**2. Mathematical Modeling**

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Let us assume a thin film flow of a nanoliquid flow comprising CNTs past a time dependent linearly stretched surface. The elastic sheet emerges from a slender slit at the Cartesian coordinate system's origin (Figure 1). The surface moves along the *x*-axis (*y* = 0) with a velocity *uw*(*x*, *t*) = *bx* (1−α*t*), with *b* and *a* being the constants in the *y*-direction and temperature *Tw*(*x, y*). The stream function ξ is considered such that *u* = ξ*y*, and *v* = −ξ*x*.

**Figure 1.** The flow geometry of the model.

The thin film is of width *h*(*x*, *y*). The flow is laminar and incompressible. A magnetic field *B*(*x*, *t*) = *Bo*(1 − *at*) −1 <sup>2</sup> , is employed normal to the extended surface. The governing unsteady conservation equations [17] under the aforementioned assumptions are appended as follows:

$$\frac{\partial^2 \xi}{\partial x \partial y} - \frac{\partial^2 \xi}{\partial y \partial x} = 0 \tag{1}$$

$$
\frac{\partial^2 \xi}{\partial t \partial y} - \frac{\partial \xi}{\partial y} \frac{\partial^2 \xi}{\partial x \partial y} - \frac{\partial \xi}{\partial x} \frac{\partial^2 \xi}{\partial y^2} = \upsilon\_{nf} \frac{\partial^3 \xi}{\partial y^3} + \frac{\sigma\_{nf}}{\rho\_{nf}} B^2(t) \frac{\partial \xi}{\partial y} \cos^2 \varepsilon,\tag{2}
$$

$$(\rho \mathbf{C}\_p)\_{nf} \left( \frac{\partial T}{\partial t} + \frac{\partial \xi}{\partial y} \frac{\partial T}{\partial \mathbf{x}} - \frac{\partial \xi}{\partial \mathbf{x}} \frac{\partial T}{\partial y} \right) + \lambda\_2 \Omega\_2 = \left( k\_{nf} + \frac{16T\_{\infty} ^3 \sigma'}{3k^\*} \right) \frac{\partial^2 T}{\partial y^2} + q''' \tag{3}$$

With the following corresponding boundary conditions

$$\begin{cases} -\xi\_{\mathbf{x}} = 0, \ \xi\_{\mathbf{y}} = \mu\_{\mathbf{w}\_{\prime}} \ T = T\_{\mathbf{s}\_{\prime}} \quad \text{at} \quad \mathbf{y} = \mathbf{0},\\ \xi\_{\mathbf{y}\mathbf{y}} = \mathbf{0}, \ -\xi\_{\mathbf{x}} = h\_{t\prime} \ T = \mathbf{0}, \quad \text{as} \quad \mathbf{y} = h(t). \end{cases} \tag{4}$$

The Cattaneo-Christov term is defined as

$$\begin{split} \Omega\_{2} = \frac{\partial^{2}T}{\partial t^{2}} + \frac{\partial u}{\partial t} \frac{\partial T}{\partial x} + 2 \frac{\underline{d}}{\partial y} \frac{\partial^{2}T}{\partial t \partial x} - 2 \frac{\underline{d}}{\partial x} \frac{\partial^{2}T}{\partial t \partial y} + \frac{\partial v}{\partial t} \frac{\partial T}{\partial y} + \frac{\underline{d}}{\partial y} \frac{\partial^{2}\underline{\xi}}{\partial x \partial y} \frac{\partial T}{\partial x} + \frac{\underline{d}}{\partial x} \frac{\partial^{2}\underline{\xi}}{\partial y \partial x} \frac{\partial T}{\partial y} \\ + \left( \frac{\underline{d}}{\partial y} \right)^{2} \frac{\partial^{2}T}{\partial x} + \left( \frac{\partial x}{\partial x} \right)^{2} \frac{\partial^{2}T}{\partial y^{2}} - 2 \frac{\underline{d}}{\partial y} \frac{\partial}{\partial x} \frac{\partial^{2}T}{\partial x \partial y} - \frac{\underline{d}}{\partial y} \frac{\partial^{2}\underline{\xi}}{\partial x^{2}} \frac{\partial T}{\partial y} - \frac{\underline{d}}{\partial x} \frac{\partial^{2}\underline{\xi}}{\partial y^{2}} \frac{\partial T}{\partial x} \end{split} \tag{5}$$

The heat source/sink "*q*---" is represented by

$$q^{\prime\prime\prime} = \frac{k\_f u\_{\rm av} (T\_s - T\_0)}{\mathbf{x} \mathbf{v}\_f} \bigg( A^\* f' + B^\* \frac{(T - T\_0)}{(T\_s - T\_0)} \right) \tag{6}$$

The thermophysical attributes (specific heat *Cp*, density ρ and thermal conductivity *k*) of the base fluid (H2O) and carbon nanotubes (SWCNTs /MWCNTs) are appended in Table 2.

**Table 2.** The thermophysical physiognomies of the fluid and CNTs [30].


The hypothetical relations are characterized as follows:

$$
\mu\_{nf} = \frac{\mu\_f}{\left(1 - \Phi\right)^{\frac{2.5}{2.5}}}, \ v\_{nf} = \frac{\mu\_{nf}}{\rho\_{nf}}.\tag{7}
$$

$$\mathfrak{p}\_{nf} = (1 - \Phi)\mathfrak{p}\_f + \Phi \mathfrak{p}\_{\mathbb{C}\mathbb{N}^f} \text{ } \mathfrak{a}\_{\mathbb{H}f} = \frac{k\_{nf}}{\mathfrak{p}\_{nf} \left(\mathfrak{c}\_p\right)\_{nf}}\tag{8}$$

$$\frac{\sigma\_{nf}}{\sigma\_f} = 1 + \frac{3\sigma\Phi - 3\phi}{\sigma + 2 - \sigma\Phi + \phi}, \sigma = \frac{\sigma\_{CNT}}{\sigma\_f},\tag{9}$$

$$\frac{k\_{nf}}{k\_f} = \frac{(1-\Phi) + 2\Phi \frac{k\_{\text{CNT}}}{k\_{\text{CNT}} - k\_f} \ln(\frac{k\_{\text{CNT}} + k\_f}{2k\_f})}{(1-\Phi) + 2\Phi \frac{k\_f}{k\_{\text{CNT}} - k\_f} \ln(\frac{k\_{\text{CNT}} + k\_f}{2k\_f})}\tag{10}$$

Using the similarity transformations

 $\eta = \frac{1}{\mathcal{S}} \left(\frac{b}{\mathbf{v}\_f (1 - at)}\right)^{\frac{1}{2}}$  $y, \text{ } \Psi = \mathcal{S} \left(\frac{b \mathbf{v}\_f}{(1 - at)}\right)^{\frac{1}{2}}$  $x f(\eta), \theta = \frac{T - T\_0}{T\_s - T\_0}$  $\eta = T - T\_0 - T\_f \left(\frac{b \mathbf{v}^2}{2 \mathbf{v}\_f}\right)$  $(1 - at)^{-1.5} \Theta(\eta),$ 

The requirement of Equation (1) is fulfilled undoubtedly and Equations (2) and (3) yield

$$f^{\prime\prime\prime} + (1 - \phi)^{2.5} (1 - \phi + \phi \frac{\text{PCN}}{\rho\_f}) \lambda \left( f f^{\prime\prime} - f^{\prime 2} - S \left( f^{\prime} + \frac{1}{2} \mathfrak{n} f^{\prime\prime} \right) \right) - (1 - \phi)^{2.5} \frac{\sigma\_{\text{\tiny if}}}{\sigma\_f} M f^{\prime} \cos^2 \varepsilon = 0 \tag{12}$$

$$\begin{split} & \frac{\left(\frac{k\_{pf}}{k\_f} + \frac{4}{3}R\right)}{\sqrt{1-\Phi+\phi}\frac{\left(\rho C\_{p}\right)\_{\text{CVI}}}{\left(\rho C\_{p}\right)\_{f}}} \Theta'' - \lambda \Big[2f'\Theta - f\Theta' + \frac{\xi}{2}\big(3\Theta + \eta\Theta'\big)\Big] + \frac{1}{P\_{r}\left[1-\Phi+\phi\frac{\left(\rho C\_{p}\right)\_{\text{CVI}}}{\left(\rho C\_{p}\right)\_{f}}\right]} \left(A^{\*}f' + B^{\*}\Theta\right) \\ & + \gamma \Big\{ - \frac{15}{3}S^{2}\Theta - \frac{7}{2}S^{2}\eta\Theta - \frac{1}{4}S^{2}\eta^{2}\Theta'' - 8Sf'\Theta - \eta Sf'\Theta - \frac{3}{2}\eta Sf'\Theta'\Big{)}\Big{]}\Theta' - \frac{3}{2}\eta Sf'\Theta'\Big{)} = 0, \\ \end{split} \tag{13}$$

Additionally, the boundary conditions of Equation (4) become

$$f(0) = 0, \ f'(0) = 1, \ \theta(0) = 1, \ f(1) = \frac{S}{2}, \ f''(1) = 0, \ \theta'(1) = 0 \tag{14}$$

The values of various non-dimensional parameters are defined as follows:

$$P\_I = \frac{\upsilon\_f}{\alpha\_f}, S = \frac{\alpha}{b}, R = \frac{4\sigma^\* T\_0^{\circ}}{k^\* k\_f}, M = \frac{\sigma\_f B\_0^{\circ 2}}{b\mathfrak{p}\_f}, \gamma = \frac{\lambda\_2 b}{1 - \alpha t}, \lambda = \beta^2 \tag{15}$$

Physical quantities like the Skin friction coefficient and the local Nusselt number are given as

$$\begin{array}{lcl} \mathrm{Nu}\_{x} = \frac{\mathrm{xq}\_{w}(x)}{k\_{f}(T\_{s} - T\_{0})}, \mathbf{C}\_{f} = \frac{\mathbf{r}\_{w}}{\rho\_{f}u\_{w}\mathbf{r}^{2}},\\ q\_{w}(\mathbf{x}) = -k\_{nf}(\frac{\partial T}{\partial y})\_{y=0}, \mathbf{r}\_{w} = \mu\_{nf}(\frac{\partial u}{\partial y})\_{y=0} \end{array} \tag{16}$$

Additionally, in dimensionless form, as follows:

$$\begin{array}{l} \mathbb{C}\_{f} \text{Re}\_{\mathbf{x}}^{-1/2} = \frac{1}{\mathfrak{G} \left(1 - \mathfrak{G}\right)^{2.5}} f'' \left(0\right), \\ \mathrm{Nu}\_{\mathbf{x}} \mathrm{Re}\_{\mathbf{x}}^{-1/2} = -\frac{1}{\mathfrak{G}} \left(\frac{k\_{\pi f}}{k\_f} + \frac{4}{3} \mathcal{R}\right) \mathfrak{G}'(0) \end{array} \tag{17}$$
