A disk turbine is used to stir a liquid in a baffled tank. To design the agitator, experiments are performed in a lab-scale model with a turbine diameter of $0.05 \:m$ and a turbine impeller speed of $600$ $rpm$. The liquid viscosity is $0.001$ $Pa$ $s$ while the liquid density is $1000$ $kg/m_{3}$. The actual application has a turbine diameter of $0.5 \:m$, an impeller speed of $600$ $rpm$, a liquid viscosity of $0.1$ $Pa$ $s$ and a liquid density of $1000$ $kg/m^{3}$. The effect of gravity is negligible. If the power required the lab-scale model is $P_{1}$ and the estimated power for the actual application is $P_{2}$, then the ratio $P_{2}/P_{1}$ is
- $10^{3}$
- $10^{4}$
- $10^{5}$
- $10^{6}$