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 be `rho` and the density of the fluid be `sigma.`
Gravitational force acting on the sphere.
`F_G = mg = 4/3 pi r^2 rho g` (downwards force)
Up thrust, `U = 4/3 pi r^3 sigma g` (upward force)
Viscous force `F = 6 pi eta rv_t`
At terminal velocity `v_t`,
Downward force = upward force
`F_G - U = F implies 4/3 pi r^3 rho g - 4/3 sigma r^2 sigma g = 6 pi eta r v_t`
`v_t = 2/9 xx (r^2 (rho - sigma))/(eta) g implies v_t prop r^2`.
Here, it should be noted that hte 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.