## 2 Problem Formulation  
We assume a three-dimensional, viscous, steady, MHD rotating Casson nanofluid flow comprising single- and multi-walled CNTs over a stretching sheet. The flows of SWCNTs and MWCNTs are taken under the influence of non-linear thermal radiation and heat absorption/generation. We adopt the coordinate structure in such a manner that the surface associated with the \(xy\)-plane and fluid is placed in \(z>0\). The stretching rate of the surface and the angular velocity of fluid are assumed as \(c>0\) and \(\Omega\), respectively. The surface temperature is a result of the convective heating process, which involves the hot fluid temperature \(T_{f}\) and coefficient of heat transfer \(h_{f}\). The geometry of the problem shown in Figure 1.  
The necessary constitutive model of a Casson fluid [19; 22; 24] is as follows:  
\[ \begin{array}{l}\frac{\tau_{min}^{c,f}}{2\tau_{min}}=\mu_{\gamma}^{c,f}+\frac{ p_{y}}{\sqrt{2\pi_{d}}};\qquad\pi_{d}>\tau_{c,f},\\ \frac{\tau_{min}^{c,f}}{2\tau_{min}}=\mu_{\gamma}^{c,f}+\frac{p_{y}}{\sqrt{2 \pi_{c,f}}};\qquad\pi_{d}<\pi_{c,f}.\end{array}  \tag{1}\]  
where \(\tau_{min}^{c,f}\) is the shear stress tensor of the Casson fluid, \(\mu_{\gamma}^{c,f}\) is the Casson fluid plastic viscosity, \(\pi_{d}\) is the rate of deformation, and \(\pi_{c,f}\) is the critical amount of current product that depends on the Casson fluid. We assume a situation in which a certain liquid requires the gradual growth of the the shear stress to sustain the constant rate of strain. In the case \(\pi_{d}>\pi_{c,f}\) the Casson fluid possesses the kinematic viscosity \(v^{c,f}\) as a function of \(\mu_{\gamma}^{c,f}\), \(\rho^{c,f}\), and Casson parameter \(\beta\), such that  
\[ \begin{array}{l}\mu^{c,f}=\mu_{\gamma}^{c,f}+\frac{p_{y}}{\sqrt{2\pi_{d}}}, \ p_{y}=\frac{\mu_{\gamma}^{c,f}\sqrt{2\pi_{d}}}{\beta},\ v^{c,f}=\frac{\mu^{c, f}}{\rho^{c,f}},\\ v^{c,f}=\frac{\mu^{c,f}\left(1+\frac{1}{\beta}\right)}{\rho^{c,f}},\ \frac{\partial\tau^{c,f}}{ \partial\tau^{c,f}}=\left(1+\frac{1}{\beta}\right)u_{zz}.\end{array}  \tag{2}\]  
Here \(\mu^{c,f}\) represents the dynamic viscosity of the Casson fluid, \(\beta=\frac{\mu_{\gamma}^{c,f}\sqrt{2\pi_{d}}}{p_{y}}\), and \(p_{y}\) shows the yield stress of the liquid. From the denotation of \(\beta\) describing the ratio of the product of \(\mu_{\gamma}^{c,f}\) and \(\sqrt{2\pi_{d}}\) to the yield stress \(p_{y}\), it can be seen that the resistivity in liquid is greater if \(\beta>o\), and the liquid presents an inviscid behavior if \(\beta<o\), which eliminates the viscosity of the Casson fluid. Taking into account the above declared rules, the primary equation takes the following form [8; 25]:  
\[u_{x}+v_{y}+w_{z}=0, \tag{3}\]  
\[\rho_{nf}\big{\{}u\ u_{x}+v\ u_{y}+w\ u_{z}-2\Omega\ v\big{\}}=\mu_{nf}\left( 1+\frac{1}{\beta}\right)(u_{zz})-\sigma_{nf}\ B_{0}^{2}\ (u), \tag{4}\]  
\[\rho_{nf}\big{\{}u\ v_{x}+v\ v_{y}+w\ v_{z}-2\ \Omega\ u\big{\}}=\mu_{nf} \bigg{(}1+\frac{1}{\beta}\bigg{)}(v_{zz})-\sigma_{nf}\ B_{0}^{2}\ (v), \tag{5}\]  
\[u\ T_{x}+v\ T_{y}+w\ T_{z}=\alpha_{nf}\left(T_{zz}\right)-\frac{\bigg{(}q_{z}^ {\text{red}}\bigg{)}}{\big{(}\rho c_{p}\big{)}_{f}}+\frac{Q_{0}(T-T_{0})}{ \big{(}\rho c_{p}\big{)}_{f}}. \tag{6}\]  
Figure 1: Schematic physical geometry. The allied boundary conditions [25] are defined as  
\[ \begin{split}& u_{w}=c\,x,\;v=0,\;w=0,\;-k_{nf}T_{z}=h_{f}\Big{(}T _{f}-T\Big{)}\;at\;z=0,\\ & u=v=w=0,\quad T=T_{\infty},\quad at\quad z=\infty.\end{split}  \tag{7}\]  
where \(u,v\) and \(w\) specify velocities in the \(x,y,z\) directions, respectively, \(\rho_{nf}\) represents the nanofluid density, \(\mu_{nf}\) the nanofluid dynamic viscosity, and \(\sigma_{nf}\) the electrical conductivity of the nanofluid. \(T\) shows the local temperature, \(k_{nf}\) is the nanofluid thermal conductivity, and \(\big{(}\rho c_{p}\big{)}_{nf}\) is the nanofluid specific heat capacity.