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

Explanation

Let us 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)pir^(3)rhog` (downward force)
Up thrust, `U=(4)/(3)pir^(3)sigmag` (upward force)
viscous force `F=6pietarv_(t)`
At terminal velocity `v_(t)`
downward force = upward force.
`F_(G)-U=FrArr(4)/(3)pir^(3)rhog-(4)/(3)pir^(3)sigmag=6pietarv_(t)`
`v_(t)=(2)/(9)xx(r^(2)(rho-sigma))/(eta)g`
`thereforev_(t)propr^(2)`