Continuity equation $($v$\_$r$/$r$)+(1/$r$)($v$\_$r$)+($v$\_$z$/$z$)=0$ Theta momentum $($v$\_$r$)($v$\_$theta$/$r$)+($v$\_$z$)($v$\_$theta$/$z$)+(1/$r$)($v$\_$r$)($v$\_$theta$)=$v$((^2$v$\_$theta$/$r$^2)+(1/$r$)($v$\_$theta$/$r$)+(^2$v$\_$theta$/$z$^2)-($v$\_$theta$/$r$^2))$ R momentum $($v$\_$r$)($v$\_$r$/$r$)+($v$\_$z$)($v$\_$r$/$z$)-($v$\_$theta$^2/$r$)=-(1/$rho$)($p$/$r$)+$v$((^2$v$\_$r$/$r$^2)+(1/$r$)($v$\_$r$/$r$)+(^2$v$\_$r$/$z$^2)-($v$\_$r$/$r$^2))$ Z momentum $($v$\_$r$)($v$\_$z$/$r$)+($v$\_$z$)($v$\_$z$/$z$)=-(1/$rho$)($p$/$z$)+$v$((^2$v$\_$z$/$r$^2)+(1/$r$)($v$\_$z$/$r$)+(^2$v$\_$z$/$z$^2))$ To obtain similarity solutions to Eq$.($A$),$ introduce the dimensionless variable $=$ sqrt$($$/$$)$z$,$ and functions v$\_$r$($r$,$ z$)=$ rF$($$($z$)),$ v$\_$$($r$,$ z$)=$ rG$($$($z$)),$ v$\_$z$($r$,$ z$)=$ sqrt$($$)$H$($$($z$)),$ p$($r$,$ z$)=$ rhoP$($$($z$)),$ where F$,$ G$,$ H$,$ and P are dimensionless, smooth functions of $.$ Substitute these dimensionless variables and functions into Eq$.($A$)$ and derive equation group Eq$.($B$)$ and the corresponding boundary conditions at $=0$ and $+$ specified in the lecture notes.