= Laminar flow
{wiki}
<Laminar flow> is a regime of <fluid flow> in which the motion remains organized, with neighboring fluid parcels following smooth, coherent paths rather than the irregular interacting fluctuations characteristic of <turbulence>. In a straight pipe, the simplest example consists of coaxial layers sliding past one another. This describes the character of the motion, not a requirement that every parcel have the same <velocity>. Nor does it require <steady flow>: a smooth profile can evolve in time as <viscosity> transmits motion from a newly moving wall into a resting <fluid>. The unsteady wall-driven solution provides an explicit example. https://notes.shreyasmandre.com/fluid-dynamics-notes/Ch3.html[Shreyas Mandre, Fluid Dynamics Lecture Notes, section 3.3]{ref}
The <Reynolds pipe-and-dye experiment> makes this distinction visible. A thin injected dye band can remain coherent in <laminar flow>, whereas irregular motion can fold it and carry dye across the pipe. However, <molecular diffusion> can spread dye even in <laminar flow>, and an inlet disturbance can move a coherent band without producing <turbulence>. A dye band is a <streakline>, not a measurement of the entire <velocity field>; in <steady flow> an ideal passive tracer's <streakline> coincides with a <streamline>. Reynolds explicitly distinguished a wandering band from one broken into eddies. https://archive.org/stream/philtrans06508925/06508925_djvu.txt[Osborne Reynolds, An Experimental Investigation (1883), section 11, pp. 941–943]{ref}
For a circular pipe of internal diameter $D$, define the mean axial <velocity> $U$ using the <volumetric flow rate> $Q$ and cross-sectional <area> $A$:
$$
A=\frac{\pi D^2}{4},\qquad U=\frac QA,\qquad
\mathrm{Re}_D=\frac{UD}{\nu}=\frac{\rho UD}{\mu}
=\frac{4Q}{\pi D\nu}.
$$
Here $\rho$ is <mass density>, $\mu$ is <dynamic viscosity> and $\nu=\mu/\rho$ is <kinematic viscosity>. This particular <Reynolds number> uses the diameter and mean <velocity>, not the radius or centreline <velocity>. Small <Reynolds number> generally favors <laminar flow>, but a single universal critical <Reynolds number> does not distinguish it from <turbulence> in all geometries and experiments. Even in a given pipe, inlet conditions and the magnitude of disturbances affect the observed onset. Reynolds demonstrated this dependence by deliberately inserting a wire coil. https://archive.org/stream/philtrans06508925/06508925_djvu.txt[Osborne Reynolds (1883), section 23, pp. 955–956]{ref}
\Image[/laminar-pipe-velocity-profile.png]
{title=Ideal laminar pipe flow: zero velocity at the wall and twice the mean velocity at the centre}
{description=An illustrative fully developed Poiseuille solution, not measured data. The arrows and curve use a 10 mm diameter pipe with mean axial velocity 0.10 m/s. The dashed line marks the mean over the circular cross-section, not the mean along a diameter.}
The quantitative example above assumes <steady flow>, <incompressible flow>, a <Newtonian fluid> with constant <dynamic viscosity>, a straight horizontal circular pipe of radius $R=D/2$, stationary walls satisfying the <no-slip boundary condition>, and <fully developed flow>. The last assumption means that the axial <velocity> profile no longer changes downstream. A positive <pressure> drop $\Delta p=p_{\rm in}-p_{\rm out}$ over a developed length $L$ gives the <Hagen-Poiseuille equation>:
$$
u(r)=\frac{\Delta p}{4\mu L}(R^2-r^2)
=2U\left(1-\frac{r^2}{R^2}\right),\qquad
Q=\int_0^R u(r)\,2\pi r\,dr
=\frac{\pi R^4\Delta p}{8\mu L}.
$$
The <velocity> is zero at the wall and maximal at the centre. Averaging over the circular <area>, whose annuli have <area> $2\pi r\,dr$, gives $U=u(0)/2$; averaging along one diameter would give a different answer. These assumptions produce a <parabola>, rather than every <laminar flow> having a parabolic profile. https://notes.shreyasmandre.com/fluid-dynamics-notes/Ch3.html[Shreyas Mandre, Fluid Dynamics Lecture Notes, section 3.2.2, equations (3.9)–(3.11)]{ref}
As a stipulated calculation, take $D=0.010\,\mathrm m$, $U=0.10\,\mathrm{m\,s^{-1}}$, $\rho=1000\,\mathrm{kg\,m^{-3}}$, $\mu=0.0010\,\mathrm{Pa\,s}$ and $L=1.0\,\mathrm m$, and assume the developed <laminar flow> above. Then $\nu=10^{-6}\,\mathrm{m^2\,s^{-1}}$, $\mathrm{Re}_D=1000$, $u(0)=0.20\,\mathrm{m\,s^{-1}}$, $Q=7.85\times10^{-6}\,\mathrm{m^3\,s^{-1}}$ (about $0.471\,\mathrm{L\,min^{-1}}$), and $\Delta p=32\mu LU/D^2=32\,\mathrm{Pa}$. These are predictions from chosen parameters, not measurements or a claim that $\mathrm{Re}_D=1000$ guarantees <laminar flow> in every apparatus.
The beauty of this example is that a spatially varying <velocity field> still gives an exact relation between two accessible measurements, <pressure> drop and <volumetric flow rate>. It predicts the <hydraulic resistance> $\Delta p/Q=8\mu L/(\pi R^4)$. Doubling the radius therefore multiplies $Q$ by sixteen at fixed <pressure> drop, length and <dynamic viscosity>, provided the model remains applicable. This makes the <laminar flow> model useful for sizing narrow tubes or determining <dynamic viscosity>. It also explains why a persistent narrow dye band can be informative without implying that the fluid is moving as a uniform plug.
Back to article page
