# Magnetohydrodynamics Flow Past a Moving Vertical Thin Needle in a Nanofluid with Stability Analysis
## 2 Problem Description and Formulation  
### Governing Equations  
Consider an incompressible laminar boundary layer flow of an electrically conducting nanofluid past a moving vertical thin needle with constant ambient temperature \(T_{\infty}\) and nanoparticle volume fraction \(C_{\infty}\). A schematic representation of the physical model and coordinate system is illustrated in Figure 1. Here, \(x\) and \(r\) are the axial and radial coordinates, respectively, in which \(r=R(x)\) is the radius of the needle. The \(x\)-axis measures the needle leading edge in the vertical direction; meanwhile, \(r\) is always normal to the \(x\)-axis. Since the needle is considered thin, the effect of its transverse curvature is important; however, the pressure gradient along the surface can be neglected ([PERSON], 1998). We assumed that the temperature of the needle surface is \(T_{w}\), where \(T_{w}>T_{\infty}\) corresponds to a heated needle (assisting flow) and \(T_{w}<T_{\infty}\) corresponds to a cooled needle (opposing flow) and the surface volume fraction is \(C_{w}\) with \(C_{w}>C_{\infty}\). The magnetic field \(B_{0}\) is imposed parallel to the direction of the flow and the induced magnetic field can be ignored due to the smallest magnetic Reynold number. The needle moves with a constant velocity \(U_{w}\), in the same or the opposite direction of the free stream flow of a constant velocity \(U_{\infty}\). Under the above assumptions, the governing boundary layer equations for mass, momentum, thermal energy and solid volume fraction can be written as:  
\[\frac{\partial}{\partial x}(ru)+\frac{\partial}{\partial r}(rv)=0, \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}=\frac{\
u}{r} \frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)-\frac{ \sigma}{\rho_{f}}B_{0}^{2}u+\frac{1}{\rho_{f}}\left[(1-C_{\infty})\rho_{f_{ \infty}}\beta(T-T_{\infty})-(\rho_{p}-\rho_{f_{\infty}})(C-C_{\infty})\right] \mathcal{S}, \tag{2}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{\alpha}{r }\frac{\partial T}{\partial r}\left(r\frac{\partial T}{\partial r}\right)+ \kappa\left(D_{B}\frac{\partial T}{\partial r}\frac{\partial C}{\partial r}+ \frac{D_{T}}{T_{\infty}}\left(\frac{\partial T}{\partial r}\right)^{2}\right), \tag{3}\]  
\[u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial r}=\frac{D_{B}}{r} \frac{\partial}{\partial r}\left(r\frac{\partial C}{\partial r}\right)+\frac{ D_{T}}{T_{\infty}}\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{ \partial r}\right). \tag{4}\]We assume that the boundary conditions of Equations (1)-(4) are (see [PERSON] and [PERSON] [31]):  
\[u=U_{w},\ \ v=0,\ \ T=T_{w},\ \ D_{B}\frac{\partial C}{\partial r}+\frac{D_{T}}{T _{\infty}}\frac{\partial T}{\partial r}=0\ \ \mbox{at}\ \ \ r=R(x),\]  
\[u\to U_{\infty},\ \ T\to T_{\infty},\ \ C\to C_{\infty}\ \ \mbox{as}\ \ \ \ r\to\infty. \tag{5}\]  
Here, \(u\) and \(v\) are velocities in the axial and radial \((x,r)\) directions, respectively, \(T\) and \(C\) are the temperature and the concentration of the nanofluid, respectively, \(D_{B}\) and \(D_{T}\) represent the Brownian diffusion and thermophoretic diffusion coefficients, \(g\) is the gravity acceleration, \(v\) is the kinematic viscosity, \(\alpha\) is the thermal diffusivity, \(\beta\) is the volumetric expansion coefficient, \(\kappa=(\rho c)_{p}/(\rho c)_{f}\) is the ratio of the effective heat capacity of nanofluid to that of the base fluid, \(\rho_{f}\) and \(\rho_{p}\) are the densities of the base fluid and of nanoparticles, respectively, \(\rho_{f_{\infty}}\) is the density of the ambient fluid and \(\sigma\) is the electrical conductivity.