## 2 Mathematical Modeling  
Assume an MHD steady, a thin needle immersed is in a nanofluid with a uniform stream velocity \(u_{\infty}\) of incompressible flow with permeable media in the existence of viscous dissipation and temperature-dependent heat sink/source. The effects of melting heat transfer and non-linear chemical reactions are also considered with the slip boundary. The strength of magnetic field \(B_{0}\) is applied in the radial direction. Here, induced magnetic and electric fields are neglected due to our presumption of a small value of Reynolds number. The needle is seen as thin and its thickness is smaller than the boundary layer formed over it. A physical interpretation of the flow can be seen in Figure 1. Here, \(x-\) coordinate is measured as the axial and \(r-\) as the radial direction of the cylinder. Assume the function \(r=\ R(x)=\ (vxc/U)^{1/2}\) determines the state of a slight needle in which \(U=u_{w}+u_{\infty}\) is the composite velocity. The boundary layer governing equations are represented as:  
\[ru_{x}+ur_{x}+vr_{r}+rv_{r}=0, \tag{1}\]  
\[uu_{x}+vu_{x}=\frac{v}{r}(ru_{rr}+u_{r}r_{r})-\frac{\sigma B_{0}^{2}u}{\rho}- \frac{v}{k^{\star}}u, \tag{2}\]  
\[uT_{x}+vT_{r}=\frac{\alpha}{r}(rT_{rr}+T_{r}r_{r})+\tau\Big{(}D_{B}T_{r}C_{r} +\frac{D_{T}}{T_{\infty}}(T_{r})^{2}\Big{)}+\frac{Q_{0}}{\left(\rho c_{p} \right)_{f}}(T-T_{m})+\frac{\sigma B_{0}^{2}u^{2}}{\left(\rho c_{p}\right)_{f }}, \tag{3}\]  
\[uC_{x}+vC_{r}=\frac{D_{B}}{r}(rC_{rr}+C_{r}r_{r})+\frac{D_{T}}{T_{\infty}} \frac{1}{r}(rT_{rr}+T_{r}r_{r})-K_{tt}(C-C_{m})^{n}. \tag{4}\]  
Figure 1: Geometrical sketch of the model. The suitable boundary conditions are described by:  
\[ \begin{array}{c}u=u_{w}+Lu_{r}|_{r=0},\ T=T_{m},\ C=C_{m},\ \text{at}\ r=R(x)\\ u\to u_{\infty},\ \ T\to T_{\infty},\ \ C\to C_{\infty},\ \text{as}\ r\to\infty,\\ K(T_{r})_{r=R}=\rho(\lambda_{1}+C_{s}(T_{m}-T_{0}))v(R,x).\end{array}  \tag{5}\]  
Similarity transformations are defined as:  
\[\psi=vxf(\eta),\ \theta(\eta)=\frac{T-T_{m}}{T_{\infty}-T_{m}},\ \varphi(\eta)=\frac{C-C_{m}}{C_{\infty}-C_{m}},\ \eta=\frac{Ur^{2}}{vx}. \tag{6}\]  
Using the above transformation, the incompressibility Equation is satisfied, and Equations (2)-(4) are transformed into:  
\[2\eta f^{\prime\prime\prime\prime}+2f^{\prime\prime\prime}+ff^{\prime\prime}- \frac{1}{2}\lambda f\prime\left(M^{2}+k_{0}\right)=0, \tag{7}\]  
\[\frac{2}{\Pr}(\theta\prime+\eta\theta^{\prime\prime})+2\eta\left(N_{b}\theta \
u\theta^{\prime}+N_{t}\theta^{\prime 2}\right)+f\theta\prime+\frac{ \lambda}{2}M^{2}f^{\prime 2}+\frac{\lambda}{2}Q\theta=0, \tag{8}\]  
\[2(\eta\eta^{\prime\prime}+\varphi\prime)+2\frac{N_{t}}{N_{B}}(\eta\theta^{ \prime\prime}+\theta)+Scf\varphi\prime-\frac{\lambda}{2}\gamma\phi^{n}=0. \tag{9}\]  
The pertinent boundary conditions are:  
\[ \begin{array}{c}\text{At}\ \eta=m;f\prime=\frac{\lambda}{2}+\frac{2}{ \sqrt{\lambda}}S_{1}\eta f^{\prime\prime},\theta=0,\varphi=0,\\ \text{As}\ \eta=\infty;f\mapsto\frac{1-\lambda}{2},\theta\to 1,\varphi\to 1,\\ M_{e}\theta\prime+\frac{\Pr}{\eta}(f-\eta f^{\prime})=0.\end{array}  \tag{10}\]  
With  
\[ \begin{array}{c}\Pr=\frac{\upsilon}{\alpha},\ S_{c}=\frac{\upsilon}{D_{s}}, \ N_{r}=\frac{\tau D_{r}\left(T_{a}-T_{m}\right)}{\upsilon T_{a}},\ N_{s}= \frac{\tau D_{s}C_{m}}{\upsilon T_{a}},\ S_{i}=L\overline{\frac{1}{\
u}},\\ \gamma=\frac{K_{w}C_{\infty}^{n-1}}{c},M^{2}=\frac{\alpha B_{0}^{2}}{\rho c}, M_{e}=\frac{\upsilon_{f}(T_{\infty}-T_{m})}{\lambda_{1}+C_{s}(T_{m}-T_{0})}A= \frac{u_{w}}{(u_{w}+u_{w})}.\end{array}