Fluid mechanics

Viscosity and the Reynolds number

Viscosity describes internal friction in a fluid; the Reynolds number compares inertial and viscous effects to characterize flow regimes.

Real fluids resist shear through viscosity. Flow is often laminar at low Reynolds number; when inertia is more dominant, fluctuations and turbulent mixing can develop.

Re=\frac{\rho vL}{\mu}=\frac{vL}{\nu},\qquad \tau=\mu\frac{\partial u}{\partial y}

Definition: Dynamic viscosity

Dynamic viscosity $\mu$ (Pa·s) relates shear stress to velocity gradient; kinematic viscosity is $\nu=\mu/\rho$ (m²/s). $L$ and $v$ are characteristic length and speed.

Vary Reynolds number to explore qualitative flow changes; the parabolic profile represents laminar Poiseuille flow in a circular pipe.

Pipe-flow regimes

In a straight circular pipe, low $Re$ generally favors laminar flow. Transition depends on disturbances and inlet conditions; $Re\approx2300$ is a common practical guide, not an exact universal threshold. At high $Re$ , inertia can sustain fluctuations and turbulent mixing.

Poiseuille profile

u(r)=\frac{\Delta p}{4\mu\ell}(R^2-r^2),\qquad Q=\frac{\pi R^4\Delta p}{8\mu\ell}

Fully developed laminar flow of a Newtonian liquid in a circular pipe has a parabolic profile: velocity is zero at the wall (no slip) and maximal on the axis. Poiseuille’s formula requires laminar flow and suitable pipe assumptions.

Example: Estimating Reynolds number

Water ( $\rho=1000\,\mathrm{kg/m^3}$ , $\mu=0.001\,\mathrm{Pa\,s}$ ) flows at $v=0.20\,\mathrm{m/s}$ in a pipe of diameter $D=0.010\,\mathrm m$ . Find $Re$ .

Solution

$Re=\rho vD/\mu=1000(0.20)(0.010)/0.001=2000$ . This is often laminar in a smooth pipe, though disturbances still affect transition.

Quick check

Example: Estimating Reynolds number in a pipe

Water with $ho=1000,mathrm{kg/m^3}$ and $mu=1.0 imes10^{-3},mathrm{Pa,s}$ flows through a pipe of diameter $D=0.020,mathrm m$ at mean speed $v=0.10,mathrm{m/s}$ . Then $Re= ho vD/mu=2000$ . This is near the commonly observed transition range in a circular pipe, so inlet disturbances and roughness can determine whether flow remains laminar or becomes unstable; a single threshold is not an absolute boundary.

If $Re$ rises because speed increases, which effect becomes relatively stronger?

Where is speed maximal in laminar Poiseuille flow?

References

Frank M. White (2016). Fluid Mechanics