## 2 Mathematical Modeling  
Let us assume an H\({}_{2}\)O-CNTs-based nanofluid flow with Hall current over a moving slender needle having speed \(u_{w}\) and radius \"\(a\)\" (Figure 1). The speed of fluid far away from the surface is taken as \(u_{\infty}\). The cylindrical coordinates (\(x\), \(r\)) are taken in such a way that \(x-\) is along the axis of the needle and \(r-\) normal to the axis. The flow containing dust particles is generated in a non-Darcy absorbent media. The associated impacts affecting the flow in the heat equation are viscous dissipation and nonlinear thermal radiation. Furthermore, \(T_{w}\) and \(T_{\infty}\) are the constant temperatures at the wall and far off from the wall with \(T_{\infty}>T_{w}\). A magnetic field with magnetic strength \(B_{0}\) is applied with the low Reynold number assumption [41], which eventually results in the induced magnetic field to be neglected. Two types of the equations, i.e., fluid phase and particle phase, comprising the envisioned mathematical model fulfilling laws of conservation are also laid down as given below:  
The above flow theory gives rise to the following boundary layer equations [41; 42; 43; 44; 45]:  
Continuity equation: Fluid phase  
\[\frac{\partial(ru)}{\partial x}+\frac{\partial(rv)}{\partial r}=0, \tag{1}\]  
Momentum equation: Fluid phase  
\[ \begin{array}{c}(1-\phi_{d})\Big{(}u\frac{\partial u}{\partial x}+v\frac{ \partial u}{\partial r}\Big{)}=(1-\phi_{d})\Big{(}\frac{u_{\infty}}{\rho_{d}} \frac{1}{r}\frac{\partial}{\partial r}\Big{(}\frac{\partial u}{\partial r} \Big{)}-\frac{v_{sf}}{k^{s}}u\\ -\frac{C_{b}}{x\sqrt{k^{s}}}u^{2}+\frac{KN}{\rho_{sf}}\Big{(}u_{p}-u\Big{)}- \frac{\alpha B_{0}}{\rho_{sf}(1+m^{2})}u,\end{array}  \tag{2}\]  
Continuity equation: Particle phase  
\[\frac{\partial\Big{(}ru_{p}\Big{)}}{\partial x}+\frac{\partial\Big{(}rv_{p} \Big{)}}{\partial r}=0, \tag{3}\]  
Momentum equation: Particle phase  
\[u_{p}\frac{\partial u_{p}}{\partial x}+v_{p}\frac{\partial u_{p}}{\partial r }=\frac{K}{m_{d}}\Big{(}u-u_{p}\Big{)}, \tag{4}\]  
Figure 1: The physical design of the flow problem. Energy equation: Fluid phase  
\[ \begin{split} u\frac{\partial T}{\partial x}+v\frac{\partial T}{ \partial r}=\frac{k_{uf}}{\left(\rho C_{p}\right)_{nf}}\frac{1}{r}\frac{ \partial}{\partial r}\Big{(}r\frac{\partial T}{\partial r}\Big{)}-\frac{1}{ \left(\rho C_{p}\right)_{nf}}\frac{\partial qr}{\partial r}+\frac{\mu_{nf}}{ \left(\rho C_{p}\right)_{nf}}\Big{(}\frac{\partial u}{\partial r}\Big{)}^{2} \\ +\frac{\mu_{nf}}{\kappa\left(\rho C_{p}\right)_{nf}}u^{2}+\frac{N_ {1}}{\tau_{w}\left(\rho C_{p}\right)_{nf}}\Big{(}u_{p}-u\Big{)}^{2}+\frac{N_{1 }\left(C_{p}\right)_{nf}}{\tau_{T}\left(\rho C_{p}\right)_{nf}}\Big{(}T_{p}-T \Big{)}\\ +\frac{\sigma B_{0}{}^{2}}{\left(\rho C_{p}\right)_{nf}(1+m^{2}) }u^{2},\end{split}  \tag{5}\]  
Energy equation: Particle phase  
\[N_{1}c_{m}\left(u_{p}\frac{\partial T_{p}}{\partial x}+v_{p}\frac{\partial T_{ p}}{\partial r}\right)=\frac{N_{1}\left(C_{p}\right)_{nf}}{\tau_{T}}\Big{(}T_{p}-T \Big{)}, \tag{6}\]  
with the boundary conditions  
\[ \begin{split} u(x,r)=u_{av},~{}~{}v(x,r)=0,u_{p}(x,r)=u_{av},~{}~ {}v_{p}(x,r)=0,\\ T(x,r)=T_{w},~{}~{}at~{}~{}r=R(x),\end{split}  \tag{7}\]  
\[u(x,r)~{}\rightarrow~{}u_{\infty},~{}~{}~{}T(x,r)\rightarrow~{}T_{\infty},~{ }T_{p}(x,r)\rightarrow T_{\infty}~{}~{}as~{}r\rightarrow\infty. \tag{8}\]  
The heat flux in simplified form after considering Rosseland approximation is:  
\[q_{r}=-\frac{4\sigma_{SB}}{3a_{r}}\frac{\partial T^{4}}{\partial r}, \tag{9}\]  
where \(a_{r}\) is the mean absorption coefficient, and \(\sigma_{SB}\) is the Stephen-Boltzman constant. For a planer boundary layer flow, the above equation can be written as:  
\[q_{r}=-\frac{16\sigma_{SB}}{3a_{r}}T^{3}\frac{\partial T}{\partial r}. \tag{10}\]  
The mathematical model proposed by [PERSON] [46] for the CNTs is given in Table 1. The thermo-physical traits of the CNTs of both types and H\({}_{2}\)O are appended in Table 2.