Class 11 > Unit # 07: Fluid Dynamics > Stoke's Law & Terminal Velocity

Talha's Physics Academy

Fluid Dynamics - Terminal Velocity of a Spherical Body

Q. Derive an expression for the terminal velocity of spherical body.

Terminal Velocity of a Spherical Body

Stokes’ law describes the viscous drag force experienced by a small spherical object moving through a viscous fluid. The drag force acting on the object is directly proportional to its velocity, radius, and the dynamic viscosity of the fluid, and is given by Stokes' equation:

$F_d = 6 \pi \eta r v$    --- (i)

Where:

• $F_d$ = Viscous drag force experienced by the object.
• $\eta$ = Dynamic viscosity of the fluid.
• $r$ = Radius of the spherical object.
• $v$ = Velocity of the object relative to the fluid ($\text{m/s}$).

Condition for Terminal Velocity

"The terminal velocity of a spherical body is the maximum constant velocity attained by the body when falling through a viscous fluid, which occurs when the upward viscous drag force plus the buoyant force equals the downward gravitational force (weight). Under these conditions, the net force on the body is zero, resulting in zero acceleration."

Considering the gravitational force (weight, $W = m g$) acting on the sphere:

$W = \frac{4}{3} \pi r^3 \rho g$    --- (ii)

Where:

• $\rho$ = Density of the spherical body ($\text{kg/m}^3$).
• $g$ = Acceleration due to gravity ($\text{m/s}^2$).
• $V = \frac{4}{3}\pi r^3$ = Volume of the spherical body.

Mathematical Derivation

At terminal velocity ($v = v_t$), the upward drag force equals the gravitational force acting downwards (ignoring fluid buoyancy for a small dense sphere, or setting drag equal to net downward weight):

$F_d = W$

Substitute the expressions for Stokes' drag force and weight into the equation:

$6 \pi \eta r v_t = \frac{4}{3} \pi r^3 \rho g$

Solving for terminal velocity ($v_t$), divide both sides by $6 \pi \eta r$:

$v_t = \frac{2 \rho g r^2}{9 \eta}$

Conclusion & Proportionalities:

This equation gives the expression for the terminal velocity of a small spherical body moving through a viscous fluid according to Stokes’ law. From the formula, terminal velocity is:

• Directly proportional to the square of the radius of the body ($r^2$).
• Directly proportional to the density of the body ($\rho$).
• Directly proportional to the acceleration due to gravity ($g$).
• Inversely proportional to the dynamic viscosity of the fluid ($\eta$).

© 2026 Talha's Physics Academy. All rights reserved.