Question

A solid cylinder of diameter D, length L, and density ρs falls due to gravity concentrically through a long vertical sleeve of diameter D₀. The clearance is filled with fluid of density ρ and viscosity µ.   a) Find an expression for the velocity at which the cylinder is travelling. b) Apply your formula to the case of a steel cylinder (ρ = 7.6g/cm³ ), D = 2 cm, D₀ = 2.04cm, L = 15 cm, with a film of SAE 30W oil at 20°C.

Answer 1

Using the force balance

Total force in downward direction = cylinder weight + force due to viscous drag

Since cylinder falls due to gravity, the attained velocity is terminal velocity.

At terminal velocity,

Total force in downward direction = 0

cylinder weight = viscous drag

Mass of cylinder x volume of cylinder = shear stress x area

ρs x g x (π/4) x D² x L = [(2μV) / (D0 - D)] x πDL

Solve for V

ρs x g x (π/4) x D² x L x (D0 - D) = (2μV) x πDL

V = ρs x g x D x (D0 - D) / 8μ

Part b

For SAE 30 oil at 20°C

Viscosity μ = 0.29 kg/m·s

ρs = 7.6*1000 = 7600 kg/m3

g = 9.81 m/s2

D = 2 cm x 1m/100cm = 0.02 m

D0 = 2.04 cm x 1m/100cm = 0.0204 m

V = 7600 x 9.81 x 0.02 x (0.0204 - 0.02) / (8 x 0.29)

= 0.596448 / 2.32

= 0.257 m/s