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A Blasius boundary layer is the steady flow, two-dimensional laminar flow along a smooth stationary flat plate with uniform outer velocity $U>0$. Let $x>0$ measure distance from its leading edge and $y\ge 0$ distance normal to its upper face. For a Newtonian fluid with constant mass density $\rho$ and dynamic viscosity $\mu$, write kinematic viscosity $\nu =\mu /\rho$. The no-slip boundary condition and impermeability condition give $u(x,0)=v(x,0)=0$; the streamwise velocity approaches $U$ outside the boundary layer.

For example, choose $U=10\mathrm{m}{\mathrm{s}}^{-1}$ and $\nu =1.5\times 10^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. At $x=0.30\mathrm{m}$, the local Reynolds number is ${\mathrm{Re}}_x=Ux/\nu =2.0\times 10^5$ and the boundary layer thickness is about $3.29\mathrm{mm}$. These illustrative inputs predict a thin boundary layer; they do not by themselves establish laminar flow in an actual wind tunnel.

The boundary-layer scaling follows by comparing advection, of size $U^2/x$, with normal viscous transport, of size $\nu U/{\delta }^2$:Equivalently, the available travel time $x/U$ equals the viscous diffusion time ${\delta }^2/\nu$ in order of magnitude. Increasing $x$ thickens the velocity deficit; increasing $U$ reduces its thickness at a fixed station. This boundary-layer scaling requires ${\mathrm{Re}}_x\gg 1$. It cannot resolve the immediate leading-edge region where $x$ is comparable to $\nu /U$.^[ref]

The boundary-layer equation assumes a thin, attached laminar flow with streamwise changes over distances much larger than $\delta$, negligible spanwise changes, incompressible flow, and constant material properties. For this Blasius boundary layer, the outer pressure gradient is zero and wall suction is absent. Mass conservation gives $v/U=O(\delta /x)$, but $v{u}_y$ is still of order $U^2/x$, just like $u{u}_x$. Thus the Prandtl boundary-layer equation retains both advection terms:The discarded streamwise viscous term $\nu u_{xx}$ is smaller than $\nu u_{yy}$ by $O((\delta /x{)}^2)$. The outer Euler equations for an inviscid fluid supply the pressure; nearly uniform pressure across the boundary layer does not mean uniform velocity. Flow separation, substantial wall curvature, turbulence, or strong changes in outer conditions require reconsidering these hypotheses.

Introduce the stream function and a dimensionless normal coordinate:Substitution gives the Blasius equation $f^{\prime \prime \prime }+f{f}^{\prime \prime }/2=0$, with $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$. Its numerical boundary value problem gives $f^{\prime \prime }(0)\simeq 0.332057$. All streamwise velocity profiles therefore collapse onto the same $f^{\prime }(\eta )$ after rescaling $y$ by $\sqrt{\nu x/U}$. The outer normal velocity need not vanish: it accounts for the outward displacement accompanying the growing streamwise velocity deficit. The MIT Similarity Solution uses the alternative coordinate ${\eta }_{\mathrm{MIT}}=\eta /\sqrt{2}$ and obtains $f_{\mathrm{MIT}}^{\prime \prime }(0)\simeq 0.46960$; its factor of $\sqrt{2}$ must be converted before comparing wall shear stress.

The historical Motion of fluids with very little viscosity by L. Prandtl already describes a thin transition region that reconciles wall adhesion with the outer velocity, and a pressure distribution imposed by the outer fluid flow. Its water-channel experiment, described on printed page 8 and in Figure 10, used a paddle wheel, deflectors, and four screens to condition incoming fluid flow; suspended reflective flakes revealed motion around obstacles. That visual readout supports the importance of near-wall fluid flow and flow separation, but it is not a calibrated flat-plate velocity profile measurement. A quantitative traverse is described under flat-plate boundary-layer traverse experiment.

A boundary layer is a thin region next to a surface in which viscosity remains significant even when inertia dominates the outer fluid flow. Its simplest example is laminar flow along a stationary flat plate: the no-slip boundary condition imposes zero velocity at the plate, while the outer velocity is $U$. The resulting thin velocity profile contains a large normal velocity gradient, so small dynamic viscosity can still produce appreciable wall shear stress. The Blasius boundary layer makes this competition quantitative and explains how a plate aligned with the incoming fluid flow acquires drag.

The same balance appears in a singular perturbation of a differential equation: large derivatives in a thin region can keep a term with a small coefficient significant. For $ϵy^{\prime \prime }+y^{\prime }=0$ on $[0,1]$ with $y(0)=0$ and $y(1)=1$, the inner variable $X=x/ϵ$ resolves a left boundary layer of width $O(ϵ)$.

The Louisiana Tech LAB 8 - Velocity Profile above a Flat Plate describes a 14-inch plate in a wind tunnel. A Pitot-static tube of 0.10-inch outside diameter traverses normally about 12 inches downstream; its screw scale has 0.001-inch increments. A water-column manometer reads the difference between stagnation pressure and static pressure. The procedure establishes a wall position, zeros the tunnel indicator, sets fan speed, and records at least ten positions inside the boundary layer, together with room temperature and barometric pressure. The instrument list includes a flow-stream thermometer implemented with a temperature-sensitive resistor. The manual requests comparison of outer velocity with the tunnel indicator and of the recovered velocity profile with laminar or turbulent predictions; it provides no observed dataset. These are apparatus specifications and instructions, not a reported confirmation of laminar flow.

Under incompressible flow conditions, an ideal aligned Pitot tube gives $q(y)=p_0(y)-p(y)=\rho u(y{)}^2/2$. Convert the manometer head $h$ to $q=({\rho }_{\mathrm{water}}-\rho )gh$, then infer $u=\sqrt{2q/\rho }$. Here $g$ is gravitational acceleration; mass density $\rho$ can be estimated from temperature and absolute pressure using the ideal gas law. A normalized velocity profile rises from the unresolved near-wall region to a plateau, whose value defines $U$; downstream profiles should broaden and collapse when plotted against $\eta$ if the Blasius boundary layer assumptions hold. Integrating a recovered velocity profile gives displacement thickness and momentum thickness.

Additional controls for interpreting this experiment follow from the measurement model: check differential-pressure calibration with equal port pressures and known liquid heads; leak-check tubing; align the Pitot tube with the incoming velocity; convert wall contact to the inlet-centre coordinate rather than treating outside contact as $y=0$; repeat traverses and outer velocity readings to detect drift. Keep the plate smooth and aligned, and check outer static pressure at multiple downstream stations before assuming zero pressure gradient. Repeating at another station or outer velocity tests the predicted boundary-layer scaling more directly than fitting one curve. These controls are proposed checks, not actions documented as completed in the manual.

Finite probe size, wall interference, flow fluctuations, and weak near-wall dynamic pressure limit inference. Even the manual's probe-based Reynolds number criterion, ${\mathrm{Re}}_d=ud/\nu >1000$ for its ideal Bernoulli equation conversion, must be checked with local $u$, not just $U$. In the illustrative model at $x=0.3048\mathrm{m}$, ${\delta }_{99}\simeq 3.32\mathrm{mm}$; the quoted outside diameter is $2.54\mathrm{mm}$, a substantial fraction of that thickness. A real inlet's sensing footprint and correction cannot be deduced from outside diameter alone. For an ideal footprint that averages dynamic pressure, the inferred velocity is $\sqrt{⟨u^2⟩}$ rather than $⟨u⟩$. Differentiating such sparse readings to obtain wall shear stress magnifies measurement error; the momentum thickness route is less sensitive to this differentiation but still needs the unresolved near-wall contribution. The no-slip boundary condition is a model constraint here, not a direct reading from a probe centred on the wall.

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. ^[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. ^[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 $A$: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. ^[ref]

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 $\Delta p=p_{\mathrm{in}}-p_{\mathrm{out}}$ over a developed length $L$ gives the Hagen-Poiseuille equation:The velocity is zero at the wall and maximal at the centre. Averaging over the circular area, whose annuli have area $2\pi rdr$, 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. ^[ref]

As a stipulated calculation, take $D=0.010\mathrm{m}$, $U=0.10\mathrm{m}{\mathrm{s}}^{-1}$, $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, $\mu =0.0010\mathrm{Pa}\mathrm{s}$ and $L=1.0\mathrm{m}$, and assume the developed laminar flow above. Then $\nu =10^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, ${\mathrm{Re}}_D=1000$, $u(0)=0.20\mathrm{m}{\mathrm{s}}^{-1}$, $Q=7.85\times 10^{-6}{\mathrm{m}}^3{\mathrm{s}}^{-1}$ (about $0.471\mathrm{L}\mathrm{mi}{\mathrm{n}}^{-1}$), 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.

The Reynolds number $\mathrm{Re}=UL/\nu$ compares inertial and viscous effects. Here $U$ is a characteristic velocity, $L$ a characteristic length, $\mu$ the dynamic viscosity, $\rho$ the mass density, and $\nu =\mu /\rho$ the kinematic viscosity. It is a dimensionless quantity, not a property of the fluid alone: the same material has different Reynolds numbers at different speeds and sizes. Specify the chosen scales when reporting it.

The comparison comes from the Navier-Stokes equation. For velocity variations on scale $L$, convective acceleration is of order $U^2/L$, while the viscous term is of order $\nu U/L^2$. Their ratio is $UL/\nu$. Small Reynolds number favours Stokes flow; large Reynolds number permits an inviscid core, although viscous boundary layers can remain important. Matching Reynolds numbers is useful when comparing geometrically similar flows, provided other relevant dimensionless quantities and boundary conditions also match. ^[ref]

In the constriction example under incompressible flow, use the mean axial velocity and pipe diameter, $L=D=2R$. Taking $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$ and a stipulated dynamic viscosity $\mu =0.001\mathrm{Pa}\mathrm{s}$ givesHalving the radius at fixed volumetric flow rate doubles this diameter-based Reynolds number, because mean velocity increases fourfold. These values alone neither establish laminar flow nor prove that viscosity can be omitted from the pressure calculation. Increasing $\mu$ to $0.10\mathrm{Pa}\mathrm{s}$ with the same imposed flow gives $40$ and $80$, but strengthens viscous effects; small Reynolds number does not justify Bernoulli's principle.

The name recalls Osborne Reynolds's 1883 experimental investigation of water motion and resistance in pipes. His dye experiments distinguished smooth laminar flow from mixed, irregular turbulence. ^{[ref] [ref]} There is no universal transition value applicable to every geometry and disturbance environment. ^[ref]

In his 1883 paper, Osborne Reynolds investigated how the character of pipe flow depends jointly on velocity, pipe diameter and viscosity. His final dye apparatus drew water from a large glass tank through a horizontal glass tube with a smooth trumpet-shaped entrance. A separate reservoir supplied strongly colored water just upstream of the entrance; a valve below the tank controlled the outflow. The glass tube was immersed in the tank, so its internal dye band could be watched. Reynolds repeated the trials with different tube diameters and water temperatures. ^[ref]

The volumetric flow rate was measured independently of the appearance of the dye. Reynolds timed a one-inch fall of the tank's water level, corresponding to a calibrated discharged volume. In modern notation, if this volume is $V$ and the elapsed time is $\Delta t$, then $Q=V/\Delta t$ and $U=4V/(\pi D^2\Delta t)$. He determined the glass tubes' internal diameters by filling them with water. For a uniform bore, a known filled length $\ell$ and measured water volume $V_{\mathrm{tube}}$ determine $D=\sqrt{4V_{\mathrm{tube}}/(\pi \ell )}$; checking bore uniformity matters because the same volume would only give an average cross-section in a tapered tube. Thermometers at different tank levels checked both the temperature and its uniformity. Reynolds compared the viscosity at different temperatures using Poiseuille's results and checked his own water by separate experiments of that type. Thus the Reynolds number estimate did not come from judging the dye band or from assuming that transition occurred at a preset value. ^[ref]

A practical repetition can instead collect a known outflow volume in a measuring vessel and time its collection, or weigh the collected water and divide its mass by mass density. Measure the internal diameter, rather than the external diameter, and obtain kinematic viscosity at the measured water temperature from independently established properties or a separate viscosity measurement. Record the dye appearance and observation position alongside $Q$, $D$, temperature and the resulting Reynolds number. These are alternative measurement procedures following $Q=V/\Delta t$, not additional historical claims about Reynolds' dye apparatus.

Begin with settled water and a small dye supply, then open the outlet valve in small steps. Reynolds saw a thin straight band at sufficiently low speeds. With a tank that had not fully settled, the band could wander while retaining its coherence. At higher discharge it could abruptly spread into the surrounding water downstream of the entrance. Short illumination by an electric spark resolved curls within what otherwise looked like a colored cloud. This connected the visible breakup with internal irregular motion rather than merely the appearance of an opaque pipe. ^[ref]

The controls are part of the result. Reynolds used a smooth entrance and let the tank rest to reduce inlet disturbances. He found that a heated tank could retain slow circulation as it cooled, so measuring one temperature was insufficient: thermal uniformity and residual motion also mattered. His wire-coil trial broke up the band below the previous onset speed, but he explicitly cautioned that once the dye was mixed he could not tell from that dye alone whether the motion became orderly again farther along the tube. In a repetition, keep the dye supply small enough to act approximately as a passive scalar, avoid a strong injection jet, and record the inlet condition and downstream observation distance. If also testing the Hagen-Poiseuille equation, allow the profile to develop and measure pressure drop between positions within the developed region; do not treat the entrance and outlet losses as part of its uniform pressure gradient. The dye experiment by itself does not establish fully developed flow. ^[ref]

The experiment's usefulness is that it compares a visible change in the motion with independently measured scales. Repeating it with several diameters and temperatures, while controlling disturbances, tests the organizing role of the Reynolds number. Its limitations are equally instructive: dye is a tracer of transport, diffusion still acts in laminar flow, and a local dye observation cannot by itself prove that turbulence persists throughout an arbitrarily long pipe.