**2. Mathematical Formulation**

We considered a 2D steady, incompressible nanoliquid flow over a curved stretching channel looped in the form of a circle with a radius *R* about the curvilinear directions *r* and *x*, as shown in Figure 1. Here, a higher value of *R* corresponds to a marginally curved surface. The stretching velocity is taken as *u* = *uw* along the *x*-direction. A magnetic field is applied normal to the fluid flow and along the *r*-direction. The electric and induced magnetic fields were overlooked owing to our assumption of small Reynolds number.

**Figure 1.** Flow geometry.

The assumed system is governed by the following equations:

$$
\frac{
\partial
}{
\partial r
} \{(R+r)v\} + R \frac{
\partial u
}{
\partial x
} = 0
\tag{1}
$$

$$\frac{\mu^2}{r+R} = \frac{1}{\rho\_{nf}} \frac{\partial p}{\partial r} \tag{2}$$

$$\frac{\partial u}{\partial r} + \frac{R u}{r + R} \frac{\partial u}{\partial \mathbf{x}} + \frac{u v}{r + R} = -\frac{1}{\rho\_{nf}} \frac{R}{r + R} \frac{\partial p}{\partial \mathbf{x}} + \frac{\mu\_{nf}}{\rho\_{nf}} (\frac{\partial^2 u}{\partial r^2} + \frac{1}{r + R} \frac{\partial u}{\partial r} - \frac{u}{\left(r + R\right)^2}) - \frac{\sigma}{\rho\_{nf}} B\_0^{-2} u \tag{3}$$

$$v\frac{\partial T}{\partial r} + \frac{Ru}{r+R}\frac{\partial T}{\partial x} = \mathfrak{a}\_{nf}(\frac{\partial^2 T}{\partial r^2} + \frac{1}{r+R}\frac{\partial T}{\partial r}) + \frac{Q\_0}{\left(\rho \mathbb{C}\_p\right)\_{nf}}(T\_{\mathbb{S}^1} - T) + \frac{1}{\left(\rho \mathbb{C}\_p\right)\_{nf}}\frac{1}{r+R}\frac{\partial}{\partial r}(r+R)q\_r \tag{4}$$

$$
\rho \frac{\partial \mathcal{C}}{\partial r} + \frac{Ru}{r + R} \frac{\partial \mathcal{C}}{\partial x} = D\_B (\frac{\partial^2 \mathcal{C}}{\partial r^2} + \frac{1}{r + R} \frac{\partial \mathcal{C}}{\partial r}) \tag{5}
$$

The system of Equations (1)–(5) is supported by the following boundary conditions:

$$\begin{aligned} \left. w \right|\_{r=0} = 0, \left. u \right|\_{r=0} = u\_w(\mathbf{x}) = \mathbf{s} \mathbf{x}, \left. k\_f \frac{\partial T}{\partial r} \right|\_{y=0} = h^\*(T\_f - T), \left. -D\_B \frac{\partial \mathcal{C}}{\partial r} \right|\_{r=0} = j\_w \\ \left. u \right|\_{r \to \infty} \to 0, \left. \frac{\partial u}{\partial r} \right|\_{r \to \infty} \to 0, \left. T \right|\_{r \to \infty} \to T\_{\infty} \left. \mathcal{C} \right|\_{r \to \infty} \to \mathcal{C}\_{\infty} \end{aligned} \tag{6}$$

The thermophysical traits of the hybrid nanoliquid are appended in Table 1.

**Table 1.** The values of *Cp*, *ρ*, and *k* for ethylene glycol and NiZnFe2O4 (nickel–zinc ferrite) [28–33].


The mathematical form of thermophysical properties are given as follows:

$$
\mu\_{nf} = \frac{\mu\_f}{\left(1 - \phi\right)^{2.5}}, \ \alpha\_{nf} = \frac{k\_{nf}}{\left(\rho C\_p\right)\_{nf}}\tag{7}
$$

$$\rho\_{nf} = (1 - \Phi)\rho\_f + \Phi\rho\_{s\prime} \left(\rho\mathbb{C}\_p\right)\_{nf} = (1 - \Phi)\left(\rho\mathbb{C}\_p\right)\_f + \Phi\left(\rho\mathbb{C}\_p\right)\_s \tag{8}$$

$$\frac{k\_{nf}}{k\_f} = \frac{(k\_s + 2k\_f) - 2\phi(k\_f - k\_s)}{(k\_s + 2k\_f) + \phi(k\_f - k\_s)}\tag{9}$$

In Equation (4), the nonlinear radiation heat flux term via Rosseland's approximation is given as follows:

$$q\_r = \frac{4\sigma^\*}{3k^\*} \frac{\partial T^4}{\partial r} = \frac{16\sigma^\* T^3}{3k^\*} \frac{\partial T}{\partial r} \tag{10}$$
