Derive an expression for the terminal velocit of a sphere falling through a viscous liquid.

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 = 6 pi 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 sigma r^3 = 6pi eta rv_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 vdocity.