Question

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

Answer 1

Expression for terminal velocity: Consider a sphere of radius r which falls freely through a highly viscous liquid of coefficient of viscosity η. Let the density of the material of the sphere be ρ and the density of the fluid be σ.

Forces acting on the sphere when it falls in a viscous liquid

Gravitational force acting on the sphere,

F_G = mg = `4/3π"r"^3ρ"g"` (downward force)

Upthrust, U = `4/3π"r"^3σ"g"` (upward force)

Viscous force, F = 6πηrv_t

At terminal velocity v_t,

Downward force = upward force

F_G = U + F

F_G − U = F ⇒ `4/3π"r"^3ρ"g" - 4/3π"r"^3σ"g"` = 6πηrv_t

v_t = `2/9 xx ("r"^2(ρ - σ))/η` g ⇒ `"v"_"t" ∝ "r"^2`

Here, it should be noted that the terminal speed of the sphere is directly proportional to the square of its radius. If a is greater than ρ, then the term (ρ – σ) becomes negative leading to a negative terminal velocity.