## 2 Mathematical Modelling  
Here, we consider a 2D laminar, incompressible, and radiative Casson hybrid nanomaterial liquid flow on a stretching cylinder in the presence of the slip and microorganism effects. The energy and mass concentration equation has been analyzed by the influence of viscous dissipation and modified Fick's law. Moreover, the stratification boundary conditions are implemented on the surface and away from the surface. A magnetic dipole is considered with a magnetic field in the direction of \(x\). The coordinates axes are \(x\) and \(r\), \(R\) is the radius of the cylinder, and \(U_{w}\) is the stretching velocity. The flow pattern is seen in Figure 1. The ambient temperature, concentration, and microorganism density are denoted by \(T_{\infty}\), \(C_{\infty}\), and \(N_{\infty}\), respectively, and at the wall they are stated by \(T_{w}\), \(C_{w}\), and \(N_{w}\), respectively. Using the above-mentioned supposition and boundary layer approximation theory, the flow equations are developed as [25; 26; 27],  
\[\frac{\partial(ru)}{\partial x}+\frac{\partial(rv)}{\partial r}=0, \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}=\frac{\mu_{lnf} }{\rho_{lnf}}\bigg{(}1+\frac{1}{\beta}\bigg{)}\bigg{(}\frac{\partial^{2}u}{ \partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}\bigg{)}+\frac{\mu_{ 0}}{\rho_{lnf}}M\frac{\partial H}{\partial x}, \tag{2}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\alpha_{lnf} \bigg{(}\frac{\partial^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{ \partial r}\bigg{)}-\mu_{0}T\frac{\partial M}{\partial T}\bigg{(}u\frac{ \partial H}{\partial x}+\frac{v}{r}\frac{\partial H}{\partial r}\bigg{)}+ \frac{1}{\big{(}\rho_{cp}\big{)}_{lnf}}\left( \begin{array}{c}\mu_{lnf}\big{(} 1+\frac{1}{\beta}\big{)}\bigg{(}\frac{\partial u}{\partial y}\bigg{)}^{2}\\ +\frac{16\sigma^{\prime}T_{\infty}^{2}}{3\delta^{2}}\bigg{(}\frac{\partial^{2} T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}\bigg{)}\end{array} \right), \tag{3}\]\[u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial r}=(D_{B})_{lnf}\bigg{(} \frac{\partial^{2}C}{\partial r^{2}}+\frac{1}{r}\frac{\partial C}{\partial r} \bigg{)}-\lambda_{c}\left( \begin{array}{c}v\frac{\partial v}{\partial r} \frac{\partial C}{\partial r}+v\frac{\partial u}{\partial r}\frac{\partial C}{ \partial x}+u\frac{\partial v}{\partial r}\frac{\partial C}{\partial x}+u\frac {\partial u}{\partial r}\frac{\partial C}{\partial x}+u\frac{\partial u}{ \partial r}\frac{\partial C}{\partial x}\\ +2 uv\frac{\partial^{2}C}{\partial x\partial r}+u^{2}\frac{\partial^{2}C}{ \partial x^{2}}+v^{2}\frac{\partial^{2}C}{\partial r^{2}}\end{array} \right), \tag{4}\]  
\[u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial r}+\frac{bW_{c}}{C _{w}-C_{0}}\frac{\partial}{\partial r}\bigg{(}N\frac{\partial C}{\partial r} \bigg{)}=(D_{m})_{lnf}\bigg{(}\frac{\partial^{2}N}{\partial r^{2}}+\frac{1}{ r}\frac{\partial N}{\partial r}\bigg{)}. \tag{5}\]  
where \(\mu_{0}M\frac{\partial H}{\partial x}\) represents the ferromagnetic force per unit volume. The associated boundary conditions are,  
\[v=0,\,\,\,u=U_{w}+w_{1}v_{lnf}\bigg{(}1+\frac{1}{\beta}\bigg{)}\frac{\partial u }{\partial r},T=T_{0}+\frac{bx}{l},C=C_{0}+\frac{dx}{l},\,\,N=N_{0}+\frac{ax}{ l},\,\text{at}\,\,r=R, \tag{6}\]  
\[u\to 0,\,\,\,T\to T_{\infty}(x)=T_{0}+\frac{cx}{l},\,\,C\to C_{\infty}(x)=C_{0}+ \frac{ex}{l},\,\,N\to N_{\infty}=N_{0}+\frac{a_{1}x}{l},\,\,\text{as}\,\,\,r \rightarrow\infty. \tag{7}\]  
In the above equations, \(u\) and \(v\) are the velocity components in the directions of \(x-\) and \(r-\), respectively.