Two parallel circular steel discs of radius face each other across a thin film of oil…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity

Fluid Dynamics, Ch Stokes flow, lubrication and viscous wetting

Problem

Two parallel circular steel discs of radius

R = 3.0\,\mathrm{cm}

face each other across a thin film of oil of viscosity

\mu = 0.10\,\mathrm{Pa\,s}

. A constant external force

W = 50\,\mathrm{N}

pushes the discs together, squeezing the film radially outward. The lubrication approximation for radial flow between the discs gives the squeeze-film force

F = 3\pi\mu R^4|\dot h|/(2h^3)

needed to close the gap at rate

\dot h

when the instantaneous gap is

h

. ((a)) Using

F = W

(constant), derive a first-order ordinary differential equation for

h(t)

and integrate it to find

h

as a function of

t

, given that

h = h_0

t = 0

. ((b)) Find the time taken for the gap to close from

h_0 = 200\,\mu\mathrm{m}

h_1 = 50\,\mu\mathrm{m}

Related theory

Ordinary differential equations

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