2. Formulation

Consider the thin film flow of Casson nanofluid elegantly through a circular cylinder of radius "a." The cylinder is supposed to be stretched along with radial direction with velocity Uw, and temperature at the surface of cylinder is taken Tw. The uniform ambient temperature is considered Tb such that Tw � Tb > 0 for assisting flow and Tw � Tb < 0 for opposing flow.

The governing equations of continuity, heat transfer, and mass transfer are

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

$$
\mu \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z} = \nu \left( \left( + \frac{1}{\beta\_1} \right) \left( \frac{\partial^2 w}{\partial r^2} + \frac{1}{r} \frac{\partial w}{\partial r} \right) \left( + g\beta^\* (T - T\_b)(1 - \mathbb{C}\_b) + \frac{1}{\rho} (\rho^\* - \rho)(\mathbb{C} - \mathbb{C}\_b) \right) \right) \tag{2}
$$

$$u\frac{\partial u}{\partial r} + w\frac{\partial u}{\partial z} = -\frac{1}{\rho}\frac{\partial P}{\partial r} + \nu \left( \left( +\frac{1}{\beta\_1} \right) \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{u}{r^2} \right) \right) \tag{3}$$

$$
\mu \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} = \alpha \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} \right) \left( \frac{\rho^\* c\_p^\*}{\rho c\_p} \quad D\_B \frac{\partial T}{\partial r} \frac{\partial \phi}{\partial r} + \frac{D\_T}{T\_b} \left( \frac{\partial T}{\partial r} \right)^2 \right) \tag{4}
$$

$$
\mu \frac{\partial \mathbb{C}}{\partial r} + w \frac{\partial \mathbb{C}}{\partial z} = D\_B \left( \frac{\partial^2 \mathbb{C}}{\partial r^2} + \frac{1}{r} \frac{\partial \mathbb{C}}{\partial r} \right) \left( \frac{D\_T}{T\_b} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} \right) \right) \tag{5}
$$

where u r<sup>ð</sup> ; <sup>z</sup><sup>Þ</sup> and w r<sup>ð</sup> ; <sup>z</sup><sup>Þ</sup> are velocity components; <sup>r</sup> is density; <sup>υ</sup> is kinematic viscosity; <sup>β</sup><sup>1</sup> is the <sup>∗</sup> constant characteristic to Casson fluid; <sup>β</sup> is the coefficient of thermal expansion; <sup>g</sup> is the gravitational acceleration along z-axis; T, Tb, C and Cb determine the temperature, ambient temperature, concentration, and ambient concentration, respectively; and α, DT, DB stands for the thermal diffusivity, thermophoresis diffusion coefficient, and Brownian diffusion coefficient.

The suitable boundary conditions are

$$\mu = \mathcal{U}\_{w\prime} \, w = \mathcal{W}\_{w\prime} \, T = T\_{w\prime} \, \mathbb{C} = \mathbb{C}\_{w} \, \text{ at } \, r = a,\tag{6}$$

$$
\mu \frac{\partial w}{\partial r} = \frac{\partial T}{\partial r} = \frac{\partial C}{\partial r} = 0,\\
u = w \frac{d\delta}{dz} \quad \text{at} \quad r = b. \tag{7}
$$

Here Uw ¼ �ca represents the suction and injection velocity, and Ww ¼ 2cz is the stretching velocity such that c represents the stretching parameter and δ is the thickness of fluid film.

The similarity transformations are used to alter the basic Eqs. (1)–(7) used in [22] as

$$u = -ca \frac{f(\eta)}{\sqrt{\eta}} \Big/ \eta = 2cz \frac{df}{d\eta}, \\ T(z) = T\_b - T\_{ref} \left(\frac{\varrho^2}{\Phi\_{\eta f}}\right) \Big/ (\eta), \\ \mathcal{C}(z) = \mathcal{C}\_b - \mathcal{C}\_{ref} \left(\frac{\varrho^2}{\Phi\_{\eta f}}\right) \phi(\eta), \tag{8}$$
 
$$\eta = \left(\frac{r}{r}\right)^2.$$

where

<sup>η</sup> <sup>¼</sup> : <sup>a</sup>

� �<sup>2</sup> In the case of the outer radius <sup>b</sup> of the flow, <sup>η</sup> <sup>¼</sup> <sup>b</sup> : <sup>a</sup>

Using these transformations in Eqs. (1), (2), (4)–(7), we obtained a set of dimensionless equations which is

$$\left( \left( + \frac{1}{\beta\_1} \right) \left( \mathfrak{d} \frac{\partial^3 f}{\partial \eta^3} + \frac{\partial^2 f}{\partial \eta^2} \right) \left( + \text{Re} \quad f \frac{\partial^2 f}{\partial \eta^2} - \left( \frac{\partial f}{\partial \eta} \right)^2 + \left( \lambda \left( \Theta + Nrq \right) \right) \right) \right) = 0,\tag{9}$$

$$
\eta \frac{\partial^2 \theta}{\partial \eta^2} + \frac{\partial \theta}{\partial \eta} + \text{Pr.Re} \left( \left( \frac{\partial \theta}{\partial \eta} - 2 \frac{\partial f}{\partial \eta} \theta \right) \left( + \eta \frac{\partial \theta}{\partial \eta} \left( \left( \delta\_t \frac{\partial \theta}{\partial \eta} + N\_b \frac{\partial \phi}{\partial \eta} \right) \right) \right) = 0,\tag{10}
$$

$$
\eta \frac{\partial^2 \phi}{\partial \eta^2} + \frac{\partial \phi}{\partial \eta} + Le. \text{Re}\left(\oint\_{\partial \eta} -2 \frac{\partial f}{\partial \eta} \phi \right) + \frac{N\_t}{N\_b} \left(\eta \frac{\partial^2 \theta}{\partial \eta^2} + \frac{\partial \theta}{\partial \eta} \right) \Big| = 0,\tag{11}
$$

where

$$\begin{split} \text{Re} &= \frac{c\boldsymbol{\alpha}^{2}}{2\boldsymbol{\upsilon}\_{\text{nf}}}, \boldsymbol{\lambda} = \frac{\operatorname{g}\mathcal{P}^{\*}\boldsymbol{a}(T\_{\text{w}} - T\_{\text{w}})(1 - \mathsf{C}\_{\text{w}})}{\mathcal{W}\_{\text{w}}^{2}}, \operatorname{N} \boldsymbol{r} = \frac{(\boldsymbol{\rho} - \boldsymbol{\rho}^{\*})(\mathsf{C}\_{\text{w}} - \mathsf{C}\_{\text{w}})}{\boldsymbol{\rho}\boldsymbol{\beta}^{\*}(T\_{\text{w}} - T\_{\text{w}})(1 - \mathsf{C}\_{\text{w}})}, \\ \text{Pr} &= \frac{\mu c\_{p}}{k}, \mathcal{N}\_{t} = \frac{\boldsymbol{\rho}^{\*}\boldsymbol{c}\_{p}^{\*}D\_{T}\Delta T}{\rho\boldsymbol{c}\_{p}\alpha T\_{b}}, \operatorname{Le} = \frac{\boldsymbol{\nu}}{D\_{\text{B}}}, \mathcal{N}\_{b} = \frac{\boldsymbol{\rho}^{\*}\boldsymbol{c}\_{p}^{\*}D\_{\text{B}}\Delta\mathsf{C}}{\rho\boldsymbol{c}\_{p}\alpha}. \end{split} \tag{12}$$

In Eq. (12) Re stands for the Reynolds number, λ is the buoyancy parameter or in other word it is the natural convection parameter, Nr stands for the buoyancy ratio, Pr represents the Prandtl number, Nt is used to represent thermophoresis parameter, Le is Lewis number, and Nb is Brownian motion parameter.

Physical conditions for momentum, thermal, and concentration fields are transformed as

$$f(1) = \frac{\partial f(1)}{\partial \eta} = \theta(1) = \phi(1) = 1,\tag{13}$$

$$\frac{\partial^2 f(\boldsymbol{\beta})}{\partial \eta^2} \neq \begin{pmatrix} f(\boldsymbol{\beta}) \ \neg \frac{\partial \boldsymbol{\partial} \boldsymbol{\theta}(\boldsymbol{\beta})}{\partial \eta} \end{pmatrix} \neq \begin{pmatrix} \partial \boldsymbol{\phi}(\boldsymbol{\beta}) \\ \frac{\partial \boldsymbol{\eta}}{\partial \eta} \end{pmatrix} \tag{14}$$

where β is the thickness of liquid film sprayed on the outer surface of the cylinder.

Integrating Eq. (3) for pressure term

$$\frac{p - p\_b}{\mu c} = \frac{\text{Re}}{\eta} f^2 - 2 \left( \left( + \frac{1}{\beta\_1} \right) \frac{\partial f}{\partial \eta} . \tag{15}$$

At the outer surface, the shear stress of the liquid film is zero, i.e.,

$$\frac{\partial^2 f(\boldsymbol{\beta})}{\partial \boldsymbol{\eta}^2} = \begin{cases} 0. & \text{if } \boldsymbol{\eta} = \boldsymbol{\eta} \\ \end{cases} \tag{16}$$

The shear stress on the cylinder is

$$
\pi = \frac{\rho v \imath c z}{a} f''(1) = \frac{4c\mu z}{a} f''(1) \tag{17}
$$

The Heat and Mass Transfer Analysis During Bunch Coating of a Stretching Cylinder by Casson Fluid 45 http://dx.doi.org/10.5772/intechopen.79772

The deposition velocity V is written as

$$-V = -ca\frac{f' \langle \beta \rangle}{\sqrt{\beta}}.\tag{18}$$

Mass flux m<sup>1</sup> is in association with the deposition per axial length which is

$$m\_1 = V 2\pi b$$

The normalized mass flux m<sup>2</sup> is

$$m\_2 = \frac{m\_1}{2\pi a^2 c} = \frac{m\_1}{4\pi \upsilon\_{\eta f} \text{Re}} = f(\beta) \tag{20}$$

The flow, temperature, and concentration rates are

$$\mathbf{S}\_{f} = \frac{2\nu\_{\eta f}}{W\_{w}} \left(\frac{\partial w}{\partial r}\right)\_{r=a} \text{,} \mathbf{N}u = -\frac{ak}{(T\_{w} - T\_{b})} \left(\frac{\partial T}{\partial r}\right)\_{r=a} \text{,} \mathbf{S}h = -\frac{a}{2(\mathsf{C}\_{w} - \mathsf{C}\_{b})} \left(\frac{\partial \mathbf{C}}{\partial r}\right)\_{r=a} . \tag{21}$$

The nondimensional forms for the abovementioned physical properties are

$$\frac{z\text{Re}}{a}S\_f = \frac{\partial^2 f(1)}{\partial \eta^2}, \text{Nu} = -2k \frac{\partial \theta(1)}{\partial \eta}, \text{Sh} = -\frac{\partial \phi(1)}{\partial \eta}. \tag{22}$$
