*2.1. Flow Modelling*

In the current work, we examine the steady and incompressible boundary layer hydro-magnetic flow of alumina-copper/water hybrid nanofluids through a permeable stretch cylinder. An external magnetic field is applied. The hybrid nanofluid flow is affected in the axial direction by the stretching of the elastic cylinder. The Darcy–Forchheimer impact is included in momentum equation. The coordinate system are selected in such a method that fluid flow is started due to elongating cylinder in the axial direction, where (*x*, *r*) shows axial and radial directions.

The energy, mass, and momentum conservation laws in boundary layer approximation can be represented as:

$$\frac{\partial}{\partial \mathbf{x}}(ru) + \frac{\partial}{\partial \mathbf{x}}(rv) = 0 \tag{1}$$

$$\left(\mu\frac{\partial}{\partial x}\mu + v\frac{\partial}{\partial r}\mu\right) = \nu\_{\text{Inf}}\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right) - \frac{1}{\rho\_{\text{Inf}f}}\left[\sigma\_0\theta\_0^2\mu + (\rho\theta)\_{\text{Inf}f}g(T - T\_{\text{co}}) - \frac{\nu\_{\text{nf}}}{k}u - Fu^2\right],\tag{2}$$

$$
\tau \left( u \frac{\partial}{\partial x} T + v \frac{\partial}{\partial r} T \right) = \frac{k\_{\rm Imf}}{\left( \rho c\_p \right)\_{\rm Imf}} \left( \frac{\partial}{\partial r} \left( r \frac{\partial}{\partial r} T \right) + \frac{1}{r} \frac{\partial}{\partial r} T \right) + \tau \left( D\_B \frac{\partial}{\partial r} \mathcal{C} \frac{\partial}{\partial r} T + \frac{D\_T}{T\_{\rm co}} \left( \frac{\partial}{\partial r} T \right)^2 \right), \tag{3}
$$

$$\left(u\frac{\partial}{\partial \mathbf{x}}\mathbb{C} + v\frac{\partial}{\partial r}\mathbb{C}\right) = D\_{\mathbb{B}}\left(\frac{\partial^2}{\partial r^2}\mathbb{C} + \frac{1}{r}\frac{\partial}{\partial r}\mathbb{C}\right) + \frac{D\_T}{T\_{\infty}}\left(\frac{\partial^2}{\partial r^2}T + \frac{1}{r}\frac{\partial}{\partial r}T\right). \tag{4}$$

Boundary conditions are:

$$\begin{aligned} r &= a \quad \mu = c\mathbf{x}, \; v = 0, \; T = T\_{w\prime} \; \mathbb{C} = \mathbb{C}\_{w\prime} \\ r &\to \infty \, \mu = 0, \; T = T\_{\mathbb{C}\mathbb{O}\_{\prime}} \; \mathbb{C} = \mathbb{C}\_{\mathbb{O}^{\otimes}} \end{aligned} \tag{5}$$

We employ the following transformations to convert the model equations into dimensionless form are [36]:

$$\begin{array}{ll} \eta = \frac{r^2 - a^2}{2a} \sqrt{\frac{c}{\nu}}, & u = c x f'(\eta), \ v = -\frac{a}{r} \sqrt{c \nu} f(\eta), \\\ \theta(\eta) = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}}, \phi(\eta) = \frac{\mathbb{C} - \mathbb{C}\_{\text{av}}}{\mathbb{C}\_w - \mathbb{C}\_{\text{av}}} \end{array} \tag{6}$$

By using Equation (7) in Equations (1) to (6), we get:

$$A\_1((1+2\eta\gamma)f''''+2\gamma f'')+A\_2(f f'' - f'^2(1-\text{Fx}))-f'(M+k\_1A\_2)+A\_3\lambda\theta = 0\tag{7}$$

$$(1 + 2\eta \gamma)\theta'' + \gamma \theta' + \Pr(1 + 2\eta \gamma) \{\mathcal{N}\_b \theta' \phi' + \mathcal{N}\_t \theta'^2\} + \Pr f \theta' = 0 \tag{8}$$

$$(1 + 2\eta \gamma)\phi'' + \gamma \phi' + \frac{N\_t}{N\_b} \mathbb{I} (1 + 2\eta \gamma)\theta'' + \gamma \theta' \mathbb{I} + \mathcal{S}cf\phi' = 0 \tag{9}$$

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

$$f(\circ \circ) = \theta(\circ \circ) = \phi(\circ \circ) = 0 \tag{11}$$

We have defined the different parameters in Equations (8)–(10) as; *<sup>k</sup>*<sup>1</sup> <sup>=</sup> <sup>ν</sup>*n f kc* , is the permeability parameter, Re = *cx*<sup>2</sup> <sup>ν</sup> is the local Reynolds number, *Nt* <sup>=</sup> <sup>τ</sup>*DT*(*Tw*−*T*∞)*x*<sup>3</sup> *<sup>T</sup>*∞<sup>ν</sup> is the thermophoresis constraint, Pr = νρ*cp <sup>k</sup>* is the Prandtl number. *Nb* <sup>=</sup> <sup>τ</sup>*DB*(*Cw*−*C*∞) <sup>ν</sup> represents the Brownian motion limitation, γ = <sup>ν</sup> *ca*<sup>2</sup> is the curvature parameter, and *<sup>M</sup>* <sup>=</sup> <sup>σ</sup>0β<sup>2</sup> 0 *<sup>c</sup>*ρ*<sup>f</sup>* is the magnetic parameter. The Schmidt number is defined by *Sc* = <sup>ν</sup> *DB* , whereas *<sup>F</sup>* = *Cbx*/ √ *K* represents the local inertia parameter.
