A bearing is lubricated with oil of dynamic viscosity and density

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

Fluids and Continuum Physics, Ch The Navier-Stokes equations and exact viscous solutions

Problem

A bearing is lubricated with oil of dynamic viscosity

\mu = 0.20\ \mathrm{Pa\,s}

and density

\rho = 880\ \mathrm{kg\,m^{-3}}

. The oil reaches the bearing through a straight horizontal circular feed duct of radius

a = 0.50\ \mathrm{mm}

and length

L = 40\ \mathrm{mm}

, across which the pressure difference is

\Delta p = 150\ \mathrm{kPa}

; the flow in the duct is steady, smooth, layered, incompressible and fully developed, with no slip at the wall. In the bearing itself the oil fills a gap of thickness

h = 0.10\ \mathrm{mm}

between two flat parallel surfaces (the curvature of the bearing is ignored). One surface moves in its own plane at

U = 4.0\ \mathrm{m\,s^{-1}}

and the other is at rest; there is no pressure gradient along the gap and gravity is negligible across it, so the film is in plane Couette flow with

u_x = Uy/h

u_y = 0

y

measured from the stationary surface. ((a)) Find the volume flow rate through the feed duct. ((b)) For the film, find the rate-of-strain component

e_{xy} = \tfrac12(\partial u_x/\partial y + \partial u_y/\partial x)

and the viscous shear stress

\tau_{xy}

. ((c)) Find the magnitude of the vorticity

\boldsymbol{\omega} = \nabla\times\mathbf{u}

in the film, and say whether the film flow is irrotational. In your working, state which terms of the Navier-Stokes equation vanish in each of the two flows and why.

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