# Magnetohydrodynamic Nanoliquid Thin Film Sprayed on a Stretching Cylinder with Heat Transfer
## 2 Materials and Methods  
### Basic Equations  
A motion of a steady and axi-symmetric sprayed liquid thin film CuO-H\({}_{2}\)O nanofluid past a stretching cylinder in two dimensions is considered in the presence of magnetic field. The magnetic field is uniform of intensity \(B_{0}\) acting in the radial way. The Reynolds number due to magnetic field (\(Re_{m}=\sigma_{nf}\mu_{e}W_{w}\delta\), where \(\sigma_{nf},\mu_{e},W_{w}\) and \(\delta\) stand for the electrical conductivity of the nanofluid, magnetic permeability, surface velocity and thickness of the fluid film respectively) is taken less in magnitude as compared to the applied magnetic field. Ignoring viscous dissipation by taking it very less in amount. When the material stretches, the thickness of the cylinder reduces but the outer radius \(a\) of the cylinder remains fixed. An axisymmetric spray in a radial direction having velocity \(V\) condenses as a film and is dragged along through the help of outer surface of the cylinder. Also, it is suppose that the base fluid and the nanoparticles are in thermal equilibrium prevailing the absence of slip. Selecting \(z\)-axis along the cylinder and \(r\)-axis is taken radially, as depicted in Figure 1.  
According to the above statement, in cylindrical coordinates (\(r\), \(z\)), the equations of continuity, momentum and energy, governing the film in scalar form are:  
\[\frac{\partial(ru)}{\partial r}+\frac{\partial(rw)}{\partial z}=0, \tag{1}\]  
\[\rho_{nf}\left(u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z} \right)=-\frac{\partial P}{\partial r}+\mu_{nf}\left(\frac{1}{r}\frac{ \partial u}{\partial r}-\frac{u}{r^{2}}+\frac{\partial^{2}u}{\partial r^{2}} \right), \tag{2}\]  
\[\rho_{nf}\left(u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z} \right)=\mu_{nf}\left(\frac{1}{r}\frac{\partial w}{\partial r}+\frac{\partial ^{2}w}{\partial r^{2}}\right)-\sigma_{nf}B_{0}^{2}w, \tag{3}\]  
\[\rho_{nf}\left(u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z} \right)=\frac{k_{nf}}{\left(\rho C_{P}\right)_{nf}}\left(\frac{1}{r}\frac{ \partial T}{\partial r}+\frac{\partial^{2}T}{\partial r^{2}}\right), \tag{4}\]  
where the components of velocity are \(u(r\), \(z)\) and \(w(r\), \(z)\), the magnetic induction is \(B=\left(B_{0},0,0\right)\), \(P\) expresses the pressure and \(T\) is the fluid temperature. Further, \(\rho_{nf}\), \(\mu_{nf}\), \((C_{P})_{nf}\) and \(k_{nf}\) are the density, viscosity, heat capacitance and thermal conductivity of the nanofluid respectively. The subscript \"\(nf\)\" is used for the nanofluid. The effective density \((\rho_{nf})\) and the heat capacitance \((\rho C_{P})_{nf}\) which are the relation between base fluid and nanoparticles are defined as  
\[\rho_{nf}=\rho_{f}(1-\phi)+\rho_{s}\phi_{s}, \tag{5}\]  
\[(\rho C_{P})_{nf}=(\rho C_{P})_{f}(1-\phi)+(\rho C_{P})_{s}\phi, \tag{6}\]  
where \(\phi\) describes the solid volume fraction and the subscript \"\(s\)\" is used for nanosolid particles. The boundary conditions are  
\[u=U_{w},\hskip 14.226378 ptw=W_{w},\hskip 14.226378 ptT=T_{w},\hskip 14.226378 ptat \hskip 14.226378 ptr=a, \tag{7}\]  
Figure 1: Geometry of the Physical Model. \[\frac{\partial w}{\partial r}=\frac{\partial\delta}{\partial r}=\frac{\partial T}{ \partial r}=0,\ \ \ \ \ u=\frac{d\delta}{dz}\ \ \ \ \ at\ \ \ \ r=b, \tag{8}\]  
where \(U_{w}=-ca\), \(W_{w}=2 cz\), \(T_{w}(z)=T_{b}-T_{ref}\Big{[}\frac{cz^{2}}{v_{nf}}\Big{]}\). The subscript \"\(w\)\" is used for condition at the surface, \(c>0\) is the stretching rate, \(a\) is the outer radius of the cylinder and \(b\) is the outer radius of the film. \(v_{nf}=\frac{\mu_{nf}}{\rho_{nf}}\) is the kinematic viscosity of the nanofluid, \(T_{b}\) is the temperature at the outer radius of the film surface and \(T_{ref}\) is the reference temperature such that \(0\leq T_{ref}\leq T_{b}\). The argument \(\frac{cz^{2}}{v_{nf}}\) in \(T_{ref}\Big{[}\frac{cz^{2}}{v_{nf}}\Big{]}\) shows the stretching velocity along \(z\)-axis [31; 32; 33; 34; 35; 36; 37; 38].