## 2 Mathematical Formulation  
Consider a two-dimensional, steady mixed convection boundary layer flow embedded in a porous medium along a permeable vertical cylinder with radius \(a\), which is filled with thermal radiation and a nanofluid, as illustrated in Figure 1.  
By considering these assumptions alongside the boundary layer and Boussinesq approximations, the respective system of continuity, momentum and energy are:  
\[\frac{\partial(ru)}{\partial x}+\frac{\partial(rv)}{\partial r}=0, \tag{1}\]  
\[\frac{\mu_{nf}}{\mu_{f}}\frac{\partial u}{\partial r}=\frac{gK\Big{[}\varphi \rho_{s}\beta_{s}+(1-\varphi)\rho_{f}\beta_{f}\Big{]}}{\mu_{f}}\frac{\partial T }{\partial r}, \tag{2}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{\alpha_{ nf}}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial T}{\partial r} \bigg{)}-\frac{1}{\big{(}\rho C_{p}\big{)}_{nf}}\bigg{(}\frac{\partial q_{r} }{\partial r}\bigg{)}, \tag{3}\]subjected to the boundary conditions as follows:  
\[ \begin{array}{l}v=v_{w},\ T=T_{\infty}+T_{0}\frac{x}{a}\ \text{on}\ r=a,\\ u\to U_{0}\frac{x}{a},\ T\to T_{\infty}\ \text{as}\ r\to\infty.\end{array}  \tag{4}\]  
Here \(r\) and \(x\) are the cylindrical coordinates measured over the cylinder axes, \(u\) and \(v\) are the components of velocity over the \(x\) and \(r\)-axes, \(g\) is gravity acceleration, \(T\) is temperature, \(\varphi\) is nanoparticle volume fraction, \(\rho\) is fluid density, \(\beta\) is the coefficient of thermal expansion, \(\alpha\) is thermal diffusivity, \(\mu\) is viscosity, \(\eta\), is the heat flux of radiative and the subscripts of \(nf,f\) and \(s\) correspond to the nanofluid, fluid and solid, respectively. Using [PERSON]'s approximation for radiation [14], we have \(q_{r}=\frac{-4\sigma}{3\kappa}\frac{\partial T^{4}}{\partial r}\) where \(k*\) is the coefficient of mean absorption and \(\sigma\) is [PERSON] constant. The temperature differences are significantly small within the flow so that \(T^{4}\) may be considered as a temperature linear function \(T\) by using a Taylor series of truncated about the temperature of free stream \(T_{\infty}\), and we get \(T^{4}\approx 4T_{\infty}^{3}T-3T_{\infty}^{4}\). Equation (3) now can be reduced to:  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{\alpha_{ nf}}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right)+ \frac{16\sigma T_{\infty}^{3}}{3k*\left(\rho C_{p}\right)_{nf}}\frac{\partial^ {2}T}{\partial r^{2}}. \tag{5}\]  
The physical properties of nanofluids as stated in Equations (3) and (5) are given by [15]:  
\[ \begin{array}{l}\mu_{nf}=\frac{\mu_{f}}{\left(1-\varphi\right)^{2s}},\ \alpha_{nf}=\frac{k_{nf}}{\left(\rho C_{p}\right)_{nf}},\ \rho_{nf}=(1-\varphi)\rho_{f}+\varphi\rho_{s},\\ \left(\rho C_{p}\right)_{nf}=\left(1-\varphi\right)\left(\rho C_{p}\right)_{ f}+\varphi\left(\rho C_{p}\right)_{s^{\prime}}\frac{k_{nf}}{k_{f}}=\frac{\left(k_{ }+2k_{f}\right)-2\varphi\left(k_{f}-k_{h}\right)}{\left(k_{h}+2k_{f}\right)+ \varphi\left(k_{f}-k_{h}\right)},\end{array}  \tag{6}\]  
where \(\rho C_{p}\) is heat capacitance. We now introduce the following similarity variables:  
\[\psi=\sqrt{2U_{0}a\alpha_{f}}xf(\eta),\ T=T_{\infty}+\frac{T_{0}x}{a}\theta( \eta),\ \eta=\frac{r^{2}-a^{2}}{2a\alpha_{f}}\sqrt{\frac{U_{0}\alpha_{f}}{2a}}. \tag{7}\]  
Figure 1: Physical model of the present problem. Substituting Equation (7) into Equations (3) and (5), we can reduce the governing system as follows:  
\[f^{\prime\prime}+B\lambda\theta^{\prime}=0, \tag{8}\] \[\left[A(1+2\xi\eta)+\frac{8}{3}\xi Rd\right]\theta^{\prime\prime}+ \left(2A\xi+\frac{8}{3}\xi Rd\right)\theta^{\prime}+2f\theta^{\prime}-2f^{ \prime}\theta=0, \tag{9}\]  
subjected to the boundary conditions:  
\[ \begin{array}{c}f(0)=S,\theta(0)=1,\\ f^{\prime}(\eta)\to 1,\theta(\eta)\to 0\text{ as }\eta\rightarrow\infty,\end{array}  \tag{10}\]  
where \(\lambda\) is the parameter of mixed convection, \(\xi\) is curvature parameter, \(Rd\) is radiation parameter and \(S\) is suction parameter.