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Question
Derive the expression for the terminal velocity of a sphere moving in a high viscous fluid using stokes force.
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Solution
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,
FG = mg = `4/3π"r"^3ρ"g"` (downward force)
Upthrust, U = `4/3π"r"^3σ"g"` (upward force)
Viscous force, F = 6πηrvt
At terminal velocity vt,
Downward force = upward force
FG = U + F
FG − U = F ⇒ `4/3π"r"^3ρ"g" - 4/3π"r"^3σ"g"` = 6πηrvt
vt = `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.
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