AcademyFluids
Academy
Fluid Flow
Level 1 - Physics topic page in Fluids.
Principle
Steady flow follows mass conservation from one cross-section of a streamtube to another.
Notation
\(Q\)
volume flow rate
\(\mathrm{m^{3}\,s^{-1}}\)
\(\dot m\)
mass flow rate
\(\mathrm{kg\,s^{-1}}\)
\(A\)
cross-sectional area
\(\mathrm{m^{2}}\)
\(v\)
flow speed through the section
\(\mathrm{m\,s^{-1}}\)
\(\rho\)
fluid density
\(\mathrm{kg\,m^{-3}}\)
\(\Delta t\)
time interval
\(\mathrm{s}\)
Method
In steady flow, the mass entering one section in a given time must equal the mass leaving another section.
Volume passed
\[\Delta V=Av\Delta t\]
Mass passed
\[\Delta m=\rho Av\Delta t\]
Mass conservation
\[\rho_1A_1v_1=\rho_2A_2v_2\]
Incompressible flow
\[A_1v_1=A_2v_2\]
This follows when the density is effectively constant.
The key model choice is whether the fluid can be treated as incompressible over the change in pressure and speed.
Rules
These are the compact steady-flow relations.
Volume flow rate
\[Q=Av\]
Mass flow rate
\[\dot m=\rho Av\]
Steady continuity
\[\rho_1A_1v_1=\rho_2A_2v_2\]
Incompressible continuity
\[A_1v_1=A_2v_2\]
Examples
Question
Water flows from a pipe of area
\[6.0\times10^{-4}\,\mathrm{m^2}\]
into a nozzle of area \[2.0\times10^{-4}\,\mathrm{m^2}\]
If the pipe speed is \[1.5\,\mathrm{m\,s^{-1}}\]
find the nozzle speed.Answer
For incompressible flow,
\[A_1v_1=A_2v_2\]
so \[v_2=\frac{A_1}{A_2}v_1=\frac{6.0\times10^{-4}}{2.0\times10^{-4}}(1.5)=4.5\,\mathrm{m\,s^{-1}}\]
Checks
- \(Q\) has units of cubic metres per second; \(\\dot m\) has units of kilograms per second.
- The incompressible form \(Av\) constant needs nearly constant density.
- A smaller cross-section gives a larger speed in the same streamtube.
- Continuity compares sections of the same steady flow.