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The Bernoulli principle relates pressure, kinetic energy per unit volume and Newtonian gravitational potential along a streamline of steady flow with inviscid flow. For constant mass density, the Bernoulli equation gives $p/\rho +u^2/2+\Phi$ constant along that streamline. Constancy across different streamlines requires further hypotheses, such as irrotational flow.

Precisely, suppose a fluid of constant positive mass density obeys the Euler equations for an inviscid fluid, its velocity field and pressure are steady, and the body force per unit mass is $-\nabla \Phi$. ThenEvery term has units of energy per unit mass. For a uniform gravitational field, Newtonian gravitational potential is $\Phi =gz$, up to an additive constant. The pressure term represents work by surrounding fluid, so acceleration can exchange pressure work with kinetic energy. Dotting the steady Euler equations with $\mathbf{u}$ gives the displayed directional derivative: this is why constancy follows a streamline. ^[ref]

Continue the horizontal water constriction example from incompressible flow. Assume negligible viscous losses, steady flow and approximately uniform flow at both measurement sections, so the streamline speeds there equal the mean speeds $0.20$ and $0.80\mathrm{m}{\mathrm{s}}^{-1}$. Equal height cancels the Newtonian gravitational potential terms. The ideal pressure drop isThis is the Venturi effect in its ideal limit: the gain in kinetic energy density is supplied by pressure work. Cambridge's treatment uses this relation to connect a measured pressure difference to volumetric flow rate. ^[ref]

The $300\mathrm{Pa}$ result is conditional, rather than a prediction of the actual viscous pipe. Mass conservation fixes $Q$ and the ratio of mean speeds even with viscosity, but nonuniform velocity profiles require an integrated energy balance, and wall friction dissipates mechanical energy. A pump would supply additional work. In particular, constant mean speed in a straight pipe does not imply constant pressure: Poiseuille flow needs a continuing pressure gradient to overcome viscosity. ^[ref]

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 conventional boundary layer thickness ${\delta }_{99}$ is the normal distance at which the streamwise velocity first reaches $0.99U$ in a monotone velocity profile. A Blasius boundary layer approaches $U$ smoothly, so its apparent edge depends on the chosen percentage; there is no discontinuity. Solving $f^{\prime }({\eta }_{99})=0.99$ in the Blasius equation gives ${\eta }_{99}\simeq 4.910$, henceThe factor $4.910$ belongs to this definition, whereas the boundary-layer scaling specifies only order of magnitude. Near this edge the velocity changes slowly, making a threshold position sensitive to measurement error; displacement thickness and momentum thickness summarize the whole velocity profile.^[ref]

Choose $\ell =1.50\mathrm{m}$, $n=4/3$, ${\lambda }_0=550\mathrm{nm}$ and $u=6.00\mathrm{m}{\mathrm{s}}^{-1}$, with $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$. These are illustrative choices, not measured values or a reconstruction of Hippolyte Fizeau's data. Each beam traverses $L=3.00\mathrm{m}$ of water. The nondispersive prediction isThe first number is the offset from rest to one flow setting; the second compares equal opposite settings. The very short equivalent delay explains why interference is useful: a measurable fraction of a wavelength replaces an unattainable direct timing measurement.

The Galilean transformation would give phase velocity magnitudes $c/n\pm u$, hence $N_{\mathrm{full}}\simeq 4\ell u{n}^2/(c{\lambda }_0)$. At the chosen values this is $0.38815$ fringe from rest, larger than the special relativity prediction by $n^2/(n^2-1)=16/7$. A model with no flow-induced change of phase velocity instead gives $N=0$. The three alternatives are experimentally distinguishable even when $u/c$ is tiny.

Hippolyte Fizeau's positive displacement favoured Augustin-Jean Fresnel's partial-drag law over zero or full drag. He explicitly distinguished support for that law from establishing its underlying luminiferous aether hypothesis.^[ref] The Fizeau experiment therefore supplies a nonzero relative-motion measurement, not a null test proving an luminiferous aether's absence. The derivation above exhibits the same first-order law within special relativity, showing why agreement with the coefficient alone cannot select an physical interpretation.

Two limitations matter for precision. The velocity profile sampled by the beam need not equal the cross-sectional average inferred from volumetric flow rate. Also, the Doppler effect gives opposite rest-frame frequencies for the counterpropagating beams. With optical dispersion, the coefficient $n^2-1$ in the interference fringe formula becomeswhere the refractive index and its derivative are evaluated for stationary water at the laboratory angular frequency $\omega =2\pi c/{\lambda }_0$. A wavelength-independent $n$ therefore defines an approximation, not a precision fit to real water.^[ref] Flow reversal removes a fixed phase difference, but a mechanical or temperature change that reverses with the flow can still imitate the signal and must be controlled.

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.

A stationary point in a non-monotone velocity profile can support a quarter-power critical layer whose phase-speed correction has positive imaginary part. The resulting normal mode grows at a rate proportional to the square root of the wavenumber. Such unbounded high-frequency amplification demonstrates linear instability of the frozen layer; nonlinear evolution and precise well-posedness consequences require separate analysis.

An incompressible flow preserves the volume of each moving fluid element. Its local condition is zero velocity divergence, $\nabla \cdot \mathbf{u}=0$. Combined with mass conservation, it says that each parcel's mass density stays unchanged as it moves. This does not require every parcel to have the same mass density: an initially nonuniform distribution can be transported without compression. Nor does incompressibility require a constant velocity or constant pressure.

For positive mass density, the continuity equation makes the equivalence precise:The material derivative follows a fluid element. Thus $D\rho /Dt=0$ if and only if the velocity field is a divergence-free vector field. The distinction between following a parcel and observing a fixed position prevents confusing incompressibility with an unchanging density field. ^[ref]

For a rigid pipe with impermeable walls, no leaks and steady flow, mass conservation at constant mass density implies a constant volumetric flow rate:Here $\mathbf{n}$ points downstream, $A$ is the section's area and $\overline{U}$ is the mean normal velocity, not necessarily the centreline speed. No assumption of inviscid flow or a uniform velocity profile is needed. If unequal amounts entered and left a fixed segment, its mass would accumulate, contradicting steady flow.

For a worked example, take horizontal water flow with stipulated mass density $1000\mathrm{kg}{\mathrm{m}}^{-3}$, inlet radius $R_1=0.010\mathrm{m}$, outlet radius $R_2=0.005\mathrm{m}$ and inlet mean velocity ${\overline{U}}_1=0.20\mathrm{m}{\mathrm{s}}^{-1}$. The circular areas giveThe radius halves, the area falls to one quarter, and the mean velocity quadruples. The mass flow rate is $\rho Q=0.02\pi \mathrm{kg}{\mathrm{s}}^{-1}$. The same amount of water passes both sections each second; narrowing changes its speed rather than compressing it. The Reynolds number and Bernoulli's principle discussions use this same example to examine viscous effects and the additional assumptions needed to calculate pressure.

The momentum thickness weights the streamwise velocity deficit by the local velocity carrying it:Thus $\rho U^2\theta$ measures a momentum flux deficit per unit span relative to transporting the same local mass flow rate at $U$. It is not $\rho \int (U^2-u^2)dy$, which compares a different reference flux. A Blasius boundary layer has $\theta \simeq 0.664115\sqrt{\nu x/U}$; the ratio of displacement thickness to momentum thickness is about $2.591$.^[ref]

For the steady flow over an impermeable flat plate at constant $U$, integrating the boundary-layer equation and using mass conservation givesThis connects growth of the momentum thickness to local wall shear stress. Consequently the drag per unit span on one face up to $x=L$ is $D^{\prime }=\rho U^2\theta (L)$ in the ideal Blasius boundary layer with $\theta (0)=0$. Measured velocity profiles can therefore estimate integrated drag without differentiating noisy near-wall readings. The identity assumes no wall suction and zero outer pressure gradient.^[ref]

The bottom-wall shear stress vanishes exactly when $u^{\prime }(0)=0$. Differentiating the velocity profile givesWith this value the velocity profile becomes $u(y)=-G{y}^2/(2\mu )$, so the volume flux per unit transverse width isFor $G>0$, the plate must move against the pressure-driven direction, and the resulting signed volume flux is negative. Zero bottom-wall shear does not mean zero flow.

Choose $x$ downslope along the rigid plane and $z$ normal to it into the fluid, with the wall at $z=0$ and the flat free surface at $z=h$. The gravitational body force per unit volume has components $(\rho g\sin \alpha ,-\rho g\cos \alpha )$. The wind's tangential traction on the top surface is $-S$ in the $x$ direction; the atmospheric normal traction is $-p_{\mathrm{atm}}$ in the $z$ direction. At the wall there is no-slip boundary condition, and the wall supplies the opposing normal force and a tangential traction whose sign depends on the wind strength.

The velocity is parallel to the plane, $(u(z),0)$. Its velocity profile is a concave quadratic, with $u(0)=0$ and negative slope at the free surface when $S>0$. The diagram uses moderate wind, for which the flow is still everywhere downslope. Stronger wind can reverse the upper portion, or the entire moving layer; the later cases quantify this.

The volume flux per unit transverse width isThusAssume a genuinely inclined plane, $0<\alpha <\pi /2$. With $Z=z/h$ and $U=\rho g{h}^2\sin \alpha /\mu$, the zero-flux velocity profile is $u/U=Z/3-Z^2/2$. It is positive for $0<Z<2/3$, zero at $Z=2/3$, and negative above that level. Its maximum is $U/18$ at $Z=1/3$, and its surface value is $-U/6$. Thus equal integrated downslope and upslope transports cancel, rather than the fluid being motionless.