Air (density at the instant considered) flows through a straight duct of constant…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Kinematics in three dimensions and Cauchy's equation of motion

Problem

Air (density

\rho=1.20\,\mathrm{kg\,m^{-3}}

at the instant considered) flows through a straight duct of constant cross-sectional area

A=0.010\,\mathrm{m^2}

along the

x

axis. At this instant the velocity is

u(x)=U_0+kx

with

U_0=2.0\,\mathrm{m\,s^{-1}}

and

k=5.0\,\mathrm{s^{-1}}

, uniform across the duct, and the density is momentarily uniform along the duct. Mass conservation for this one-dimensional flow gives

\partial\rho/\partial t=-\rho\,\partial u/\partial x

at this instant. Consider the fixed control volume between

x=0

and

x=1.0\,\mathrm m

. ((a)) Use

\partial\rho/\partial t

to compute

\mathrm d/\mathrm dt\int_V\rho\,\mathrm dV

for this fixed volume directly. ((b)) Compute the same rate independently as the net mass flux into the volume through its two fixed end faces. ((c)) Confirm the two results agree.

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