Scholarly article on topic 'Effects of magnetic field and partial slip on unsteady axisymmetric flow of Carreau nanofluid over a radially stretching surface'

Effects of magnetic field and partial slip on unsteady axisymmetric flow of Carreau nanofluid over a radially stretching surface Academic research paper on "Mathematics"

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{"Unsteady axisymmetric flow" / "MHD Carreau nanofluid" / "Velocity slip condition" / "Convective boundary condition" / "Numerical solutions"}

Abstract of research paper on Mathematics, author of scientific article — M. Azam, M. Khan, A.S. Alshomrani

Abstract The unsteady magnetohydrodynamic (MHD) axisymmetric flow of Carreau nanofluid over a radially stretching sheet is investigated numerically in this article. Recently devised model for nanofluid namely Buongiorno’s model incorporating the effects of Brownian motion and thermophoresis is adopted here. Additionally, partial velocity slip and convective boundary condition are considered. Mathematical problem is modeled with the help of momentum, energy and nanoparticles concentration equations using suitable transformation variables. The numerical solutions for the transformed highly nonlinear ordinary differential equations are computed for both shear thinning and shear thickening fluids. For numerical computations, an effective numerical approach namely the Runge-Kutta Felhberg integration scheme is adopted. Effects of involved controlling parameters on the temperature and nanoparticles concentration are examined. Numerical computations for the local Nusselt number and local Sherwood number are also performed. It is interesting to note that the strong magnetic field has the tendency to enhance the thermal and concentration boundary layer thicknesses. Additionally, the local Nusselt and Sherwood numbers depreciate by improving values of unsteadiness parameter, magnetic parameter, velocity slip parameter and thermophoresis parameter in shear thickening and shear thinning fluids.

Academic research paper on topic "Effects of magnetic field and partial slip on unsteady axisymmetric flow of Carreau nanofluid over a radially stretching surface"

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Effects of magnetic field and partial slip on unsteady axisymmetric flow of Carreau nanofluid over a radially stretching surface

PII: DOI:

Reference:

M. Azam, M. Khan, A.S. Alshomrani

S2211-3797(17)30965-8 http://dx.doi.org/10.1016/j-rinp.2017.07.025 RINP 798

To appear in:

Results in Physics

Please cite this article as: Azam, M., Khan, M., Alshomrani, A.S., Effects of magnetic field and partial slip on unsteady axisymmetric flow of Carreau nanofluid over a radially stretching surface, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.07.025

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Effects of magnetic field and partial slip on unsteady axisymmetric flow of

2 Carreau nanofluid over a radially stretching surface

3 M. Azam"'1, M. Khan" and A.S. Alshomranib

4 "Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan

5 ^Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah

6 21589, Saudi Arabia

7 Abstract: The unsteady magnetohydrodynamic (MHD) axisymmetric flow of Carreau

8 nanofluid over a radially stretching sheet is investigated numerically in this article. Recently de-

9 vised model for nanofluid namely Buongiorno's model incorporating the effects of Brownian motion

10 and thermophoresis is adopted here. Additionally, partial velocity slip and convective boundary

11 condition are considered. Mathematical problem is modeled with the help of momentum, energy

12 and nanoparticles concentration equations using suitable transformation variables. The numeri-

13 cal solutions for the transformed highly nonlinear ordinary differential equations are computed for

14 both shear thinning and shear thickening fluids. For numerical computations, an effective numeri-

15 cal approach namely the Runge-Kutta Felhberg integration scheme is adopted. Effects of involved

16 controlling parameters on the temperature and nanoparticles concentration are examined. Numer-

17 ical computations for the local Nusselt number and local Sherwood number are also performed.

18 It is interesting to note that the strong magnetic field has the tendency to enhance the thermal

19 and concentration boundary layer thicknesses. Additionally, the local Nusselt and Sherwood num-

20 bers depreciate by improving values of unsteadiness parameter, magnetic parameter, velocity slip

iprovine lophores

21 parameter and thermophoresis parameter in shear thickening and shear thinning fluids.

22 Keywords: Unsteady axisymmetric flow, MHD Carreau nanofluid, velocity slip condition,

23 convective boundary condition, numerical solutions.

[ntroduction

25 Recently, many explorations have been made to explore the nanofluid technology in the area

26 of heat transfer enhancement and new generation cooling technology. This covers the limi-

1 Correspondance author's E-mail: mazam@math.qau.edu.pk

tation of several conventional heat transfer fluids such as ethylene glycol mixture, water and

28 oil. Due to tiny size of nanoparticles and the low volume fraction nanoelements required

29 for the enhancement of thermal conductivity, nanofluids are very stable and have no extra

30 problems [1], like as additional pressure drop, sedimentation and non-Newtonian behavior.

31 Several schemes have been attempted to develop modern heat transfer fluids by suspending

32 nanoparticles in liquids [2]. The broad range of present and future applications for nanoflu-

33 ids can be seen in the recent book [3]. The concept of nanofluid was devised by Choi and

34 Eastman [4] as rout for improving the performance of the rate of heat transfer in liquids.

35 The nanometer size elements possess unique chemical and physical characteristics. They

36 can flow easily in micro-channels without clogging. Due to modern developed technology,

37 nanofluids involve broad applications in distinct aspects such as thermal absorption, trans-

38 poration, heating and cooling processing of energy, electronic devices, nuclear reactor and so

39 forth. Transport characteristics of nanofluids can be analyzed by two models proposed by

40 Buongiorno [5] and Tiwari and Das [6]. Whereas, we are adopting Buongiorno's model to

41 analyze the heat transfer characteristics in nanofluids. Buongiorno proposed a model which

42 ignores the limitations of dispersion and homogeneous models. He demonstrated the seven

43 slip mechanisms that generate a parallel velocity between the nanoparticles and base fluid.

44 These include Magnus effect, inertia, Brownian diffusion, fluid drainage, thermophoresis,

45 gravity and diffusiophoresis. He concluded that the Brownian diffusion and thermophore-

46 sis are two important slip mechanisms in nanofluids. On the basis of these features, he

47 reported two-component four-equation nonhomogeneous equilibrium model regarding mass,

48 momentum and heat transport in nanofluids. From all these features, several investigations

49 on nanofluids were reported by many authors. The study of nanofluid flow past a stretching

50 surface was reported by Khan and Pop [7]. They noticed that reduced Sherwood number is

51 a growing function of higher Prandtl number. Turkyilmazoglu [8] reported the analytical

52 solutions of the single and multi-phase models in nanofluids. He noticed that the water

53 based nanofluid in the presence of nanoparticles Ag is the best heat transferring mixture.

54 Mustafa et al. [9] reported a numerical as well as analytical study for the axisymmetric

55 flow of nanofluid. They observed that enhancement in Schmidt number relates to a thinner

56 nanoparticle volume fraction boundary layer. Khan and Azam [10] reported a numerical

study for unsteady heat and mass transfer in Carreau nanofluid over a stretching sheet.

58 They noted that the local Nusselt and Sherwood numbers are depreciating function of the

59 thermophoresis parameter. The study of MHD Maxwell and Casson fluid was conducted by

60 Kumaran et al. [11]. They examined that there is significance enhance in the rate of heat

61 and mass transfer Maxwell fluid when compared to Casson fluid.

62 On the other hand, Sakiadis [12] seems to be the first amongst the researchers to initiate

63 the work on boundary layer flow over a solid surface and modeled boundary layer equations of

64 two-dimensional axisymmetric flow. Several investigations have been made by the researchers

65 on this problem from different point of view. However, a literature survey reveals that less

66 attention has been paid regarding the axisymmetric flows induced by radially stretching sur-

67 face. Ariel [13] studied the problem of axisymmetric flow over a radially stretching surface

68 and computed the exact, numerical, perturbation and asymptotic solutions of the problem.

69 Martins et al. [14] reported the effects of inertia and shear thinning on the axisymmetric

70 flows of Carreau fluids by adopting Galerkin least square method. Sajid et al. [15] discussed

71 the problem of unsteady axisymmetric flow past a radially stretching surface and computed

72 the series solutions by employing the homotopy analysis method. They noticed that the

73 temperature as well as thermal boundary layer thickness diminish by improving the values

74 of Prandtl number. Sahoo [16] examined the effects of partial slip on axisymmetric flow of

75 viscoelastic fluid induced by a radially stretching surface by adopting the Broyden's method

76 and finite difference method. Khan and Shahzad [17] also discussed the axisymmetric flow

77 in Sisko fluid induced by a radially stretched surface. Turkyilmazoglu [18] presented the

78 multiple solutions for MHD slip flow of viscoelastic fluid. He noted that the shear stress

79 for the first branch of solutions improves with improving magnetic and suction parameters

80 but it diminishes for the second branch. Sheikh and Abbas [19] discussed the effects of

81 heat generation/absorption on MHD flow in the presence of thermophoresis and chemical

82 reactive species. They observed that concentration as well as concentration boundary layer

83 thickness depreciate with improving values of Schmidt number. The study of nanofluid in

84 the presence of magnetic field and mixed convection was conducted by Hsiao [20]. He found

85 that heat transfer effect is significant for larger values of stagnation parameter. Narahari et

86 al. [21] reported a work on unsteady flow of nanofluid over a vertical plate. They noticed

that the local Nusselt number improved for improving values of Brownian motion parameter.

88 Makinde et al. [22] presented the numerical solutions for the problem of stagnation point

89 flow of MHD nanofluid past a stretching surface in the presence of buonyancy effects and

90 convective condition. Babu and Sandeep [23] analyzed the impact of melting phenomena on

91 upper convected Maxwell fluid in the presence of solutal and thermal effects. They observed

92 that the temperature as well as nanoparticle concentration fileds were looking large in New-

93 tonian fluid. Hsiao [24] utilized Carreau nanofluid to improve the activation energy system

94 by adopting parameters control technique. He noted that growing values of Schmidt number

95 relates to higher diffusion effects. The problem of heat generation/absorption and melting

96 phenomena in Falkner-Skan flow of Carreau nanofluid past a wedge was considered by Khan

97 et al. [25]. Their results revealed that the nanoparticles concentration as well as tempera-

98 ture fields were decreased by growing values of melting parameter. Makinde and Aziz [26]

99 reported a numerical study of nannofluid flow over a convectively heated stretching surface.

100 They concluded that nanoparticles concentration field was an enhancing function of the Biot

101 number. The study of Cattaneo-Christov heat flux model for MHD Casson fluid flow over dif-

102 ferent geometries in the presence of non-linear radiation was investigated by Ali and Sandeep

103 [27]. Their study revealed that for the thermal relaxation parameter, heat transfer perfor-

104 mance is high in wedge flow as compared to plate and cone. The work of Micropolar fluid in

105 the presence of viscous dissipation and MHD was conducted by Hsiao [28]. He noticed that

106 temperature is an improving function of the Eckert number. A numerical investigation on

107 unsteady stagnation point flow of Carreau nanofluid past an expanding/contracting cylinder

108 in the presence of time dependent magnetic field and non-linear radiation was reported by

109 Azam et al. [29]. Their investigation revealed that Nusselt number was a depreciating func-

110 tion of the thermophoresis, radiation and unsteadiness parameters. Shateyi and Makinde

111 [30] demonstrated a work on MHD stagnation point flow past a radially disk with convective

112 condition. They noted that heat transfer rate was incresed by increasing the Biot number.

113 Kumaran et al. [31] discussed MHD Maxwell and Casson fluid flows with cross diffusion.

114 They showed that heat and mass transfer rates are high in Maxwell fluid when compared

115 with Casson fluid. Hsiao [32] analyzed the impact of viscous dissipation and thermal ra-

116 diation in Maxwell fluid. He noticed the greater heat transfer effects for larger values of

viscoelastic number. Khan et al. [33] worked on unsteady wedge flow of Carreau nanofluid

118 in the presence of MHD and convective condition Their study revealed that temperature is

119 a growing function of the Brownian motion and thermophoresis parameter. Khan et al. [34]

120 persued the work on mass and heat transfer analysis in third grade nanofluid over a perme-

121 able vertical surface in the presence of partial slip. They presented the numerical solutions

122 of the considered study. Sandeep [35] reported a study on aligned magnetic field on thin film

123 flow of nanofluid by using Runge Kutta method. He demonstrated the numerical solutions

124 of assumed study. Hsiao [36] analyzed the combined effects of thermal radiation and mixed

125 convection on nanofluid. Khan et al. [37] studied MHD Blasius flow of power law nanofluid

126 and presented numerical solutions by spectral relaxation method. They resulted that shear

127 thinning nanofluids have larger skin friction and lower Nusselt number as compared to shear

128 thickening fluids. Raju and Sandeep [38] demonstrated a numerical study of MHD Falkner-

129 Skan flow of Carreau fluid over a wedge. Their study revealed that rate of heat transfer is

130 high in case of accelerating as compared to decelerating case. Ibrahim and Makinde [39]

131 assumed a problem of stagnation pont flow of MHD power law nanofluid past a surface in

132 the presence of velocity and convective conditions. They predicted that skin friction coef-

133 ficient is an enhancing function of power law index n. Kumaran and Sandeep [40] worked

134 on MHD parabolic flow of Williamson and Casson fluids in the presence of nanofluid. They

135 noted that heat and mass transfer performance is high in Casson fluid when compared to

136 Williamson fluid. Ibrahim and Makinde [41] considered a problem of stagnation pont flow of

137 MHD Casson nanofluid past a surface in the presence of velocity and convective conditions.

138 They presented the numerical solutions by RK45 shooting method.

139 The purpose of the current study is to explore the problem of unsteady axisymmetric

140 flow of Carreau nanofluid over a radially stretching sheet in the presence of time dependent

141 magnetic field. Additionally, convective boundary condition is considered at the boundary

142 which leads to a more realistic physical problem. Buongiorno model accounting the effects

143 of thermophoresis and Brownian motion is adopted. It is important to state that Carreau

144 viscosity model is an important class of generalized Newtonian fluid. The Carreau viscosity

145 model reduces to viscous fluid when n =1 and We = 0. Furthermore, velocity partial

146 slip condition is also implemented at the surface. Suitable transformations are utilized to

alter the highly nonlinear partial differential equations into nonlinear ordinary differential

148 equations. After that these highly nonlinear system is solved numerically by adopting an

149 effective numerical scheme namely Runge-Kutta Fehlberg integration scheme. The numer-

150 ical solutions of the problem are demonstrated for both cases of shear thinning and shear

151 thickening fluids.

152 2 Model development

153 The problem of unsteady axisymmetric two-dimensional flow of an incompressible Carreau

154 nanofluid over a convectively heated radially stretching sheet in the presence of time de-

155 pendent magnetic field is considered. The sheet is stretched in the radial direction with

156 stretching velocity Uw (r,t) = yari, which is linearly proportional to the distance r from the

157 origin with a and c as positive constants having the dimensions of (time)-1. The sheet

158 is coinciding with the plane z = 0 and flow appears in the upper half plane z > 0. For

159 mathematical description, we assume the cylindrical polar coordinate system (r,9,z). A

160 non-uniform transverse magnetic field of strength B(t) = yprgt is implemented in z— direc-

161 tion, where B0 is a constant related to magnetic field strength (see Fig. 1). The magnetic

162 Reynolds number is taken to be small enough so that the induced magnetic field can be

163 neglected. Recently devised model for nanofluid incorporating the effects of Brownian mo-

164 tion and thermophoresis is utilized. Additionally, the velocity partial slip condition at the

165 surface is also implemented. A heated fluid under the surface of the sheet with temperature

166 Tw is used to change the temperature of the sheet by convective heat transfer mode which

167 provides the heat transfer coefficient hf. Moreover, the surface of the sheet is at constant

168 concentration Cw with Cw > C^.

169 For the unsteady two-dimensional axisymmetric flow, the velocity, temperature and

170 nanoparticles concentration fields are choosen in the following manner

V =[u(r,z,t), 0, w(r,z,t)], T = T (r, z, t), C = C (r,z,t). (1)

171 Under the boundary layer analysis and the aforesaid assumptions, the governing equations

172 of mass, momentum, energy and nanoparticles concentration for the Carreau nanofluid in

the presence of time dependent magnetic field can be read as [42 — 43]

du + u + dw о dr r dz '

du du du d2u dt dr dz dz2

i + r2f ^

п-1)Г2(du2

dz2 \ dz

n — 3

i+гм du

aB2(t)u

dT dT dT d2T — + u— + w— = am^—r + t dt dr dz dz2

^ dCdT Dt [dT

Tж \ dz

de dC dC = D д-C Dtl d^r

dt dr dz B dz2 T™ dz

T 2 ■

where (u,w) represent the velocity components in (r, z) directions, respectively, Г the material constant, n the power law index, v the kinematic viscosity, p the density, t (pc)p / (pc)f^ the ratio of the effective heat capacity of the nanoparticle to the effective heat capacity of the base fluid, am pr) the thermal diffusivity with k the thermal conductivity, cp the specific heat, DT the thermophoresis diffusion coefficient, DB the Brownian diffusion coefficient and t the time.

The boundary conditions for velocity, temperature and nanoparticles concentration are

for veloc

u = Uw(r, t) + uslip, w = 0,

k— = -hf (Tw - T)

C = Cw at z = 0,

u ^ 0,

T ^ Tn

where T^ and Cœ are the temperature and concentration at infinity, respectively. Additionally, the velocity partial slip condition is assumed to be of the form

1 + 1Ч du

183 where l is the slip length having dimension of length.

184 The non-dimensional suitable variables can be demonstrated in the following manner

n —1

ACCEPTED MANUSCRIPT

z T — T C — C

V = - Re1/2, ¥(r,-,t) = —r2Uw Re-1/2 f (V), d(V) = —, 0 (n) = C-Co

r T w T oo

185 where ^ is the Stokes stream function having the property (u, w) = (— , 1 f^r), (0, 0) the

186 dimensionless temperature and nanoparticle concentration respectively and rq the indepen-

187 dent variable. The velocity components are represented in following way

u = Uw f (n), w = - 2UW Re-1/2 f (n). (10)

188 Substituting Eqs. (9) into Eqs. (3), (4) and (5), yealds the following nonline ar ordinary

189 differential equations

{1 + nWe2(f ")2} {1 + We2(f ")2} ^ f ' + 2ff' '- (f )2 - Af + 2 f}-M2f = 0, (11)

e" + Pr <1 2fd' - A+ Nb O'0' + Nt (0')2\ = 0, (12)

f + 2Sc f0' - ASc + ^0'' = 0, (13)

........

f (0) = 0, f'(0) = 1 + Lf''(0){1 + We (f '(0))2} 2 , 0'(0) = -7 (1 - 0 (0)), 0 (0) = 1,

f M ^ 0, 0(œ) ^ 0, 0(œ) ^ 0, (15)

190 The dimensionless parameters appearing in Eqs. (11) - (13) are the local Weissenberg

191 number We, the unsteadiness parameter A, the Prandtl number Pr, the Schmidt number

192 Sc, the thermophoresis parameter Nt, the Brownian motion parameter Nb, the magnetic

193 parameter M, the generalized Biot number y and the velocity slip parameter L. Note that

194 L = 0 corresponds to no slip case. They are respectively defined as

We2 = , A = Pr= A Sc = f, Nt = • d«)

V(1 - ct)3 a am Db vl^

ACCEPTED MANUSCRIPT

nb = tdb (cw - c^ ) m2 = a_b

^ Re-1/2, L — - Re1/2, k r

The important mechanisms of flow, heat and mass transfer are the local skin friction coefficient Cf, the local Nusselt number Nur and the local Sherwood number Shr which are

197 written as

Cf —

T w \z=0

Nur —

rQw \z=0

k(TW TQO )

r qm\z=0

DB (Cw — Coo )

where tw, qw and qm are the wall shear stress, wall heat flux and wall mass flux respectively, having the following expressions

T w — ^0

1 + ri £

qw — -k ( — J , qm

- (19)

200 Using Eqs. (9), (18) and (19), the drag, heat and mass transfer rates get the following

201 form:

Re1/2 Cf = ///(0)[1 + We2(f''(0))2]^, Re-1/2 Nu = —0'(0), Re-1/2 Sh = —0'(0). (20)

202 Where Re (= is the local Reynolds number.

203 3 Numerical

procedure

204 In general, it is difficult to find the exact solution of the system of highly non-linear ordinary

205 differential Eqs. (11) — (13) with the boundary conditions (14) and (15). Therefore, the par-

206 tially coupled highly nonlinear ordinary differential equations involving momentum, energy

207 and nanoparticles concentration along with the boundary conditions are solved numerically

208 by utilizing the Runge-Kutta Felhberg integration scheme. This scheme is adopted to solve the initial value problems of the following form

ï=f (x,y),

y(xi) = Vi. (21)

In this scheme, the differential Eqs. (11) — (13) are first converted into a system of seven first order differential equations. To solve this system by adopting RK45 scheme, we need seven initial conditions but three initial conditions each in f (r), 6(rj) and 0(r) are unknowns.

These three end conditions are used to develop three unknown initial conditions with the

214 help of shooting scheme. An important factor of this scheme is to choose the most suitable

215 finite value of Thus, we have made some initial guesses with the help of Newton-Raphson

216 method for missing conditions so that the conditions f (to) = 0, 6 (to) = 0 and = 0

217 are satisfied. In the current problem, the value of rq = ^ is taken to be 10 and step-size is

218 taken to be Arq = h = 0.01 with relative error tolerance 10-5. Consequently, the non-linear

219 equations and the corresponding boundary conditions are converted into a system of first

220 order equations as

, , , y22 - 2yiys + A(y2 + 2ys) + M2y2r

Vi = y2, y2 = ys, ys = ---„ n-3 (22)

{1 + nWe2y2 } {1 + We2yg} ~

v4 = y5, v5 = Pr <¡ -2ViV5 + AW5 - Nb y5y7 - Nt (y5)2 \ , (23)

v6 = V7, v7 = -2Scvi V7 + ASe n V7- NbУ'5, (23)

where the unknowns are stated as

f = yi, f = y 2, f" = V3, e = y 4, e' = y 5, 0 = y6, 0' = y7, (24)

222 with the boundary conditions taking the following form

Vi(0) = 0, V2(0) = 1 + Lys(0) {1 + We2(ys(0))2} 2

V5(0) = -Y(1 - V4(0)), V6(0) = 1, (25)

V2(œ) ^ 0, V4(œ) ^ 0, V6(œ) ^ 0. (26)

4 Discussion of numerical results

A numerical analysis for unsteady axisymmetric flow of Carreau nanofluid over a convectively heated radially stretching sheet in the presence of time dependent magnetic field is performed.

The numerical computations have been performed by adopting an effective numerical scheme

tudy, ed by

227 namely the shooting technique along with fourth-fifth order Runge-Kutta integration scheme

228 for several values of involved parameters namely the power law index n, magnetic parameter

229 M, local Weissenberg number We, velocity slip parameter L, generalized Biot number 7,

230 unsteadiness parameter A, Brownian motion parameter Nb, thermophoresis parameter Nt,

231 Prandtl number Pr and Schmidt number Sc. For the verification of current numerical stu

232 the numerical results of skin friction coefficient are compared with those investig

233 Ariel [44] for several values of velocity slip parameter L when We = A = M = 0 and n =1

234 (see table 1). Additionally, the numerical results of skin friction coefficient are also compared

235 with those reported by Makinde et al. [45] for some values of magnetic paramet er M when

236 We = A = L = 0 and n = 1(see table 2). It is observed that present numerical investigation

237 is in excellent agreement with the existing literature.

238 The numerical computations for the local skin friction coefficient, local Nusselt and Sher-

239 wood numbers have been performed for both cases of shear thinning (0 < n < 1) and shear

240 thickening (n > 1) fluids through tables 3, 4 and 5. From these tables, it is noted that

241 the magnitude of local skin friction coefficient is an enhancing function of the unsteadiness

242 and magnetic parameters both in shear thinning and shear thickening fluids. Additionally,

243 the magnitude of local skin friction coefficient decreases for growing values of velocity slip

244 parameter in both cases. Furthermore, the magnitude of local skin friction coefficient depre-

245 ciates in shear thinning fluid but enhances in shear thickening fluid. On the basis of these

246 tables, it is noticed that both the local Nusselt and Sherwood numbers depreciate for en-

247 hancing the values of unsteadiness parameter, magnetic parameter, velocity slip parameter

248 and thermophoresis parameter in both cases. It is also noticed that the local Nusselt number

249 depreciates with the increment of Brownian motion parameter but opposite behavior can be

250 noticed in local Sherwood number in shear thickening and shear thinning fluids. Addition-

251 ally, the generalized Biot number 7 is an enhancing function of the local Nusselt number

252 but diminishing function of the local Sherwood number in both cases. Furthermore, both

253 the local Nusselt and Sherwood numbers decrease for growing values of local Weissenberg

254 number in shear thinning fluid but opposite trend can be seen in shear thickening fluid.

255 We have also conducted a comparative study of numerical values of local skin friction coef-

ficient, local Nusselt number and local Sherwood number between two different numerical

257 approaches namely shooting RK45 and bvp4c and found to be in excellent agreement. Note

258 that the Carreau fluid reduces to Newtonian fluid when n =1 and We = 0. It is important

259 to state that the suitable transformations in Eq. (9) give rise to some pertinent parameters

260 which are not truely independent from the temporal/spatial variables. It means the current

261 model is only a local approximation.

262 Figs. 2(a,b, c, d) depict the variation of temperature d(n) and nanoparticles concentra-

263 tion 0(rq) with different values of unsteadiness parameter A in both shear thinning and shear

264 thickening fluids. It is noted that the temperature and nanoparticle concentration are grow-

265 ing functions of unsteadiness parameter for both cases. Additionally, the associat ed thermal

266 and nanoparticle concentration boundary layer thicknesses are also the enhancing function

267 of unsteadiness parameter. Infact, enhancement in unsteadiness has the tendency to improve

268 the thermal as well as concentration boundary layer thicknesses.

269 Figs. 3(a,b,c,d) indicate the variation of velocity f (n), temperature 0(rq) and nanopar-

270 ticles concentration 0(q) with different values of local Weissenberg number in shear thinning

271 and shear thickening fluids. On the behalf of these Figs., it is noted that the velocity of fluid

272 decreases in shear thinning case but increases in shear thickening case for growing values of

273 local Weissenberg number. From these Figs., it can be seen that the temperature as well

274 as nanoparticles concentration enhance for improving values of local Weissenberg number in

275 shear thinning fluid but opposite trend can be noticed in shear thickening fluid. Further-

276 more, the associated thermal and concentration boundary layer thicknesses uplift in shear

277 thinning fluid whereas opposite results can be revealed in shear thickening fluid.

278 The variation of velocity f'(n), temperature 0(rq) and nanoparticles concentration 0(rq)

279 with different values of magnetic parameter M are depicted through Figs. 4(a, b, c, d) in both

280 cases regarding shear thinning and shear thickening fluids. From these Figs., it is observed

281 that the velocity of fluid depreciates by improving values of magnetic parameter. On the basis

282 of these Figs., it can also be predicted that the temperature, nanoparticle concentration and

283 their associated thermal and concentration boundary layer thicknesses enhance for improving

284 values of magnetic parameter. Note that M = 0 is the case of hydrodynamic flow and M > 0

285 demonstrates the hydromagnetic flow. Infact, the strong magnetic field has the tendency to

enhance the thermal and concentration boundary layer thicknesses.

287 Figs. 5(a, b, c, d) elucidate the variation of velocity f ' (n), temperature 9(n) and nanoparti-

288 cles concentration ф(ц) with different values of velocity slip parameter in both cases regarding

289 shear thinning and shear thickening fluids. From these Figs., it is clear that the velocity of

290 fluid diminishes by improving values of velocity slip parameter. From these Figs., it can also

291 be examined that temperature, nanoparticle concentration and their associated thermal and

292 concentration boundary layer thicknesses are increasing functions of the generalized velocity

293 slip parameter. Physically, with the increased velocity slip, as a result of depreciate in the

294 tendency of fluid to remove the heat away from the plate an improve in temperature and

295 nanoparticles concentration is noticed.

296 Figs. 6(a, b, c, d) reveal the variation of temperature 9(n) and nanoparticles concentration

297 ф(п) with different values of generalized Biot number in shear thinning and shear thickening

298 fluids. On the evidence of these Figs., it is clear that the temperature, nanoparticles concen-

299 tration and their associated thermal and concentration boundary layer thicknesses grow for

300 improving values of generalized Biot number. The generalized Biot number 7 is the ratio of

301 internal thermal resistance of a solid to boundary layer thermal resistance. When 7 = 0, the

302 surface of sheet is totally insolated. The internal thermal resistance of the surface of sheet

303 is very large and there is no convective heat transfer from the surface of sheet to the cold

304 fluid far away from the sheet.

305 Figs. 7(a,b,c,d) represent the variation of temperature 9(n) and nanoparticles concen-

306 tration ф(п) with different values of Brownian motion parameter in shear thinning and shear

307 thickening fluids. On the evidence of these Figs., it is clear that the temperature and its

308 associated thermal boundary layer thickness grows for growing values of Brownian motion

309 parameter but opposite results can be observed in nanoparticle concentration field. Brow-

310 nian motion appears due to the presence of nanoparticles and resulted in the decrement of

311 the nanoparticles concentration thickness.

312 The variation of nanoparticles concentration ф(п) with different values of thermophoresis

313 parameter can be visualized through Figs. 8(a,b,c,d) in shear thinning and shear thicken-

314 ing fluids. It is observed that the nanoparticle concentration and its related concentration

315 boundary layer thickness enhance with the enhancement of the thermophoresis parameter.

Actually, temperature difference between ambient and surface enhances for higher ther-

317 mophoresis which grows the temperature and concentration of the fluid. Physically, the

318 thermophoresis force grows with the improvement of which tends to move nanoparticles

319 from hot portion to cold portion and hence enhances the magnitude of the nanoparticle

320 concentration profile.

321 The variation of nanoparticles concentration with different values of Schmidt number

322 can be observed through Figs. 9(a, b, c, d) in shear thinning and shear thickening fluids. It is

323 observed that the nanoparticle concentration and its related concentration b ry layer

324 thickness depreciate with the enhancement of Schmidt number. In fact, Sd number

325 is the ratio of viscosity to mass diffusivity. When Schmidt number^increa m mass

326 diffusivity decreases and results in reduction in fluid concentratio:

327 Table 1: A comparison of computation results of -/"(0) for several values of the velocity

328 slip parameter L when We = A = M = 0 and n =1.

-f "(0)

Exact [44]

1.173721

1.153472

1.134017

1.079949

1.001834

0.878425

0.650528

0.462510

0.299050

0.149393

0.082912

0.044368

0.018732

0.009594

HPM [44]

1.178511

1.157311

1.136998

1.080820

1.000308

0.874453

0.645304

0.458333

0.296534

0.148454

0.082532

0.044228

0.018698 0.009583

Perturbation [44]

1.173721

1.153481

1.134090

1.081010

1.009522

0.930213

1.201623

Asymptotic [44]

0.3107.

10753 .149590

0.082833

0.044337 0.018727

0.009593

329 Table 2: A comparison of computation results of /''(0) for several values of the magnetic

330 parameter M when We = A = L = 0 and n = 1.

/hen We

M 2 Makinde et al. [45] Present results

0.0 -1.17372 -1.17372

0.5 -1.36581 -1.36581

1.0 -1.53571 -1.53571

2.0 -1.83049 -1.83049

3.0 -2.08484 -2.08485

331 Table 3: Numerical computations of the local skin friction coefficient - Re1/2 Cf for

332 selected values of A, M, We and L.

Parameters — Re1/2 Cf (bvp4c) results — Re1/2 Cf (shooting) results

333 Table 4: Numerical computations of the local Nusselt number Re 1/2 Nu for selected

334 values of A, M, We, L, y, Nt and Nb when Pr = 2.5 and Sc = 2.

Parameters Re 1/2 Nu(bvp4c) results Re 1/2 Nu (shooting) results

A M We L Y Nt Nb n = 0.5 n =1.5 n = 0.5 n = 1.5

0.1 0.2 1.0 0.1 0.1 0.2 0.1 0.092321 0.092519 0.0923206 0.0925186

0.2 0.092075 0.092306 0.0920749 0.0923062

0.3 0.091793 0.092067 0.091793 0.0920672

0.2 0.0 1.0 0.1 0.1 0.2 0.1 0.092107 0.092330 0.0921072 0.0923301

0.4 0.091978 0.092235 0.0919776 0.0922349

0.8 0.091587 0.091956 0.0915874 0.0919561

0.2 0.2 0.2 0.1 0.1 0.2 0.1 0.092209 0.092221 0.0922089 0.092221

1.0 0.092075 0.092306 0.0920749 0.0923062

1.6 0.091908 0.092369 0.0919083 0.092369

0.2 0.2 1.0 0.2 0.1 0.2 0.1 0.091719 0.091887 0.0917189 0.0918872

0.3 0.091385 0.091512 0.0913845 0.0915119

0.4 00.091068 0.091168 0.0910685 0.0911679

0.2 0.2 1.0 0.1 0.1 0.2 0.1 |0.092075 0.092306 0.0920749 0.0923062

0.2 0.170379 0.171183 0.170379 0.171183

0.3 0.237523 0.239105 0.237523 0.239105

0.2 0.2 1.0 0.1 0.1 0.2 0.1 0.092075 0.092306 0.0920749 0.0923062

0.3 0.092033 0.092266 0.0920326 0.0922657

0.4 0.091990 0.092225 0.0919898 0.0922246

0.2 0.3 1.0 1.2 0.1 0.2 0.1 0.092075 0.092306 0.0920749 0.0923062

£ 0.2 0.091131 0.091386 0.0911308 0.0913855

X 0.3 0.090041 0.090322 0.0900413 0.0903222

335 Table 5: Numerical computations of the local Sherwood number Re 1/2 Sh for selected

336 values of A, M, We, L, 7, Nt and Nb when Pr = 2.5 and Sc = 2.

Parameters Re 1/2 Sh(bvp4c) results Re 1/2 Sh(shooting) results

A M We L Y Nt Nb n = 0.5 n = 1.5 n = 0.5 n = 1.5

0.1 0.2 1.0 0.1 0.1 0.2 0.1 1.074825 1.115492 1.07482 1.11549

0.2 1.028389 1.073354 1.02839 1.07335

0.3 0.977571 1.027777 0.977571 1.02778

0.2 0.0 1.0 0.1 0.1 0.2 0.1 1.034539 1.078220 1.03454 1.07822

0.4 1.010224 1.059022 1.01022 1.05902

0.8 0.941778 1.005378 0.941778 1.00538

0.2 0.2 0.2 0.1 0.1 0.2 0.1 1.053693 1.056039 1.05369 1.05604

1.0 1.028389 1.073354 1.02839 1.07335

1.6 0.998357 1.087086 0.998357 1.08709

0.2 0.2 1.0 0.2 0.1 0.2 0.1 0.975535 1.006120 0.975535 1.00612

0.3 0.929771 0.951642 0.929771 0.951642

0.4 00.889646 0.905889 0.889646 0.905889

0.2 0.2 1.0 0.1 0.1 0.2 0.1 |1.028389 1.073354 1.02839 1.07335

0.2 0.929443 0.976614 0.929443 0.976614

0.3 0.845049 0.893804 0.845049 0.893804

0.2 0.2 1.0 0.1 0.1 0.2 0.1 1.028389 1.073354 1.02839 1.07335

0.3 0.970229 1.016750 0.970228 1.01675

0.4 0.912315 0.960404 0.912315 0.960404

0.2 °.3 1.0 1.2 0.1 0.2 0.1 1.028389 1.073354 1.02839 1.07335

£ 0.2 1.090037 1.133641 1.09004 1.13364

X 0.3 1.110817 1.153984 1.11082 1.15398

5 Conclusions

EPTED MANUSCRIPT

338 A numerical analysis for the unsteady axisymmetric flow of an MHD Carreau nanofluid

339 over a convectively heated radially stretching sheet with velocity slip condition has been

340 performed. The Buongiorno's model involving the Brownian motion and thermophoresis

341 effects has been considered. Numerical computations were carried out for several values

342 of emerging parameters using Runge-Kutta Felhberg integration scheme and their effects

343 were examined graphically on heat and mass transfer characteristics. The computed results

344 showed that the temperature, nanoparticles concentration and their associated thermal and

345 concentration boundary layer thicknesses were enhanced by increasing values of general-

346 ized Biot number in shear thickenings and shear thinning fluids. More importantly, results

347 revealed that the temperature and nanoparticles concentration were improved for growing

348 values of local Weissenberg number in shear thinning fluid but opposite behavior was noted

349 in shear thickening fluid. Further, results showed that the effects of Schmidt number was

350 to decrease the nanoparticles concentration and associated concentration boundary layer

351 thickness in all cases.

352 Acknowledgements: The authors wish to convey their genuine thanks to the re-

353 viewers for their essential suggestions and comments to progress the superiority of this

354 manuscript. The work of Dr. Alshomrani was partially supported by Deanship of Scientific

355 Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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ACCEPTED MANUSCRIPT

Fig. 1: Flow configuration.

Fig. 2: Variation of temperature 9(n) and nanoparticles concentration 0(n) with different values of A.

Fig. 3: Variation of velocity f '(n), temperature 0(rq) and nanoparticles concentration 0(n) with different values of We.

Fig. 4: Variation of velocity f '(n), temperature 9(n) and nanoparticles concentration 0(n) with different values of M.

Fig. 5: Variation of velocity f '(n), temperature 6(n) and nanoparticles concentration 0(n) with different values of L.

Fig. 6: Variation of temperature 6(n) and nanoparticles concentration 0(n) with different values of y.

Fig. 7: Variation of temperature 6(n) and nanoparticles concentration 0(n) with different values of Nb.

Fig. 8: Variation of nanoparticle concentration 0(n) with different values of Nt.

Fig. 9: Variation of nanoparticles concentration 0(n) with different values of Sc.

ACCEPTED MANUSCRIPT

Unsteady axisymmetric flow of Carreau nanofluid is modeled.

Magnetic nanofluid is considered.

Brownian motion and thermophoresis are accounted.

Velocity slip condition is incorporated.

Convective boundary condition is assumed.

Numerical solutions are developed through Shooting RK45 technique.