Derive the expression for the terminal velocity of a sphere moving in a high viscous fluid using stokes force.

Explanation

Expression for terminal velocity : Consider a sphere of radius r which falls freely through a highly viscous liquid of coefficient of viscosity `eta`. Let the density of the material of the sphere be `rho` and the density of the fluid be `sigma`.
Gravitational force acting on the sphere.
`F_(G) = mg = (4)/(3)pi r^(3)rho g` (downward force)
Up thrust, `U = (4)/(3)pi r^(3) sigma g` (upward force)
viscous force `F= 6pi eta r v_(t)`
At terminal velocity `v_(t)`
downward force = upward force
`F_(G) - U = F rArr (4)/(3) pi r^(3) rho g - (4)/(3) pi r^(3) sigma g = 6pi eta r v_(t)`
`v_(t) = (2)/(9) xx (r^(2)(rho - sigma))/(eta) g rArr v_(t) prop r^(2)`
Here, it should be noted that the terminal speed of the sphere is directly proportional to the square of its radius. If `sigma` is greater than `rho`, then the term `(rho - sigma)` becomes negative leading to a negative terminal velocity.