A circular disc of radius R rotates with angular speed ω parallel to a fixed horizontal plate. A thin Newtonian liquid film of uniform thickness d and viscosity η separates the disc and the plate. Assuming a linear velocity gradient in the film, what is the viscous torque opposing the rotation?
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Question

A circular disc of radius R rotates with angular speed ω parallel to a fixed horizontal plate. A thin Newtonian liquid film of uniform thickness d and viscosity η separates the disc and the plate. Assuming a linear velocity gradient in the film, what is the viscous torque opposing the rotation?

Options

G1 Physics · Accepted Answer

Consider an annular element of the rotating disc at radius r and width dr. Its speed is v = ωr. The shear stress is τ = ηv/d = ηωr/d. Area of the annulus: dA = 2πr dr. Viscous force on the annulus: dF = τ dA = [ηωr/d](2πr dr) = 2πηωr²dr/d. The corresponding torque is dΓ = r dF = 2πηωr³dr/d. Integrating from 0 to R, Γ = [2πηω/d]∫₀ᴿr³dr = [2πηω/d][R⁴/4]. Therefore, Γ = πηωR⁴/(2d). Correct option: A.