**2. Description of Problem**

We consider steady and incompressible two-dimensional thin film Casson nanofluids flow along a stretching upright cylinder of radius *a*. The *z*-axis represents along the surface of the cylinder and the *r*-axis is that taken radially, as shown in Figure 1. The cylinder is supposed to electrically conduct with constant *B* (magnetic field) of strength *B*0. Here, *Tw* = *Ta* is the surface temperature, while *T<sup>δ</sup>* = *Tb* is the free surface temperature of the cylinder. In this scenario, the tube surface is stretching with velocity *Ww* = 2*s z* along the *z*-axis. Here, *s* > 0 is used for extension of the cylinder surface, while for contraction, *s* < 0 is used. Additionally, the thermal field for the present problem is [11]

$$T = T\_b - T\_r \left(\frac{c \ z^2}{\upsilon\_{nf}}\right) \Theta(\eta),\tag{7}$$

where *Tr* is the reference temperature. Furthermore, the human blood-based nanoliquid comprises two sorts of CNTs (SWCNTs and MWCNTs) [24]. Viscous dissipation and natural convection have been involved in nanofluid flow. The stress tensor of the Casson fluid model [36,37] is implemented as

$$\begin{aligned} \pi\_{mn}^{sf} &= 2\varepsilon\_{mn}\mu\_a^{df} + 2\varepsilon\_{mn}\frac{p\_y}{\sqrt{2\pi}\_d}, \text{ where } \pi\_d \ge \pi\_{cr\_r} \text{ and }\\ \pi\_{mn}^{sf} &= 2\varepsilon\_{mn}\mu\_a^{sf} + 2\varepsilon\_{mn}\frac{p\_y}{\sqrt{2\pi}\_d}, \text{ where } \pi\_d \prec \pi\_{cr}. \end{aligned} \tag{8}$$

In the above expression, the share stress along *m*th and *n*th components is *τs f mn*, the deformation rate is *πd*, deformation rate components *m*th and *n*th are *emn*, the critical value represented by *πcr* is focused on the non- Newtonian fluid model, *μs f <sup>a</sup>* is the plastic dynamic viscosity of Casson fluid, and the produce stress of the fluid is *py*.

By applying the order analysis, the suggested boundary film equations of carbon nanotubes fluid are [11]

$$\frac{\partial(ru)}{\partial r} + \frac{\partial(rw)}{\partial z} = 0,\tag{9}$$

$$\rho\_{\rm nf} \left[ u \left( \frac{\partial w}{\partial r} \right) + w \left( \frac{\partial w}{\partial z} \right) \right] = \mu\_{\rm nf} \left( 1 + \frac{1}{\beta} \right) \left[ \frac{\partial^2 w}{\partial r^2} + \frac{1}{r} \left( \frac{\partial w}{\partial r} \right) \right] + \left( \rho \theta^{\odot} \right)\_{\rm nf} (T - T\_b) \text{g} - \sigma\_{\rm nf} B\_0^2 w \,, \tag{10}$$

$$
\rho\_{nf} \left[ u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} \right] = -\frac{\partial p}{\partial r} + \mu\_{nf} \left( 1 + \frac{1}{\beta} \right) \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} - \frac{u}{r^2} \right), \tag{11}
$$

$$
\left(\rho c\_p\right)\_{nf} \left(\mu \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z}\right) = k\_{nf} \left(\frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r}\right) + \mu\_{nf} \left(\frac{\partial w}{\partial r}\right)^2. \tag{12}
$$

Here, *r*, *z* are the radial and axial coordinates, respectively. Additionally, *u*(*r*, *z*) and *w*(*r*, *z*) are the velocity elements in the *<sup>r</sup>* and *<sup>z</sup>* directions. *<sup>β</sup>* <sup>=</sup> *<sup>μ</sup>s f a* <sup>√</sup>2*πcr <sup>τ</sup>mn* is the material parameter (Casson parameter); the local pressure and temperature are specified by *p* and *T*, respectively; the specific density of the nanofluid is *ρn f* ; the dynamic viscosity of the nanofluid is *μn f* ; *β*<sup>⊗</sup> *n f* is the thermal expansion coefficient of nanoparticles; the electrical conductivity of the nanofluid is *σn f* ; the thermal conductivity of the nanofluid is *kn f* ; and the specific heat capacity of the nanofluid is *ρcp n f* .

**Figure 1.** Schematic diagram of flow model and coordinate system.

The subjected boundary conditions for the present analysis are as follows [11]:

$$u = \mathcal{U}\_{\mathcal{W}\_{\prime}} \; w = \mathcal{W}\_{\mathcal{W}\_{\prime}} \; T = T\_{\mathcal{W}\_{\prime}} \; at \; r = a\_{\prime} \tag{13}$$

$$
\mu \frac{\partial w}{\partial r} = 0,\\
\frac{\partial T}{\partial r} = 0,\\
\text{ }u = w \frac{d\delta}{dz}, \text{ at } r = b. \tag{14}
$$

where *b* is the outer radius which display the thickness of the liquid film, the expression of suction and injection velocity is *Uw*, and *Ww* is the extended velocity of the cylinder surface.

*Non-Dimensional Parameters*

With the aid of the following suitable conversions [11]:

$$\eta = \frac{r^2}{d^2}, \ u = \frac{-sa}{\sqrt{\eta}} [f(\eta)], \ w = 2sz \left[ \frac{df(\eta)}{d\eta} \right], \Theta = \frac{T - T\_b}{T\_a - T\_b}. \tag{15}$$

The transformed equations for momentum and energy arise are

$$\begin{split} & \left( 1 + \frac{1}{P} \right) \left[ \eta \left( \frac{d^2 f(\eta)}{d\eta^2} \right) + \frac{d^2 f(\eta)}{d\eta^2} \right] + \\ & \left( (1 - q) + \eta \frac{\rho\_{\rm CAT}}{\rho\_f} \right) (1 - \varrho)^{2.5} \left[ \text{Re} \left( f(\eta) \left( \frac{d^2 f(\eta)}{d\eta^2} \right) - \left( \frac{df(\eta)}{d\eta} \right)^2 \right) + Gr \Theta(\eta) - Mf(\eta) \frac{df(\eta)}{d\eta} \right] = 0, \end{split} \tag{16}$$
 
$$\begin{split} & \frac{k\_{\rm f}}{k\_f} \left( 2\eta \frac{d^2 \Theta(\eta)}{d\eta^2} + \frac{d \Theta(\eta)}{d\eta} \right) + \text{PrRe} \left( (1 - \varrho) + \eta \frac{\{\rho c\_f\}\_{\rm CAT}}{\left( \rho c\_p \right)\_f} \right) [f(\eta) \frac{d \Theta(\eta)}{d\eta} - 2 \Theta(\eta) \frac{df(\eta)}{d\eta} \\ & \quad + E\_{\zeta} \left( \frac{d^2 f(\eta)}{d\eta^2} \right)^2 \Big] = 0. \end{split} \tag{17}$$

The resultant transformed dimensionless boundary conditions are

$$f(\eta) = 1, \ \frac{df(\eta)}{d\eta} = 1, \Theta(\eta) = 1 \text{ at } \eta = 1,\tag{18}$$

$$\frac{d^2 f(\eta)}{d\eta^2} = 0, \; \frac{d\Theta(\eta)}{d\eta} = 0, \; at \; \eta = a. \tag{19}$$

The solid particle volume fraction is *ϕ*, and in non-dimensional form, the variable thickness is

$$
\kappa = \frac{b^2}{a^2} = \eta\_b.\tag{20}
$$

Here, *a*, *b*, *α* represent the radius of the cylinder, the external radius of the thin layer, and the dimensionless thickness of the thin layer, respectively.

$$\text{Re} = \frac{\text{sa}^2}{2\upsilon\_f},\\\text{Pr} = \frac{\mu\_f (\upsilon\_p)\_f}{k\_f},\\\text{M = } \frac{\upsilon\_f B\_0^2 a^2}{4\mu\_f},\\\text{Gr} = \frac{a^2 \text{g} (T - T\_0) (\not p \,\rho)\_f}{4\mathcal{W}\_0 \mu\_f},\\E\_c = \frac{\mathcal{W}\_w^2 a^2}{\Delta T (\ c\_p)\_f}. \tag{21}$$

Re is the Local Reynolds number, Pr is the prandtl number, *M* is the magnetic parameter, *Gr* is the Grashof number, and *Ec* is the Eckert number in dimensionless form defined as in [11].

Evaluating the pressure distribution term from Equation (11):

$$\frac{p - p\_b}{\mu c\_f} = -\frac{\text{Re}}{\eta} \left( (1 - \varphi) + \varrho \frac{\rho\_{\text{CNT}}}{\rho\_f} \right) (1 - \varrho)^{2.5} f^2(\eta) - 2 \left( 1 + \frac{1}{\beta} \right) \frac{df(\eta)}{d\eta}.\tag{22}$$

Now, the shear stress at the free surface of the fluid film is zero, which means that

$$\frac{d^2f(a)}{d\eta^2} = 0.\tag{23}$$

Also, the corresponding shear stress at the cylinder surface is

$$\tau\_w = \frac{4s \ z(\rho v)\_{nf}}{a} \left( 1 + \frac{1}{\beta} \right) \left[ \frac{d^2 f(1)}{d \eta^2} \right] = \frac{4s \ z \mu\_{nf}}{a} \left( 1 + \frac{1}{\beta} \right) \left[ \frac{d^2 f(1)}{d \eta^2} \right]. \tag{24}$$

The non-dimensional forms of *Cf* , *Nu* (skin friction and Nesselt number, respectively) are expressed as [11]

$$\frac{zR\varepsilon}{a}\Big|\mathcal{C}\_f = \left(1 + \frac{1}{\beta}\right) \left[\frac{d^2f(1)}{d\eta^2}\right] (1-\rho)^{-2.5}, \text{ Nu} = -2\frac{k\_{\it\ell}}{k\_f} \left[\frac{d\Theta(1)}{d\eta}\right].\tag{25}$$

Here, *Re* = *sa*<sup>2</sup> <sup>2</sup> *<sup>υ</sup><sup>f</sup>* denotes the Reynolds number.

 

#### **3. Solution Methodology**

In this paper, we use the HAM technique. The HAM scheme was initially planned by Liao [32,33] and he construed the idea of Homotopy. With the help of HAM, Equations (16) and (17) are solved along with the suggested boundary condition in Equations (18) and (19). To control and improve the convergence of the problem, we used the auxiliary constant ђ. A selection of initial gasses is

$$f\_0(\eta) = \frac{\kappa}{2(\kappa - 1)^3} \left[ \eta^3 - 3\alpha \eta^2 - (3 - 6\alpha)\eta + (2 - 3\alpha) \right] + \eta , \ \Theta\_0(\eta) = 1. \tag{26}$$

L*<sup>f</sup>* and L<sup>Θ</sup> are linear operators such that

$$\mathcal{L}\_f = \frac{d^4 f(\eta)}{d\eta^4} \text{ and } \mathcal{L}\_\Theta = \frac{d^2 \Theta(\eta)}{d\eta^2},\tag{27}$$

The general result of L*<sup>f</sup>* and L<sup>Θ</sup> is

$$\mathbb{L}\_f \left\{ \mathbf{K}\_1 + \mathbf{K}\_2 \boldsymbol{\eta} + \mathbf{K}\_3 \boldsymbol{\eta}^2 + \mathbf{K}\_4 \boldsymbol{\eta}^3 \right\} = 0 \text{ and } \mathbb{L}\_\Theta \{ \mathbf{K}\_5 + \mathbf{K}\_6 \boldsymbol{\eta} \} = 0. \tag{28}$$

For velocity and temperature distribution, the Taylor's expansions have been applied as follows:

$$f(\eta;\rho) = f\_0(\eta) + \sum\_{\vec{\xi}=1}^{\infty} f\_{\vec{\xi}}(\eta)\rho^{\vec{\xi}}\,\,\,\,\tag{29}$$

$$\Theta(\eta;\rho) = \Theta\_0(\eta) + \sum\_{\vec{\xi}=1}^{\infty} \Theta\_{\vec{\xi}}(\eta)\rho^{\vec{\xi}}.\tag{30}$$

But

$$f\_{\tilde{\xi}}(\eta) = \frac{1}{\tilde{\xi}!} \frac{df(\eta; \rho)}{d\eta} \Big|\_{\rho=0} \text{ and } \Theta\_{\tilde{\xi}}(\eta) = \frac{1}{\tilde{\xi}!} \frac{d\Theta(\eta; \rho)}{d\eta} \Big|\_{\rho=0}. \tag{31}$$

For Equations (16) and (17), the *ξ*th order system is as follows [11]:

$$\mathcal{L}\_f \left[ f\_{\vec{\xi}}(\eta) - \tilde{\mathcal{N}}\_{\vec{\xi}} f\_{\vec{\xi}-1}(\eta) \right] = \lambda\_f \mathcal{R}\_{\vec{\xi}}^f(\eta), \tag{32}$$

$$\mathcal{L}\_{\Theta} \left[ \Theta\_{\tilde{\xi}}(\eta) - \tilde{\mathcal{N}}\_{\tilde{\xi}} \Theta\_{\tilde{\xi}-1}(\eta) \right] = \lambda\_{\Theta} R\_{\tilde{\xi}}^{\Theta}(\eta). \tag{33}$$

where

$$
\tilde{\mathcal{N}}\_{\tilde{\xi}} = \begin{cases} \ 1, \text{ if } \ \rho > 1 \\\ 0, \text{ if } \ \rho \le 1 \end{cases} . \tag{34}
$$
