2. Mathematical model

of micro/nano-sized particles in fluids have also been tried since many decades. However, the word 'nanofluids' was primarily introduced by Choi [1]. He has studied the thermal conductive effects on convectional fluids subject to suspension of metallic nanoparticles. He has observed that with suspension of nanoparticles, the thermal conductivity of convectional fluids enhances. In these directions, numerous investigations have therefore been carried out in the past few decades, seeking to wide applications and developments, some interesting

To study the heat transfer rate of water based nanofluids, the various types of nanoparticles like titanium dioxide, alumina, silica diamond, zinc-oxide and copper [5–7] have been considered. It has been depicted that the thermal conductivity enhances with suspension of the nanoparticles. Moreover, CNTs have also received great interest due to significant enhancement of thermal conductivity, unique structure and physical (mechanical and electrical) properties [8, 9]. Ding et al. [8] have prepared a nanofluids with suspension of CNTs in distilled water and measured the thermal conductivity and viscosity of CNTs nanofluids. They have reported the enhancement of thermal conductivity depends on CNTs concentration, and pH level. They also concluded that nanofluids with 0.5 wt.% CNTs, the maximum enhancement is over 350% at Re = 800. Ko et al. [9] experimentally reported the flow characteristics of CNTs nanofluids in a tube and summarized that the friction factor for CNT nanofluid is low in compare to water. Another experimental study for thermophysical properties of CNTs nanofluids with base fluid as mixture of water and ethylene glycol has been presented by Kumaresan and Velraj [10]. And this study noted that the maximum thermal conductivity enhances up to 19.75% for the nanofluid containing 0.45 vol.% MWCNT at 40C. In addition, few more experimental investigations [11, 12] with applications in a tubular heat exchanger of various lengths for energy efficient cooling/heating system and turbulent flow heat exchanger are performed. With CNTs nanofluids, some more numerical simulations [13, 14] for flow characteristics are presented in literature and discussed the physiological flows application.

The second key point is boundary layer flow over a circular stretching sheet. The boundary layer flow past a stretching sheet was initially investigated by Crane [15]. He has discussed its applications such as annealing and tinning of copper wires, polymer extrusion in melt spinning process, manufacturing of metallic sheets, paper production, and glass, fiber and plastic production etc. The heat transfer rate play important role in all manufacturing, productions and fabrication processes. This model has been explored by many investigators for various physical aspects. However, some interesting extensions of Crane's model for boundary layer flow of nanofluids have recently investigated, see examples [16–30]. The first boundary layer flow model for nanofluids flow past stretching sheet was presented by Khan and Pop [16] which was extended for a convective boundary condition [17], nonlinear stretching sheet [18], unsteady stretching surface [19], micropolar nanofluid flow [20], magneto-convective non-Newtonian nanofluid slip flow over permeable stretching sheet [21], Non-aligned MHD stagnation point flow of variable viscosity nanofluids [22], Stagnation electrical MHD mixed convection [23], exponential temperature-dependent viscosity and buoyancy effects [24], thermo-diffusion and thermal radiation effects on Williamson nanofluid flow [25], magnetic dipole and radiation effects on viscous ferrofluid flow [26], transient ferromagnetic liquid flow [27], magnetohydrodynamic Oldroyd-B nanofluid [28],

reviews [2–4] have been reported.

274 Numerical Simulations in Engineering and Science

We consider the laminar incompressible flow of CNTs nanofluids over a circular sheet aligned with the rθ-plane in the cylindrical coordinate system (r, θ, z). The schematic representation and coordinate system are depicted in Figure 1.

Figure 1. Schematic representation of CNTs nanofluids flow over circular stretching sheet.

The flow regime is considered as in the half space z ≥ 0 and the sheet is stretched in radial direction with polynomial variation. The temperature at sheet is fixed as Tw and ambient temperature is considered as T∞. Considering the boundary layer flow assumptions, the governing equations for conservation of mass, momentum, and energy can be expressed as:

$$
\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial z} = 0,\tag{1}
$$

Introducing the following similarity transformations:

Reynolds model of viscosity expression can be taken as:

ð Þ <sup>η</sup> , <sup>θ</sup> <sup>¼</sup> <sup>T</sup> � <sup>T</sup><sup>∞</sup>

μfð Þ¼ θ e

Tw � T<sup>∞</sup>

, w ¼ �ar

Using Eqs. (6) and (7) in Eqs. (1)–(5), the governing equations and boundary conditions are

� � <sup>1</sup> � <sup>ϕ</sup> <sup>þ</sup> <sup>ϕ</sup> <sup>ρ</sup>cp

ρf ! ð Þ <sup>n</sup> <sup>þ</sup> <sup>3</sup>

!

∂u ∂z � �

z¼0

∂T ∂z � �

z¼0

00ð Þ0

� � s ρcp � � f

‴ <sup>þ</sup> <sup>1</sup> � <sup>ϕ</sup> <sup>þ</sup> <sup>ϕ</sup> <sup>ρ</sup><sup>S</sup>

Cfx <sup>¼</sup> <sup>μ</sup>nfð Þ <sup>T</sup> ρ<sup>f</sup> uw<sup>2</sup>

Nux <sup>¼</sup> �xknf

The dimensionless form of skin friction coefficients can be obtained as:

ð Þ Re<sup>x</sup> <sup>1</sup>=<sup>2</sup>

And the dimensionless form of skin friction coefficients can be obtained as:

ð Þ Re<sup>x</sup> <sup>1</sup>=<sup>2</sup>

kfð Þ Tw � T<sup>∞</sup>

Cfx <sup>¼</sup> <sup>μ</sup>fð Þ <sup>θ</sup>ð Þ<sup>0</sup> <sup>f</sup>

Nux ¼ � knf

kf θ0

2

ð Þ¼ 0 1, f <sup>0</sup>

ð Þ n�1 2

ffiffiffiffi νf a <sup>r</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>3</sup>

Numerical Simulation of Nanoparticles with Variable Viscosity over a Stretching Sheet

<sup>2</sup> <sup>f</sup>ð Þþ <sup>η</sup> ð Þ <sup>n</sup> � <sup>1</sup>

�ð Þ αθ <sup>¼</sup> <sup>1</sup> � ð Þþ αθ <sup>O</sup> <sup>α</sup><sup>2</sup> � �, (7)

<sup>2</sup> f f <sup>00</sup> � n f <sup>0</sup> � �<sup>2</sup> � �

ð Þ¼ ∞ 0, θð Þ¼ 0 1, θð Þ¼ ∞ 0, (10)

� �: (6)

http://dx.doi.org/10.5772/intechopen.71224

<sup>2</sup> nf <sup>0</sup>

ð Þ η

277

� Grθ ¼ 0, (8)

f θ<sup>0</sup> ½ �¼ ð Þ 0, (9)

: (11)

: (12)

<sup>1</sup> � <sup>ϕ</sup> � �<sup>2</sup>:<sup>5</sup> : (13)

ð Þ0 : (14)

0

‴ <sup>þ</sup> �αθ<sup>0</sup> ð Þ <sup>1</sup> � <sup>ϕ</sup> � �<sup>2</sup>:<sup>5</sup> <sup>f</sup>

> knf kf

The skin friction coefficients (Cfx) is defined as:

And the local Nusselt number (Nux) is defined as:

� �θ<sup>00</sup> <sup>þ</sup> Pr <sup>n</sup> <sup>þ</sup> <sup>3</sup>

fð Þ¼ 0 0, f <sup>0</sup>

is the Prandtl number.

η ¼

ffiffiffiffi a νf s r ð Þ n�1

transformed as:

where Pr <sup>¼</sup> <sup>μ</sup><sup>0</sup> ð Þ cp <sup>f</sup>

kf

1 � ð Þ αθ <sup>1</sup> � <sup>ϕ</sup> � �<sup>2</sup>:<sup>5</sup> <sup>f</sup>

<sup>2</sup> z, u <sup>¼</sup> arnf

where α is the viscosity parameter.

$$\mathcal{J}\left(\mu\frac{\partial u}{\partial r} + w\frac{\partial u}{\partial z}\right) = \frac{1}{\rho\_{\eta\zeta}}\frac{\partial}{\partial z}\left(\mu\_{\eta\zeta}(T)\frac{\partial u}{\partial z}\right) \pm g\beta(T - T\_{\circ\circ}),\tag{2}$$

$$
\left(\mu \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z}\right) = \alpha\_{\text{nf}} \frac{\partial^2 T}{\partial z^2} \tag{3}
$$

where u and w are the velocity components along r- and z-directions respectively, T is the temperature. Here the "+" sign in Eq. (2) corresponds to the assisting flow while the "�" sign corresponds to the opposing flow, respectively. Here, ρnf is the nanofluid density, kf is the thermal conductivity of the fluid, Tw is the wall temperature, μnf(T) is the temperature dependent viscosity of nanofluid and αnf is the thermal diffusivity of nanofluid which are defined [31–34] as:

$$
\mu\_{nf} = \frac{\mu\_0 e^{-a\theta}}{\left(1 - \phi\right)^{2.5}}\tag{4a}
$$

$$
\alpha\_{\rm nf} = \frac{k\_{\rm nf}}{\left(\rho c\_p\right)\_{\rm nf}}, \qquad \rho\_{\rm nf} = \left(1 - \phi\right)\rho\_f + \phi \rho\_{s'} \tag{4b}
$$

$$\left(\rho c\_{\mathfrak{p}}\right)\_{\mathfrak{n}'} = \left(1 - \phi\right) \left(\rho c\_{\mathfrak{p}}\right)\_{\mathfrak{f}} + \phi \left(\rho c\_{\mathfrak{p}}\right)\_{\mathfrak{s}'} \tag{4c}$$

$$\left(\left(\rho\gamma\right)\_{\rm nf} = \left(1-\phi\right)\left(\rho\gamma\right)\_{\rm f} + \phi\left(\rho\gamma\right)\_{\rm S'}\tag{4d}$$

$$k\_{nf} = k\_f \left(\frac{k\_s + 2k\_f - 2\phi(k\_f - k\_s)}{k\_s + 2k\_f + 2\phi(k\_f - k\_s)}\right). \tag{4e}$$

Here, ρ<sup>f</sup> is density of the base fluid, ρ<sup>s</sup> is density of the nanoparticles, kf is thermal conductivity of the base fluid, ks is thermal conductivity of the nanoparticles, γnf is the thermal expansion coefficient, γ<sup>f</sup> is the thermal expansion coefficient of base fluid, ϕ is the nanoparticle volume fraction, and γ<sup>s</sup> is the thermal expansion coefficient of the nanoparticles.

The following boundary conditions are to be employed:

$$u = u\_w(r) = ar^n, \ T = T\_{w\nu} \text{ at } z = 0,\tag{5a}$$

$$
\mu \to 0, T \to T\_{\ast \ast} \quad \text{as} \ z \to \simeq,\tag{5b}
$$

in which uw is the wall velocity, a > 0 is the stretching constant and n > 0 is the power-law index.

Introducing the following similarity transformations:

$$\eta = \sqrt{\frac{a}{\nu\_f}} r^{\frac{(n-1)}{2}} z,\\ u = a r^{\eta} f'(\eta),\\ \theta = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}},\\ w = -a r^{\frac{(n-1)}{2}} \sqrt{\frac{\nu\_f}{a}} \left(\frac{(n+3)}{2} f(\eta) + \frac{(n-1)}{2} \eta f'(\eta)\right). \tag{6}$$

Reynolds model of viscosity expression can be taken as:

$$
\mu\_f(\theta) = e^{-(a\theta)} = 1 - (a\theta) + O(a^2),
\tag{7}
$$

where α is the viscosity parameter.

The flow regime is considered as in the half space z ≥ 0 and the sheet is stretched in radial direction with polynomial variation. The temperature at sheet is fixed as Tw and ambient temperature is considered as T∞. Considering the boundary layer flow assumptions, the governing equations for conservation of mass, momentum, and energy can be

<sup>∂</sup><sup>z</sup> <sup>μ</sup>nfð Þ <sup>T</sup> <sup>∂</sup><sup>u</sup>

where u and w are the velocity components along r- and z-directions respectively, T is the temperature. Here the "+" sign in Eq. (2) corresponds to the assisting flow while the "�" sign corresponds to the opposing flow, respectively. Here, ρnf is the nanofluid density, kf is the thermal conductivity of the fluid, Tw is the wall temperature, μnf(T) is the temperature dependent viscosity of nanofluid and αnf is the thermal diffusivity of nanofluid which are defined [31–34] as:

<sup>μ</sup>nf <sup>¼</sup> <sup>μ</sup>0e�αθ

nf <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> � � <sup>ρ</sup>cp

nf <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> � � ργ

� �

� �

ks þ 2kf � 2ϕ kf � ks

!

ks þ 2kf þ 2ϕ kf � ks

Here, ρ<sup>f</sup> is density of the base fluid, ρ<sup>s</sup> is density of the nanoparticles, kf is thermal conductivity of the base fluid, ks is thermal conductivity of the nanoparticles, γnf is the thermal expansion coefficient, γ<sup>f</sup> is the thermal expansion coefficient of base fluid, ϕ is the nanoparticle volume

in which uw is the wall velocity, a > 0 is the stretching constant and n > 0 is the power-law index.

<sup>f</sup> þ ϕ ρcp � � s

<sup>f</sup> þ ϕ ργ � �

� �

� �

<sup>u</sup> <sup>¼</sup> uwð Þ¼ <sup>r</sup> arn, T <sup>¼</sup> Tw, at z <sup>¼</sup> <sup>0</sup>, (5a)

u ! 0, T ! T∞, as z ! ∞, (5b)

� �

¼ αnf

∂z

∂<sup>2</sup>T

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>0</sup>, (1)

<sup>1</sup> � <sup>ϕ</sup> � �<sup>2</sup>:<sup>5</sup> , (4a)

, (4c)

<sup>S</sup>, (4d)

: (4e)

, <sup>ρ</sup>nf <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> � �ρ<sup>f</sup> <sup>þ</sup> ϕρs, (4b)

� gβð Þ T � T<sup>∞</sup> , (2)

<sup>∂</sup>z<sup>2</sup> , (3)

∂u ∂r þ u r þ ∂w

¼ 1 ρnf ∂

� �

u ∂T ∂r þ w ∂T ∂z

u ∂u ∂r þ w ∂u ∂z

276 Numerical Simulations in Engineering and Science

� �

<sup>α</sup>nf <sup>¼</sup> knf ρcp � � nf

> ρcp � �

> > ργ � �

knf ¼ kf

fraction, and γ<sup>s</sup> is the thermal expansion coefficient of the nanoparticles.

The following boundary conditions are to be employed:

expressed as:

Using Eqs. (6) and (7) in Eqs. (1)–(5), the governing equations and boundary conditions are transformed as:

$$\frac{1 - \left(a\theta\right)}{\left(1 - \phi\right)^{2.5}} f' + \frac{\left(-a\theta'\right)}{\left(1 - \phi\right)^{2.5}} f' + \left(1 - \phi + \phi \frac{\rho\_S}{\rho\_f}\right) \left\{\frac{\left(n + 3\right)}{2} f f'' - n \left(f'\right)^2\right\} \pm G\_r \theta = 0,\tag{8}$$

$$
\left(\frac{k\_{\rm nf}}{k\_{\rm f}}\right)\theta'' + \Pr\left(\frac{n+3}{2}\right)\left(1 - \phi + \phi \frac{\left(\rho c\_p\right)\_s}{\left(\rho c\_p\right)\_f}\right)[\left(f\theta'\right)] = 0,\tag{9}
$$

$$f(0) = 0, f'(0) = 1, f'(\circ \circ) = 0, \theta(0) = 1, \theta(\circ \circ) = 0,\tag{10}$$

where Pr <sup>¼</sup> <sup>μ</sup><sup>0</sup> ð Þ cp <sup>f</sup> kf is the Prandtl number.

The skin friction coefficients (Cfx) is defined as:

$$\mathbf{C}\_{\text{fr}} = \frac{\mu\_{nf}(T)}{\rho\_f \mu\_w^2} \left(\frac{\partial \mu}{\partial z}\right)\_{z=0}. \tag{11}$$

And the local Nusselt number (Nux) is defined as:

$$N\mu\_x = \frac{-\chi k\_{\eta f}}{k\_f (T\_w - T\_\ast)} \left(\frac{\partial T}{\partial \mathbf{z}}\right)\_{z=0}.\tag{12}$$

The dimensionless form of skin friction coefficients can be obtained as:

$$\left(\text{Re}\_x\right)^{1/2}\text{C}\_{\text{f}\text{x}} = \frac{\mu\_f(\theta(0))f''(0)}{\left(1-\phi\right)^{2.5}}.\tag{13}$$

And the dimensionless form of skin friction coefficients can be obtained as:

$$\left(\mathrm{Re}\_{\mathrm{x}}\right)^{1/2}\mathrm{Nu}\_{\mathrm{x}} = -\frac{k\_{\mathrm{uf}}}{k\_{\mathrm{f}}}\theta'(0). \tag{14}$$
