Following energy through a flow
Fluid dynamics links conservation of mass and energy to the internal friction that makes real flow different from an ideal streamline picture.
Learning goals
- relate volume and mass flow rates;
- apply continuity to steady incompressible flow;
- use Bernoulli’s equation only under its stated assumptions;
- describe viscosity, no-slip layers, laminar flow and turbulence;
- use Reynolds number and Poiseuille’s law and recognise real energy loss.
1. Describing flow
Density is $\rho=m/V$ and pressure is normal force per area, $P=F/A$. Volume flow rate is $Q=dV/dt=Av$ when the average speed $v$ is normal to area $A$; mass flow rate is $\dot m=\rho Q$. A streamline is tangent to the instantaneous velocity. In steady flow, the velocity field at each position does not change with time.
2. Continuity: conservation of mass
Mass cannot accumulate in a steady pipe: $\rho_1A_1v_1=\rho_2A_2v_2$. For an incompressible fluid, $A_1v_1=A_2v_2$. A narrower section therefore has greater average speed, but only when the assumptions and the same flow path apply.
3. Bernoulli’s equation
For steady, incompressible, non-viscous flow along one streamline with no pump, turbine or heat transfer, mechanical energy per volume is constant:
$P+\tfrac12\rho v^2+\rho gh=constant$.