Expression for terminal velocity:

When a small spherical sphere of radius \( r \), falls through a liquid of coefficient of viscosity \( \eta \) with terminal velocity \( v \), then the effective weight of the spherical ball in the liquid is balanced by the viscous force \( F \) of the liquid given by,

\( F=6\pi\eta{rv}\tag{1} \)

If \( \rho \) be the density of the material of the body, then downward force due to gravity on the ball is \( \frac{4}{3}\pi{r^3}\rho{g} \) and the upward force due to the buoyancy is \( \frac{4}{3}\pi{r^3}\sigma{g} \) , where \( \sigma \) is the density of the liquid and \( g \) is the acceleration due to gravity.

The resultant downward force on the spherical ball is \( \frac{4}{3}\pi{r^3}(\rho-\sigma)\tag{2} \)

From equations (1) and (2), we get

\( 6\pi\eta{rv}=\frac{4}{3}\pi{r^3}g(\rho-\sigma) \)

or, \( v=\displaystyle{\frac{2{r^2}g(\rho-\sigma)}{9\eta}} \)

This is the expression of the terminal velocity.

Correction for wall effect and end effect:

The viscous drag of a small spherical body falling in a liquid of finite extent is given by, \( 6\pi\eta{rv} \).

But in practical case, when the liquid is confined in a cylindrical vessel, correction should be applied due to the boundaries of the walls and bottom of the cylinder and the corrected formula for \( \eta \) when the liquid is confined in a cylinder is given by,

\( \displaystyle{\eta=\frac{2}{9} \frac{{r^2}(\rho-\sigma)g}{v(1+2.4\frac{r}{R})(1+3.3\frac{r}{h})}} \)

Where, \( R \) is the radius of the cylinder and \( h \) is the height of the liquid in the cylinder. Here \( v(1+2.4\frac{r}{R}) \) is the velocity of the spherical body in the liquid of infinite width and \( v(1+3.3\frac{r}{h}) \) is the velocity in the liquid of the infinite depth and \( v \) is the observed velocity.