## 2 Mathematical Modeling  
We assumed a time-independent dusty flow of nanofluid past a rotating disk with an angular velocity of \(\Omega\). The disk was inclined at angle \(\beta\). The dusty particles present in the fluid were assumed to be uniformly distributed. These particles were initially at rest, with a consistent density throughout the stream. Moreover, the particles were uniform in size. A nanofluid film of thickness \(\delta\) was formed by spraying with velocity \(\alpha\). The disk radius was assumed to be large in comparison to the film thickness, such that the end effects were assumed to be negligible. The force of gravity \(\overline{g}\) was acting vertically downward. \(\overline{T}_{w}\) was the temperature of the disk, and \(\overline{T}_{0}\) was the temperature of the surface of the film. Pressure was taken as a function of \(z\) only (Figure 1). The phenomenological laws for CNTs are outlined below.  
The governing equations based on the above suppositions are given by [38; 39]:  
**Nanofluid Phase:**  
\[\
abla\cdot V=0, \tag{1}\]  
\[(V\cdot\
abla u)=\frac{\mu_{nf}}{\rho_{nf}}\Big{(}\
abla^{2}u\Big{)}+\overline{ g}\sin\left(\frac{\beta}{\Omega}\right)+\frac{\rho_{p}}{\rho_{nf}\tau_{v}}(u_{p}-u), \tag{2}\]  
\[(V\cdot\
abla v)=(\frac{\mu_{nf}}{\rho_{nf}})(\
abla^{2}v)+\frac{\rho_{p}}{ \rho_{nf}}\tau_{v}(v_{p}-v), \tag{3}\]  
\[(V\cdot\
abla w)=(\frac{\mu_{nf}}{\rho_{nf}})(\
abla^{2}w)+\overline{g}\sin \left(\frac{\beta}{\Omega}\right)+\frac{\rho_{p}}{\rho_{nf}}\tau_{v}(w_{p}-w), \tag{4}\]  
\[V\cdot\
abla\overline{T}=\frac{K_{nf}}{\rho_{nf}C_{nf}}\Big{(}\
abla^{2} \overline{T}\Big{)}+\frac{1}{\rho_{nf}C_{nf}}\frac{\partial q_{r}}{\partial z }+\frac{\Big{(}\rho_{p}C_{p}\Big{)}}{\rho_{nf}C_{nf}\tau_{T}}\Big{(} \overline{T}_{p}-\overline{T}\Big{)}+\frac{\mathcal{Q}_{0}}{\rho_{nf}C_{nf}} \Big{(}\overline{T}-\overline{T}_{\infty}\Big{)}. \tag{5}\]  
**Dust Phase:**  
\[\
abla\cdot V_{p}=0, \tag{6}\]  
\[(V\cdot\
abla u_{p})=\overline{g}\sin\left(\frac{\beta}{\Omega}\right)+\frac{ 1}{\tau_{v}}(u-u_{p}), \tag{7}\]  
\[(V\cdot\
abla v_{p})=\frac{1}{\tau_{v}}(v-v_{p}), \tag{8}\]  
\[(V\cdot\
abla w_{p})=\overline{g}\sin\left(\frac{\beta}{\Omega}\right)+\frac{ 1}{\tau_{v}}(w-w_{p}), \tag{9}\]  
\[V\cdot\
abla\overline{T}=\frac{1}{\tau_{T}}\Big{(}\overline{T}-\overline{T}_ {p}\Big{)}, \tag{10}\]  
Figure 1: Geometrical sketch of the model.  
The Rosseland approximation of radiative heat flux \(q_{r}\) is given by:  
\[q_{r}=\frac{16\psi}{3\overline{k}}\frac{\partial\overline{T}^{4}}{\partial z}, \tag{11}\]  
where \(\psi\) is the Stefan-Boltzmann constant, and \(\overline{k}\) is the average of absorption coefficient. Expanding \(\overline{T}^{4}\) about \(\overline{T}_{0}\) with the help of [PERSON]'s series, we have  
\[\overline{T}^{4}=\overline{T}_{0}{}^{4}+4(\overline{T}-\overline{T}_{0}) \overline{T}_{0}{}^{3}+\dots \tag{12}\]  
\[\overline{T}^{4}=\overline{T}_{0}{}^{4}+4\overline{T}_{0}{}^{3}\overline{T}-4 \overline{T}_{0}{}^{4}. \tag{13}\]  
Simplifying this, we obtain  
\[\overline{T}^{4}=4\overline{T}\overline{T}_{0}{}^{3}-3\overline{T}_{0}{}^{4}. \tag{14}\]  
From Equations (11) and (14), we get  
\[\frac{\partial q_{r}}{\partial z}=\ -\frac{16\overline{T}_{0}{}^{3}\psi}{3 \overline{k}}\frac{\partial^{2}\overline{T}^{4}}{\partial z^{2}}, \tag{15}\]  
with boundary conditions  
\[u=u_{p}=-\Omega y,v=v_{p}=\Omega x,\ \ w=w_{p}=0,\overline{T}=\overline{T}_{w }=\ \overline{T}_{0}+ax,\ \ \text{at}\ z=0. \tag{16}\]  
\[\frac{\partial u}{\partial z}=\frac{\partial u_{p}}{\partial z}=0,\frac{ \partial v}{\partial z}=\frac{\partial v_{p}}{\partial z}=0,w=w_{p}\ =-W,\overline{T}=\overline{T}_{\infty}=\ \overline{T}_{0}+bx\,\ \text{at}\ z=h. \tag{17}\]  
The thermophysical properties of water and both types of carbon nanotubes i.e., SWCNTs and MWCNTs, are given in Table 1.