Oil of dynamic viscosity and density flows along a straight horizontal pipe of…

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

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

Problem

Oil of dynamic viscosity

\mu = 0.080\ \mathrm{Pa\,s}

and density

\rho = 870\ \mathrm{kg\,m^{-3}}

flows along a straight horizontal pipe of internal radius

a = 4.0\ \mathrm{mm}

. The flow is steady, smooth, layered, incompressible, unidirectional and fully developed, and the oil obeys the no-slip condition at the wall, so the axial velocity is

u(r) = (G/4\mu)(a^{2}-r^{2})

, where

r

is the distance from the axis and

G = -\mathrm{d}p/\mathrm{d}x

is the constant axial pressure gradient, here

G = 2.5\times10^{4}\ \mathrm{Pa\,m^{-1}}

. At the section considered the absolute pressure is

p = 8.0\times10^{5}\ \mathrm{Pa}

. In the plane containing the axis take

x

along the flow and

y

across it measured from the axis, so that the flow there is planar with

u_x = u(y)

u_y = 0

. ((a)) Find the volume flow rate

Q

. ((b)) Find the traction (force per unit area) the oil exerts on the pipe wall at

y = +a

, giving the component along the flow and the component normal to the wall. Use the book's convention: with

\mathbf{n}

the unit normal pointing out of the wall into the oil,

\mathbf{t}(\mathbf{n}) = \boldsymbol{\sigma}\mathbf{n}

with

\sigma_{ij} = -p\,\delta_{ij} + \tau_{ij}

is the force per unit area exerted on the surface element by the oil. ((c)) Find the magnitude of the vorticity

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

at the wall. In your working, state which terms of the Navier-Stokes equation vanish for this flow and why.

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