Question #42106

A solid sphere of radius $R$ and density $\rho_s$ falls from rest under gravity in a fluid of density $\rho_f$ and viscosity $\eta$. Due to strong molecular cohesion, the fluid does not obey Stokes' law. Instead, a boundary layer of thickness $\delta$ forms around the sphere, where $\delta = R \left(\frac{v_0}{v}\right)$, with $v$ being the instantaneous velocity of the sphere and $v_0$ a characteristic fluid velocity constant. The velocity profile of the fluid within this boundary layer, from the sphere's surface ($y=0$) to the edge of the layer ($y=\delta$), is given by $u(y) = v\left(1 - \frac{y}{\delta}\right)^2$. Assuming the drag force is entirely due to the viscous shear stress at the sphere's surface, what is the terminal velocity $v_T$ of the sphere?