Þ, magnetic parameter ðMÞ, wall transpiration parameter ðV w Þ, mixed convection parameter ðkÞ, unsteadiness parameter ðAÞ, velocity ratio parameter ðeÞ and Eckert number ðEcÞ.

References [1] Choi US. Enhancing thermal conductivity of fluids with nanoparticles. The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, CA. ASME; 1995. p. 99–105. FED 231/MD 66. [2] Masuda H, Ebata A, Teramae K, Hishinuma N. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 1993;7:227–33.

F. Mabood et al. / Results in Physics 7 (2017) 280–287 [3] Xuan Y, Roetzel W. Conceptions for heat transfer correlation of nanofluids. Int J Heat Mass Transf 2000;43(19):3701–7. [4] Oztop HF, Abu-Nada E. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int J Heat Fluid Flow 2008;29:1326–36. [5] Xuan Y, Li Q. Heat transfer enhancement of nanofluid. Int J Heat Fluid Flow 2000;21(1):58–64. [6] Hwang KS, Jang SP, Choi SUS. Flow and convective heat transfer characteristics of water-based Al2O3 nanofluids in fully developed laminar flow regime. Int J Heat Mass Transf 2009;52(1–2):193–9. [7] Qiang Li, Yimin Xuan. Convective heat transfer and flow characteristics of Cuwater nanofluid. Sci China Ser E: Technol Sci 2002;45(4):408–16. [8] Chamka AJ, Aly AM. MHD free convection flow of a nanofluid past a vertical plate in the presence of heat generation or absorption effects. Chem Eng Commun 2010;198:425–41. [9] Ahmad S, Pop I. Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Int Commun Heat Mass Transf 2010;37:987–91. [10] Rana P, Bhargava R. Numerical study of heat transfer enhancement in mixed convection flow along a vertical plate with heat source/sink utilizing nanofluids. Commun Nonlinear Sci Numer Simul 2011;16:4318–34. [11] Tiwari RK, Das MK. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transf 2007;50:2002–18. [12] Sheremet MA, Grosan T, Pop I. Free convection square cavity filled with porous medium saturated by nanofluid using Tiwari-Das nanofluid model. Tranp Porous Media 2015;106:595–610. [13] Ahmad S, Rohni AM, Pop I. Blasius and Sakiadis problems in nanofluids. Acta Mech 2011;218:195–204. [14] Au-Nada E. Applications of nanofluids for heat transfer enhancement of separated flow encountered in a backward facing step. Int J Heat Fluid Flow 2008;29:242–9.

287

[15] Muthtamilselvan M, Kandaswamy P, Lee J. Heat transfer enhancement of copper-water nanofluids in a lid-driven enclosure. Commun Nonlinear Sci Numer Simul 2010;15:1501–10. [16] Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingler’s Polytech J 1911;326:321–4. [17] Mahapatra TR, Gupta AS. Heat transfer in stagnation point flow towards a stretching sheet. Heat Mass Transf 2002;38:517–21. [18] Bhattacharya K, Mukhopadhay Swati, Layek GC. Slip effects on boundary layer flow and heat transfer towards a shrinking sheet. Int J Heat Mass Transf 2011;54:308–13. [19] Mabood F, Khan WA. Approximate analytic solutions for influence of heat transfer on MHD stagnation point flow in porous medium. Comput Fluids 2014;100:72–8. [20] Bachok N, Ishak A, Pop I. Stagnation-point flow over a stretching sheet in a nanofluid. Nanoslcae Res Lett 2011;6:2–10. [21] Bachok N, Ishak A, Nazar R, Senu N. Stagnation-point flow over a permeable stretching/shrinking sheet in a copper-water nanofluid. Boundary Value Prob 2013;39:2–10. [22] Motsumi TG, Makinde OD. Effects of thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable moving flat plate. Phys Scr 2012;86:1–11. [23] Habibi Matin M, Dehsara M, Abbassi A. Mixed convection MHD flow of nanofluid over a non-linear stretching sheet with effects of viscous dissipation and variable magnetic field. Mechanika 2012;18:415–23. [24] Ishak A, Nazar R, Bachok N, Pop I. MHD mixed convection flow near the stagnation-point on a vertical permeable surface. Phys A 2010;389:40–6. [25] Dinarvand S, Hosseini R, Pop I. Homotopy analysis method for unsteady mixed convective stagnation-point flow of a nanofluid using Tiwari-Das nanofluid model. Int J Numer Meth Heat Fluid Flow 2016;26:40–62.