# Stability Analysis of Mixed Convection Flow towards a Moving Thin Needle in Nanofluid
## Problem Formulation  
### Basic Equation  
An incompressible laminar boundary layer flow and heat transfer of a nanofluid passing through a moving vertical thin needle of uniform ambient temperature \(T_{\infty}\) is considered. The inset of Figure 1 illustrates the vertical slender paraboloid needle whose radius is given by \(r=R(x)\) where \(x\) and \(r\) are the axial and radial coordinates, respectively. The \(x\)-axis measures the needle's leading edge in the vertical direction, whereas \(r\) is always normal to the \(x\)-axis. The needle is considered thin when its thickness does not exceed that of the boundary layer over it. Since the needle is assumed thin, the influence of its transverse curvature is important; however, the pressure gradient along the surface due to the presence of the needle can be ignored ([PERSON] and [PERSON], 2015). Furthermore, the needle is subjected to a variable surface temperature \(T_{w}\) where \(T_{w}>T_{\infty}\) corresponds to an assisting flow (heated surface) and \(T_{w}<T_{\infty}\) corresponds to an opposing flow (cooled surface). The needle moves with a uniform velocity, \(U_{w}\), in the same or reverse direction to the fluid flow of uniform velocity, \(U_{\infty}\). Based on the above assumptions, the governing equations for the nanofluid model proposed by [PERSON] and [PERSON] (2015) take the following form of cylindrical coordinates:  
\[\frac{\partial}{\partial x}(ru)+\frac{\partial}{\partial r}(rv)=0, \tag{1}\]  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u }{\partial r}=\frac{u_{nf}}{\rho_{nf}}\frac{1}{r}\frac{\partial}{\partial r} \left(r\frac{\partial u}{\partial r}\right)+\frac{(\rho\beta)_{nf}}{\rho_{nf} }g(T-T_{\infty}), \tag{2}\]  
\[\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T }{\partial r}=\frac{\alpha_{nf}}{r}\frac{\partial}{\partial r}\left(r\frac{ \partial T}{\partial r}\right) \tag{3}\]  
Assume the initial and boundary conditions of Equations (1)-(3) are:  
\[t<0:u=v=0,\;\;T=T_{\infty}\;\;for\;any\;\;x,r,\]  
\[t\geq 0:u=U_{w},\;\;v=0,\;\;\;T=T_{w}\;\;at\;\;r=R(x),\]  
\[u\to U_{\infty},\;\;T\to T_{\infty}\;\;as\;\;r\to\infty, \tag{4}\]  
where \(u\) and \(r\) are the velocity components along the \(x\) and \(r\) axis, respectively, and \(T\) is the temperature of the nanofluid on the boundary layer. \(\mu\) represents the viscosity; \(\rho\) is the density; \(\alpha\) is the thermal diffusivity; and \(\beta\) is the thermal expansion coefficient in which the subscripts '\(nf\)', '\(f\)' and '\(s\)' represent 'nanofluid', 'fluid' and'solid', respectively. The relation between these parameters is given by ([PERSON] and [PERSON], 2015):  
\[\rho_{nf}=\left(1-\phi\right)\rho_{f}+\phi\rho_{s},\;\;(\rho C_{p})_{nf}= \left(1-\phi\right)\left(\rho C_{p}\right)_{f}+\phi\left(\rho C_{p}\right)_{s },\;\;(\rho\beta)_{nf}=\left(1-\phi\right)\left(\rho\beta\right)_{f}+\phi \left(\rho\beta\right)_{s},\]  
\[\mu_{nf}=\frac{\mu_{f}}{(1-\phi)^{25}},\;\;\alpha_{nf}=\frac{k_{nf}}{(\rho C _{p})_{nf}},\;\;\frac{k_{nf}}{k_{f}}=\frac{(k_{s}+2k_{f})-2\phi(k_{f}-k_{s})} {(k_{s}+2k_{f})+\phi(k_{f}-k_{s})} \tag{5}\]  
Here, \(k\) and \((\rho C_{p})\) represent the thermal conductivity and heat capacity, respectively. In addition, \(\phi\) represents the nanoparticle volume fraction for the nanofluid.