Equation with cases and numbering and including text

Question

I am trying to create cases in the equation with numbering to individual cases and trying to push the text which is in \mbox{...} to the right. But I wasn't able to succeed. I request your kind help and suggestions on the same. Thank you very much for your help in advance. Kindly the refer the below figure for better understanding visually.

Thanks and Regards, Arjun

\begin{equation}
 d_{50} = 
  \begin{cases}
    \left(\frac{18\eta U_{r0}}{\rho_p U_{\theta0}^2}\right)^{1/2} \frac{r}{r_0^{1/2}} = C\frac{r}{r_0^{1/2}}   \mbox{ m=1(Free vortex)}\\
    \label{Eq:DpCFreV}

    \left(\frac{18\eta U_{r0}}{\rho_p U_{\theta0}^2}\right)^{1/2} \frac{r_0^{3/2}}{r} =   C\frac{r_0^{3/2}}{r}   \mbox{ m=-1(Forced vortex)}
    \label{Eq:DpCForcV}
\end{cases}
\end{equation}

Answer 1

If I understand correctly what you want, you can do that easily with the cases package, which defines a numcases and a subnumcases environment, depending on the type of numbering you want:

\documentclass{article}
\usepackage{cases}

\begin{document}

\begin{numcases}{ d_{50} =}
\left(\frac{18\eta U_{r0}}{\rho_p U_{\theta0}^2}\right)^{1/2} \frac{r}{r_0^{1/2}} = C\frac{r}{r_0^{1/2}} &\quad $ m = 1 $ (Free vortex) 
\label{Eq:DpCFreV}\\
\left(\frac{18\eta U_{r0}}{\rho_p U_{\theta0}^2}\right)^{1/2} \frac{r_0^{3/2}}{r} = C\frac{r_0^{3/2}}{r} &\quad $ m=-1 $ (Forced vortex) 
\label{Eq:DpCForcV}
\end{numcases}

\begin{subnumcases}{ d_{50} =}
\left(\frac{18\eta U_{r0}}{\rho_p U_{\theta0}^2}\right)^{1/2} \frac{r}{r_0^{1/2}} = C\frac{r}{r_0^{1/2}} &\quad $ m = 1 $ (Free vortex)
\label{Eq:DpCFreV}\\
\left(\frac{18\eta U_{r0}}{\rho_p U_{\theta0}^2}\right)^{1/2} \frac{r_0^{3/2}}{r} = C\frac{r_0^{3/2}}{r} &\quad $ m=-1 $ (Forced vortex)
\label{Eq:DpCForcV}
\end{subnumcases}

\end{document}