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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.

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]

For constant-mass density incompressible flow, the displacement thickness measures the lost volume flux per unit span divided by the outer velocity:An outer inviscid flow encountering the same volume flux deficit behaves, at leading order, as though the wall were displaced outward by ${\delta }^{\ast }$. In a Blasius boundary layer, numerical integration gives ${\delta }^{\ast }\simeq 1.720788\sqrt{\nu x/U}$. This equivalent displacement is smaller than the boundary layer thickness and is not the position of a physical surface.^[ref]

A stationary viscous wall imposes the no-slip boundary condition, $u=v=0$, whereas the outer inviscid flow has $u=Ax$. At large Reynolds number, the transition is confined to a thin viscous boundary layer, where normal velocity gradients amplify the otherwise small viscous term.

Take a horizontal scale $L$, an outer speed $U=AL$, and a boundary layer thickness $\delta \ll L$. The continuity equation gives $v=O(U\delta /L)$. Write $ϵ=\delta /L$ and ${\mathrm{Re}}_L=UL/\nu$. The tangential Navier-Stokes equation has scalesBalancing convective acceleration with normal viscous diffusion gives $ϵ=O({\mathrm{Re}}_L^{-1/2})$. The leading tangential equation and incompressible flow constraint areHere and below $\nu$ is the kinematic viscosity, while $u$ is the horizontal velocity.

For the normal equation, first remove the hydrostatic pressure by writing $P=p+\rho gy$. Normal inertia and normal viscous terms are $O(\rho U^2\delta /L^2)$, so the change of $P$ across the layer is only $O(\rho U^2{ϵ}^2)$. Matching to the outer pressure therefore makes the leading $O(\rho U^2)$ excess pressure independent of $y$:The outer expression similarly has $(p_{\infty }{)}_y=-\rho g-\rho A^{2}y$. Thus the printed conditionis a leading-order boundary layer matching statement, with the smaller normal dynamic terms discarded on both sides. It is not an exact identity at the next order. In particular, retaining the outer $-\rho A^{2}y$ while dropping normal inertia and normal viscous terms inconsistently mixes orders. The tangential driving term is unambiguous: $-(p_{\infty }{)}_x/\rho =A^{2}x$.

The Hiemenz flow profile changes on $\eta =O(1)$, so its boundary layer thickness isA specified edge convention, such as $u/(Ax)=0.99$, introduces an order-one numerical multiplier requiring the profile to be solved. Scaling alone does not determine that multiplier. The result has no separate dependence on $x$, $y$, $\rho$ or $p_0$ at fixed kinematic viscosity. At fixed dynamic viscosity, it is $\sqrt{\eta /(\rho A)}$.

The viscous diffusion time across a thickness $\delta$ is ${\delta }^2/\nu$; the imposed strain has time scale $A^{-1}$. Their balance sets the thickness. Increasing kinematic viscosity spreads the layer, while increasing the imposed velocity gradient presses the layer toward the wall. The outward tangential velocity increases as $Ax$, so the decreasing travel time per unit distance offsets the usual downstream broadening.

For a conventional estimate of transition to turbulence, form the local horizontal Reynolds number:If transition is assumed to occur at an empirical value ${\mathrm{Re}}_{\mathrm{crit}}$, thenA rough borrowed threshold of order $10^5$–$10^6$ would give $l$ of order $10^2$–$10^3\sqrt{\nu /A}$. This is a heuristic estimate, not a universal stability prediction: the accelerating outer flow and favorable pressure gradient can delay transition, and disturbance amplitudes and wall roughness matter. Simply setting ${\mathrm{Re}}_x\sim 1$ locates failure of the thin-layer approximation near the stagnation point, not onset of turbulence.

For a fluid occupying $z<0$, match an outer stagnation point flow $u=x,w=-z$ to a flat impermeable interface with dimensionless shear stress $\beta x$. With boundary layer thickness $\delta ={\mathrm{Re}}^{-1/2}$, inner variable $\zeta =z/\delta$ and $\gamma =\delta \beta$, the leading boundary-layer equation admits $u=x{f}^{\prime }(\zeta )$ and $w=-\delta f(\zeta )$. Incompressibility is automatic, and tangential momentum conservation gives the displayed ordinary differential equation, with $f(0)=0$, $f^{\prime \prime }(0)=\gamma$ and $f^{\prime }\to 1$ at $\zeta \to -\infty$. The sign of this far-field endpoint is fixed by the fluid domain. This is a prescribed-traction interface; it does not impose a no-slip boundary condition.

For a stationary flat plate in a Newtonian fluid, the streamwise shear stress exerted by the fluid on its upper face isThe sign here is positive when the fluid pulls the plate downstream; the plate exerts the opposite tangential traction on the fluid. In a Blasius boundary layer, ${\tau }_w\simeq 0.332057\rho U^2/\sqrt{{\mathrm{Re}}_x}$. A growing boundary layer thickness therefore accompanies decreasing local wall shear stress, because its near-wall velocity gradient becomes gentler. The formal divergence as $x\to 0$ is integrable, but that leading-edge limit lies outside the thin-layer hypothesis.^[ref]