Question

A lead sphere is steadily sinking in glycerin whose viscosity is equal to $\eta =13.9P$. What is the maximum diameter of the sphere at which the flow around that sphere still remains laminar? It is known that the transition to the turbulent flow corresponds to Reynolds number $Re=0.5$. (Here the characteristic length is taken to be the sphere diameter.)

Answer 1

$WehaveR=v{\rho }_{0}d/\eta andvisgivenby6\pi \eta rv=\frac{4\pi }{3}{r}^2(\rho -{\rho }_0)g(\rho =densityoflead,{\rho }_0=densityofglycerine.)v=\frac{2}{9\eta }{r}^2(\rho -{\rho }_0)g=\frac{1}{18\eta }{d}^2(\rho -{\rho }_0)gThus\frac{1}{2}=\frac{1}{18\eta }{d}^3(\rho -{\rho }_0){\rho }_{0}gandd={\left(\frac{9\eta }{(\rho -{\rho }_0){\rho }_{0}g}\right)}^{1/3}=5.2mm$