**2. Mathematical formulation**

In the coordinate plane, assume that the cylinder is taken in the vertical direction along the *z-*axis, and the *r-*axis is normal to the axis of the cylinder. Consider the fluid is moving with surface velocity,

$$U\_w = \frac{bz}{1 - at}$$

In the direction of stretching cylinder under the external magnetic field defined by

$$B(t) = \frac{B\_0}{\sqrt{1 - at}}$$

Further, suppose *u* = *u*(*r, z, t*) and *w* = *w*(*r, z, t*) be the velocity components along the respective axes of the coordinate plane, while *T* = *T*(*r, z, t*) denotes the temperature of the nanofluid, as shown in **Figure 1**.

**Figure 1.** *Representation of the dilemma.*

*Effect of Titanium Oxide Nanofluid over Cattaneo-Christov Model DOI: http://dx.doi.org/10.5772/intechopen.106900*

Let the base fluid (water) and platelet-shaped NPS in thermal equilibrium. Under these assumptions, the equation of continuity, momentum equation, and energy equations are obtained, which are as follows:

$$\frac{\partial(ru)}{\partial r} + \frac{\partial(rw)}{\partial \mathbf{z}} = \mathbf{0} \tag{1}$$

$$+\frac{\mu}{\partial r}\frac{\partial w}{\partial r} + \frac{w}{\partial \mathbf{z}}\frac{\partial w}{\partial \mathbf{z}} = \frac{\nu}{r}\frac{\partial}{\partial r}\left(\frac{r\,\partial w}{\partial r}\right) - \frac{\sigma}{\rho}\,\,\mathrm{B}\_{\circ}^{\;2}w + \mathrm{g}\mathfrak{R}(T - T\infty) \tag{2}$$

$$\begin{aligned} \frac{\partial T}{\partial t} + \frac{u}{\partial r} \frac{\partial T}{\partial r} + \frac{w}{\partial \mathbf{z}} &= \frac{\nu}{Cp} \left( \frac{\partial w}{\partial r} \right)^2 + \lambda\_1 \left[ \frac{2\nu}{Cp} \left( \frac{\partial w}{\partial r} \right) \left( \frac{\partial^2 w}{\partial t \partial r} + u \frac{\partial^2 w}{\partial r^2} \right) - \left( \frac{\partial^2 T}{\partial t^2} \right) \right] \\ &+ u^2 \frac{\partial^2 T}{\partial r^2} + w^2 \frac{\partial^2 T}{\partial z^2} 2u \frac{\partial^2 T}{\partial t \partial r} + 2w \frac{\partial^2 T}{\partial t \partial z} + 2uw \frac{\partial^2 T}{\partial z \partial r} \\ &+ \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} \right) \frac{\partial T}{\partial r} + \left( \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial r} + u \frac{\partial w}{\partial z} \right) \frac{\partial T}{\partial z} \Bigg) \Bigg]. \end{aligned} \tag{3}$$

Subject to the boundary conditions

$$\mathbf{u} = \mathbf{0}, \mathbf{w} = U\_{\mathbf{w}}, \frac{\partial T}{\partial r} = \mathbf{0} \text{ at } \mathbf{r} = \mathbf{R},$$

$$w \to o, T \to T\_{\infty} \text{ as } r \to \infty \tag{4}$$

The thermophysical properties of density (*ρnf* ), dynamic viscosity (*μnf* ), electric conductivity (*σnf* ), diffusivity (*αnf* ), and heat capacity (*ρCp*) can be defined in Refs. [7, 8], while the ratio of thermal conductivity of nanofluid and base fluid is given by the following equation:

$$\frac{k\_{\eta f}}{k\_f} = \left[ \frac{k\_s + (m - 1)k\_f + (m - 1)\left(k\_s - k\_f\right)\phi}{k\_s + (m - 1)k\_f - \left(k\_s - k\_f\right)\phi} \right] \tag{5}$$

where *ϕ* denotes volume-fraction of NPS.

The thermophysical properties of titanium nanofluid with base fluid as water [9] are given in **Table 1**, while the viscosity coefficients *A*1*, A*2*,* and shape factor *m* values of TiO2 nanofluid [10] are listed in **Table 2**.


**Table 1.**

*Thermophysical properties of base fluid and TiO2 nanoparticles.*


**Table 2.**

*Viscosity and shape factor values of platelet-shaped nanoparticles.*

Introducing the transformations, as

$$\mathbf{T} = \mathbf{T}\_{\circ\circ} + \left(\mathbf{T}\mathbf{w} - \mathbf{T}\_{\circ\circ}\right)\theta(\eta), \eta = \left(\sqrt{\frac{c}{\nu(1-\alpha t)}}\right)\left(\frac{r^2 - R^2}{2R}\right), \Psi = \left(\sqrt{\frac{c\nu}{(1-\alpha t)}}\right)x\eta^\sharp(\eta)\,\partial\Psi\tag{6}$$

where ψ is the stream function (describes the flow pattern) and is defined as u = �1 *r ∂ψ <sup>∂</sup><sup>r</sup>* and w = <sup>1</sup> *r ∂ψ <sup>∂</sup>r*. The governing Eqs. (2)–(5) have been transformed to Eqs. (8)–(10) using similarity variables in Eq. (7), as

$$\begin{aligned} \mathbf{e}\_1(\mathbf{1} + 2\mathbf{C}\boldsymbol{\eta})\mathbf{f}'''(\boldsymbol{\eta}) + 2\boldsymbol{\epsilon}\_1 \mathbf{C}\mathbf{f}''(\boldsymbol{\eta}) - \boldsymbol{\epsilon}\_3 \mathbf{M} \, \mathbf{f}(\boldsymbol{\eta}) \\ + \left[\mathbf{f}(\boldsymbol{\eta})\mathbf{f}''^{(\boldsymbol{\eta})} - \mathbf{f}'^2(\boldsymbol{\eta}) - \mathbf{S}\left(\mathbf{f}'(\boldsymbol{\eta}) + \frac{\boldsymbol{\eta}}{2}\mathbf{f}''(\boldsymbol{\eta})\right)\right] + \lambda\boldsymbol{\theta}(\boldsymbol{\eta}) = \mathbf{0} \end{aligned} \tag{7}$$

ð Þ <sup>1</sup> <sup>þ</sup> 2C<sup>η</sup> <sup>ϵ</sup>1Ecf<sup>00</sup><sup>2</sup> ð Þþ η ϵ2 Pr <sup>θ</sup>00ð Þ<sup>η</sup> � � <sup>þ</sup> 2ϵ<sup>2</sup> Pr <sup>C</sup>θ0ð Þ<sup>η</sup> <sup>þ</sup> <sup>f</sup>ð Þ<sup>η</sup> <sup>θ</sup>0ð Þ<sup>η</sup> � S 2θ ηð Þþ <sup>η</sup> 2 f 00ð Þη � � þβ <sup>ϵ</sup>1Ec 1ð Þ <sup>þ</sup> <sup>2</sup><sup>η</sup> 3Sf<sup>00</sup><sup>2</sup> ð Þþ η Sηf 00ð Þη f 000ð Þ� η 2fð Þη f 00ð Þη f 000ð Þ<sup>η</sup> � �2ϵ1EcCfð Þ<sup>η</sup> <sup>f</sup> 002 ð Þη � <sup>S</sup><sup>2</sup> <sup>6</sup>θ ηð Þþ <sup>11</sup> 4 θ00ð Þη fð Þþ η S 5f<sup>0</sup> ð Þ<sup>η</sup> θ ηð Þ� <sup>11</sup> 2 fð Þη θ<sup>0</sup> ð Þ� η ηfð Þη θ00ð Þη <sup>þ</sup> <sup>η</sup> � <sup>1</sup> 2 � �<sup>f</sup> 0 ð Þη θ00ð Þþ η η 2 f 00ð Þη θ ηð Þ 0 BBB@ 1 CCCA þf 0 ð Þη θ00ð Þ� η fð Þη f 0 ð Þη θ<sup>0</sup> ð Þ� η fð Þη f 00ð Þη θ ηð Þþ f 02 ð Þη θ ηð Þ 0 BBBBBBB@ 1 CCCCCCCA 0 BBBBBBB@ 1 CCCCCCCA 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 �f 0ð Þ<sup>η</sup> θ ηð Þ*:* (8)

where β is the thermal relaxation parameter and is given by,

$$\mathfrak{J1} = \frac{c\mathbb{11}}{(1 - at)}\tag{9}$$

Under the boundary condition,

$$f(\mathbf{0}) = \mathbf{0}, \boldsymbol{f}^{\prime}(\mathbf{0}) = \mathbf{1}, \boldsymbol{\theta}^{\prime}(\mathbf{0}) = \frac{-k\_f}{k\_{\eta^f}} \ \boldsymbol{\eta}(\mathbf{1} - \boldsymbol{\theta}(\mathbf{0})) \text{ at } \eta = R$$

$$\boldsymbol{f}^{\prime}(\boldsymbol{\eta}) = \mathbf{0}, \boldsymbol{\theta}(\mathbf{0}) \text{ as } \eta \to \infty \tag{10}$$

Now the dimensionless constants, such as *Ec*, *Pr*, *ϕ*, *M,* and *S,* and that of *ϵ*1, *ϵ*2, and *ϵ*3 are used frequently in the above equations, defined in Ref. [11]. For various

values of dimensionless parameters, the value of the local Nusselt number is shown in **Table 3**. Nusselt number can be defined as,

$$\text{Nu} = \frac{zk\_{\text{nf}}}{k\_f(T\_{w-}T\_{\infty})} \left[ \left. \frac{\partial T}{\partial r} \right|\_{\text{r=R}} \right]\_{\text{r=R}} \tag{11}$$

The non-dimensionless form of Eq. (11), using Eq. (6), as

$$Re^{\frac{-1}{\mathbb{Z}}}Nu = -\frac{k\_{\eta^f}}{k\_f}\,\dot{\theta}^{'}(0)\tag{12}$$
