Prandtl's Boundary-Layer Equations

Derivation

Prandtl's Boundary-Layer Equations

D-329 Home PU-307 Threads force · fields Depends on Navier-Stokes from the Newtonian Constitutive Law, The Reynolds Number from Nondimensionalisation

Statement

For steady, two-dimensional, incompressible flow of a Newtonian fluid past a solid wall at high Reynolds number, the Navier–Stokes equations reduce inside a thin near-wall layer of thickness \(\delta \sim L\,Re^{-1/2}\) to Prandtl's boundary-layer equations: continuity \(\partial_x u + \partial_y v = 0\), streamwise momentum \(u\,\partial_x u + v\,\partial_y u = -\tfrac{1}{\rho}\,dp/dx + \nu\,\partial_y^2 u\), and the wall-normal momentum balance \(\partial_y p = 0\), so that the pressure is impressed on the layer by the external inviscid flow through \(-\tfrac{1}{\rho}\,dp/dx = U\,dU/dx\).

Why it matters

The full Navier–Stokes equations are second order in every spatial direction, so viscosity acts everywhere at once and the equations are elliptic. Prandtl's 1904 insight was that at high Reynolds number viscous effects are confined to a thin sheet next to the wall; matching that sheet to an outer inviscid flow turns an intractable elliptic problem into a parabolic one that marches downstream. This single idea resolved d'Alembert's paradox, gave the first quantitative theory of skin-friction drag, and founded modern aerodynamics.

The reduction is also the archetype of singular perturbation theory: the small parameter \(1/Re\) multiplies the highest derivative, a boundary layer forms, and inner and outer expansions are matched. The same scaling logic reappears in heat and mass transfer, magnetohydrodynamic Hartmann layers, and lubrication theory.

Assumptions

High Reynolds number, \(Re = U_\infty L/\nu \gg 1\).If \(Re\) is order unity the layer is not thin, \(\delta/L\) is not small, and the neglected terms are the same size as those retained — the reduction is invalid and the full elliptic equations must be solved.

Steady, incompressible, constant-property flow.Unsteadiness restores \(\partial_t u\); compressibility couples density and temperature through an energy equation and adds \(\nabla\!\cdot\!\mathbf{u}\neq 0\), changing continuity and the momentum balance.

Two-dimensional flow over a slender body with small wall curvature (\(\delta \ll R\)).Strong transverse curvature adds centrifugal terms \(\sim u^2/R\) to the normal-momentum equation, so \(\partial_y p = 0\) no longer holds and pressure varies across the layer.

The layer stays attached: the streamwise pressure gradient is not strongly adverse.A sufficiently adverse \(dp/dx>0\) reverses the near-wall flow, the boundary-layer assumption \(v \ll u\) breaks at separation, and the parabolic marching problem becomes ill-posed (Goldstein singularity).

Scale separation: a single outer length \(L\) and velocity \(U_\infty\) set the outer flow.Without a clean separation between the outer scale and the layer thickness, the two-term matched expansion has no small parameter and the leading-order truncation carries uncontrolled error.

Derivation

\[ u\,\partial_x u + v\,\partial_y u = -\frac{1}{\rho}\partial_x p + \nu\left(\partial_x^2 u + \partial_y^2 u\right), \qquad u\,\partial_x v + v\,\partial_y v = -\frac{1}{\rho}\partial_y p + \nu\left(\partial_x^2 v + \partial_y^2 v\right), \qquad \partial_x u + \partial_y v = 0 \]

Start from the steady, incompressible, two-dimensional Newtonian Navier–Stokes and continuity equations (assumed prior result: navier-stokes-newtonian-constitutive), body force absorbed into the pressure. A

\[ x = L\,\tilde{x}, \quad u = U_\infty\,\tilde{u}, \quad y = \delta\,\tilde{y}, \quad v = V\,\tilde{v}, \quad p = \rho U_\infty^2\,\tilde{p} \]

Introduce anisotropic scales: the streamwise scales are the outer ones \(L,\,U_\infty\); the wall-normal length is the unknown layer thickness \(\delta\ll L\) and \(V\) is an unknown transverse velocity scale. Pressure is scaled by the inertial (dynamic) pressure, anticipating an inviscid outer balance. A

\[ \frac{U_\infty}{L}\,\partial_{\tilde x}\tilde u + \frac{V}{\delta}\,\partial_{\tilde y}\tilde v = 0 \quad\Longrightarrow\quad V = U_\infty\,\frac{\delta}{L} \]

Substitute into continuity. For both terms to balance (neither can be dropped, or the flow would not conserve mass) the coefficients must match, fixing the transverse velocity scale. Because \(\delta\ll L\), this shows \(v\ll u\). A

\[ \frac{U_\infty^2}{L}\Big(\tilde u\,\partial_{\tilde x}\tilde u + \tilde v\,\partial_{\tilde y}\tilde u\Big) = -\frac{U_\infty^2}{L}\,\partial_{\tilde x}\tilde p + \nu\left(\frac{U_\infty}{L^2}\,\partial_{\tilde x}^2\tilde u + \frac{U_\infty}{\delta^2}\,\partial_{\tilde y}^2\tilde u\right) \]

Insert the scalings into the \(x\)-momentum equation, using \(V/\delta = U_\infty/L\) so both convective terms share the coefficient \(U_\infty^2/L\). Purely algebraic substitution. A

\[ \tilde u\,\partial_{\tilde x}\tilde u + \tilde v\,\partial_{\tilde y}\tilde u = -\partial_{\tilde x}\tilde p + \frac{\nu}{U_\infty L}\,\partial_{\tilde x}^2\tilde u + \frac{\nu}{U_\infty L}\frac{L^2}{\delta^2}\,\partial_{\tilde y}^2\tilde u = -\partial_{\tilde x}\tilde p + \frac{1}{Re}\,\partial_{\tilde x}^2\tilde u + \frac{1}{Re}\frac{L^2}{\delta^2}\,\partial_{\tilde y}^2\tilde u \]

Divide through by \(U_\infty^2/L\) and identify the Reynolds number \(Re = U_\infty L/\nu\) (assumed prior result: reynolds-number-from-nondimensionalization). All terms are now dimensionless with order-unity tilded fields. A

\[ \frac{1}{Re}\frac{L^2}{\delta^2} = O(1) \quad\Longrightarrow\quad \frac{\delta}{L} = Re^{-1/2}, \qquad \frac{1}{Re}\,\partial_{\tilde x}^2\tilde u \to 0 \;\; (Re\to\infty) \]

Distinguished limit: viscosity must survive to enforce the no-slip condition, so the wall-normal diffusion term cannot vanish — its coefficient is held order unity, which fixes \(\delta/L = Re^{-1/2}\). With \(\delta/L\) so chosen, the streamwise diffusion coefficient \(1/Re\) is smaller by a factor \((\delta/L)^2\) and is dropped at leading order. C

\[ \frac{V^2}{\delta}\Big(\tilde u\,\partial_{\tilde x}\tilde v + \tilde v\,\partial_{\tilde y}\tilde v\Big)\;\Big/\;\frac{U_\infty^2}{\delta}\;\sim\;\frac{V^2}{U_\infty^2} = \left(\frac{\delta}{L}\right)^2 = \frac{1}{Re}\quad\Longrightarrow\quad \partial_{\tilde y}\tilde p = O\!\left(Re^{-1}\right)\to 0 \]

Repeat the scaling analysis on the \(y\)-momentum equation. Every convective and viscous term carries a factor \((\delta/L)^2 = 1/Re\) relative to the dominant pressure term \(\partial_{\tilde y}\tilde p\); at leading order the transverse momentum balance collapses to \(\partial_{\tilde y}\tilde p = 0\). C

\[ \partial_y p = 0 \;\Longrightarrow\; p = p(x) = p_e(x), \qquad U\frac{dU}{dx} = -\frac{1}{\rho}\frac{dp_e}{dx} \]

Because pressure does not vary across the layer, its value equals that impressed by the external inviscid stream \(U(x)\) at the wall. Evaluating the outer Euler/Bernoulli balance (inviscid \(x\)-momentum with no wall-normal variation) at the edge gives the pressure-gradient closure. B

\[ \partial_x u + \partial_y v = 0, \qquad u\,\partial_x u + v\,\partial_y u = U\frac{dU}{dx} + \nu\,\partial_y^2 u \]

Restore dimensions in the surviving leading-order equations and substitute the pressure closure. This is the closed parabolic system, solved with \(u=v=0\) at \(y=0\) (no slip) and \(u\to U(x)\) as \(y\to\infty\) (matching to the outer flow). B

Result

\[ \boxed{\;\partial_x u + \partial_y v = 0,\qquad u\,\partial_x u + v\,\partial_y u = U\frac{dU}{dx} + \nu\,\partial_y^2 u,\qquad \partial_y p = 0,\qquad \frac{\delta}{L}\sim Re^{-1/2}\;} \]

Reading. Near a wall at high Reynolds number the flow splits into a thin viscous sheet and an inviscid outer stream. Inside the sheet, streamwise inertia is balanced by the impressed pressure gradient and by wall-normal viscous diffusion only; the cross-stream momentum equation degenerates to "pressure is constant across the layer," so the outer flow dictates \(p(x)\) while the layer supplies the no-slip correction. The layer thickness grows like \(Re^{-1/2}\), so it vanishes as \(Re\to\infty\) yet always exists.

Units check. Each term of the momentum equation has dimensions of acceleration, \(\mathrm{m\,s^{-2}}\): \(u\,\partial_x u \sim (\mathrm{m\,s^{-1}})(\mathrm{s^{-1}}) = \mathrm{m\,s^{-2}}\); \(U\,dU/dx \sim \mathrm{m\,s^{-2}}\); and \(\nu\,\partial_y^2 u \sim (\mathrm{m^2\,s^{-1}})(\mathrm{m^{-1}\,s^{-1}}\cdot\mathrm{m^{-1}}) = \mathrm{m\,s^{-2}}\). The thickness relation is dimensionless: \(\delta/L\) and \(Re^{-1/2}\) are both pure numbers.

Limiting cases

Zero pressure gradient (\(U=\text{const}\), flat plate): the closure term vanishes and the equations admit the Blasius similarity solution \(u/U = f'(\eta)\), \(\eta = y\sqrt{U/(\nu x)}\), with \(\delta \approx 5.0\sqrt{\nu x/U}\).
Favourable gradient (\(dU/dx>0\), accelerating flow): the term \(U\,dU/dx>0\) energises the near-wall fluid, thins the layer and stabilises it against separation.
Falkner–Skan wedge flows \(U = C x^m\): a one-parameter similarity family, \(m>0\) accelerating, \(m<0\) decelerating; separation appears at \(m\approx-0.0904\).
Very high \(Re\): \(\delta/L\to 0\), the outer flow approaches the inviscid solution everywhere except in a vanishing sheet — recovering potential flow away from the wall.
Stagnation-point flow (\(U = a x\), Hiemenz): \(\delta\) becomes independent of \(x\); a finite viscous layer persists even at the stagnation streamline.

Breaks when

Separation under strong adverse pressure gradient. When \(dp/dx>0\) is large the wall shear \(\tau_w = \mu\,\partial_y u|_0\) reaches zero and reverses; \(v\) is no longer small, the parabolic marching problem develops the Goldstein singularity, and the boundary-layer approximation collapses. Interacting (triple-deck) theory is required.
Low Reynolds number, \(Re\lesssim O(10)\). The layer is no longer thin, \(\delta/L\) is not small, streamwise diffusion \(\partial_x^2 u\) is comparable to \(\partial_y^2 u\), and the elliptic full Navier–Stokes equations must be retained.
Transition and turbulence. Above a critical \(Re_x\) the laminar layer becomes unstable (Tollmien–Schlichting waves) and turns turbulent; the instantaneous equations still hold but the laminar similarity structure and the simple \(\delta\sim Re^{-1/2}\) scaling do not — Reynolds-averaged closures are needed.
Strong wall or streamline curvature (\(\delta\sim R\)). Centrifugal terms \(\sim u^2/R\) enter the normal-momentum balance, so \(\partial_y p\neq0\) and the pressure is no longer purely impressed; Görtler vortices may appear on concave walls.
Compressible / high-speed flow. When Mach number is order unity, density and temperature vary across the layer, viscous dissipation heats the fluid, and the incompressible closure \(U\,dU/dx\) must be replaced by a coupled energy equation.

Failure modes

Dropping the wrong diffusion term. Students often keep \(\nu\,\partial_x^2 u\) and discard \(\nu\,\partial_y^2 u\); it is the reverse — cross-stream diffusion survives because \(\partial_y^2\sim1/\delta^2\) is huge, while \(\partial_x^2\sim1/L^2\) is negligible.
Setting \(v=0\). Because \(v\) is small (\(\sim U_\infty\,\delta/L\)) some set it to zero, but \(v\,\partial_y u\) is order-one and mandatory; only then does continuity close the system.
Treating \(p\) as unknown inside the layer. Forgetting \(\partial_y p=0\) leaves the system underdetermined. Pressure is given by the outer flow through \(U\,dU/dx\), not solved for within the layer.
Confusing \(\delta\sim Re^{-1/2}\) with \(\delta\sim Re^{-1}\). Misreading the distinguished-limit balance (\(1/Re\cdot L^2/\delta^2 = O(1)\)) gives the wrong power; the correct exponent is \(-1/2\).
Applying the equations through separation. Continuing to march downstream past the point of zero wall shear produces a spurious singularity and nonphysical results.
Using Bernoulli inside the viscous layer. Bernoulli's constant holds along outer inviscid streamlines only; inside the layer viscous dissipation destroys total pressure.

Discussion

The heart of the argument is a distinguished limit. Naively taking \(Re\to\infty\) in the Navier–Stokes equations deletes every viscous term and yields the inviscid Euler equations, which cannot satisfy the no-slip condition — a classic singular perturbation, since the small parameter \(1/Re\) multiplies the highest derivative. The resolution is to rescale the wall-normal coordinate so that one viscous term is retained at leading order. Requiring \(\nu\,\partial_y^2 u\) to balance inertia forces \(\delta/L = Re^{-1/2}\); the layer is the inner region of a matched asymptotic expansion, and the outer region is the inviscid flow. The two are stitched together by the matching condition \(u\to U(x)\).

Physically, the boundary layer is where the fluid pays its "no-slip debt." The outer flow slides frictionlessly, but the wall demands zero velocity; the layer accommodates the entire velocity drop over a distance \(\delta\), generating the wall shear \(\tau_w=\mu\,\partial_y u|_0\) that is felt as skin-friction drag. This resolves d'Alembert's paradox: inviscid theory predicts zero drag, but the thin viscous layer — however thin — produces finite drag and can separate, creating a wake and pressure drag that inviscid theory entirely misses.

The change of type is profound. Full Navier–Stokes is elliptic: information travels upstream and downstream, so the whole flow is coupled. Dropping \(\partial_x^2 u\) makes the boundary-layer system parabolic in \(x\); \(x\) plays the role of a time-like coordinate and the solution marches downstream from an initial profile. This is why boundary layers are computationally cheap and why separation — where downstream information wants to travel upstream — is exactly where the parabolic assumption fails and interacting boundary-layer (triple-deck) theory restores a limited ellipticity.

The same scaling machinery generalises far beyond hydrodynamics. Any dissipative term multiplying a high derivative with a large parameter produces a thin layer: thermal boundary layers scale with the Prandtl number as \(\delta_T/\delta\sim Pr^{-1/3}\); Hartmann layers in magnetohydrodynamics scale with the Hartmann number; Ekman layers in rotating flows scale with \(E^{1/2}\). Prandtl's construction is the canonical example of matched asymptotic expansions, and Van Dyke's matching principle formalises the leading-order truncation as the first term of an inner expansion whose validity is controlled by the ratio \(\delta/L = Re^{-1/2}\).

Common misconceptions. The boundary layer is not a surface of discontinuity or a "slip layer" — \(u\) varies smoothly and continuously from \(0\) at the wall to \(U\) at the edge. Its "edge" at \(y=\delta\) is a convention (often \(u=0.99U\)), not a sharp interface. And a thin layer does not mean a negligible one: however small \(\delta\), it sets the entire drag and controls whether the flow separates.

Worked examples

Blasius layer on a flat plate. Air at \(U_\infty = 20~\mathrm{m\,s^{-1}}\), \(\nu = 1.5\times10^{-5}~\mathrm{m^2\,s^{-1}}\), plate length \(L = 0.5~\mathrm{m}\). Find \(Re_L\), the layer thickness at the trailing edge, and check that the boundary-layer approximation is valid.

Set up the relevant quantities symbolically first. A

\[ Re_L = \frac{U_\infty L}{\nu} = \frac{(20)(0.5)}{1.5\times10^{-5}} = 6.7\times10^{5} \]

Compute the Reynolds number; \(Re_L\gg1\) so a thin laminar layer is expected (and it is below the flat-plate transition value \(\sim5\times10^{5}\!-\!3\times10^{6}\), so laminar is reasonable). A

\[ \delta(x) = 5.0\sqrt{\frac{\nu x}{U_\infty}} \quad\Longrightarrow\quad \delta(L) = 5.0\sqrt{\frac{(1.5\times10^{-5})(0.5)}{20}} = 5.0\times(1.94\times10^{-4})\approx 3.1\times10^{-3}~\mathrm{m} \]

Apply the Blasius thickness formula (the \(m=0\) similarity result), then insert numbers. B

\[ Re_L \approx 6.7\times10^{5},\qquad \delta(L)\approx 3.1~\mathrm{mm},\qquad \frac{\delta}{L}\approx 6\times10^{-3}\ll 1 \]

Reading. The viscous layer at the trailing edge is only about \(3~\mathrm{mm}\) thick over a half-metre plate; the ratio \(\delta/L\approx6\times10^{-3}\) confirms the thin-layer assumption is well satisfied, and indeed \(\delta/L\approx5\,Re_L^{-1/2}=5/819\approx6\times10^{-3}\), consistent with the scaling \(\delta\sim Re^{-1/2}\).

Units check. \(\sqrt{\nu x/U} = \sqrt{(\mathrm{m^2\,s^{-1}})(\mathrm{m})/(\mathrm{m\,s^{-1}})} = \sqrt{\mathrm{m^2}} = \mathrm{m}\). Correct.

Wall shear and skin-friction drag. Same flat plate and air as Example 1, width \(b = 0.3~\mathrm{m}\), density \(\rho = 1.2~\mathrm{kg\,m^{-3}}\). Estimate the local wall shear stress at the trailing edge and the total friction drag on one side.

Identify the Blasius wall-shear and drag coefficients symbolically. A

\[ \tau_w(x) = 0.332\,\rho\,U_\infty^2\,Re_x^{-1/2},\qquad Re_L = 6.7\times10^{5} \;\Rightarrow\; Re_L^{-1/2} = 1.22\times10^{-3} \]

Use the Blasius local skin-friction relation \(c_f = 0.664\,Re_x^{-1/2}\) with \(\tau_w = \tfrac12\rho U_\infty^2 c_f\); evaluate at \(x=L\). B

\[ \tau_w(L) = 0.332\,(1.2)(20)^2(1.22\times10^{-3}) = 0.332\times480\times1.22\times10^{-3} \approx 0.194~\mathrm{Pa} \]

Insert numbers into the wall-shear expression. A

\[ C_D = \frac{1.328}{\sqrt{Re_L}} = 1.328\times1.22\times10^{-3} = 1.62\times10^{-3},\qquad F_D = C_D\cdot\tfrac12\rho U_\infty^2\,(bL) \]

The total (length-averaged) drag coefficient is twice the trailing-edge \(c_f\); apply it to the plate area \(bL\). B

\[ F_D = (1.62\times10^{-3})\,\tfrac12(1.2)(20)^2\,(0.3\times0.5) = (1.62\times10^{-3})(240)(0.15) \approx 0.058~\mathrm{N} \]

Substitute the numbers; \(\tfrac12\rho U_\infty^2 = 240~\mathrm{Pa}\) and area \(=0.15~\mathrm{m^2}\). A

\[ \tau_w(L)\approx 0.19~\mathrm{Pa},\qquad C_D\approx 1.6\times10^{-3},\qquad F_D\approx 0.058~\mathrm{N} \]

Reading. Skin friction on a streamlined plate is tiny — a fraction of a newton — which is exactly why boundary-layer theory matters: this small viscous force, invisible to inviscid theory, is the entire friction drag. The wall shear falls as \(x^{-1/2}\), largest at the leading edge.

Units check. \(\tau_w = [\,\mathrm{kg\,m^{-3}}][\mathrm{m^2\,s^{-2}}] = \mathrm{kg\,m^{-1}\,s^{-2}} = \mathrm{Pa}\). \(F_D = [\mathrm{Pa}][\mathrm{m^2}] = \mathrm{N}\). Correct.

Problems

Water (\(\nu = 1.0\times10^{-6}~\mathrm{m^2\,s^{-1}}\)) flows at \(U_\infty = 2~\mathrm{m\,s^{-1}}\) over a flat plate. At what distance \(x\) from the leading edge does the laminar boundary layer reach \(\delta = 2~\mathrm{mm}\)? Use \(\delta = 5.0\sqrt{\nu x/U_\infty}\).
Solution
Rearranging, \(x = (\delta/5.0)^2\,U_\infty/\nu\). With \(\delta = 2\times10^{-3}~\mathrm{m}\): \((\delta/5.0)^2 = (4\times10^{-4})^2 = 1.6\times10^{-7}~\mathrm{m^2}\). Then \(x = 1.6\times10^{-7}\times(2/1.0\times10^{-6}) = 1.6\times10^{-7}\times2\times10^{6} = 0.32~\mathrm{m}\). Check \(Re_x = U_\infty x/\nu = (2)(0.32)/10^{-6} = 6.4\times10^{5}\), laminar, consistent. \(x\approx0.32~\mathrm{m}\).
Starting from the distinguished-limit condition \(\dfrac{1}{Re}\dfrac{L^2}{\delta^2} = O(1)\), derive the scaling of the transverse velocity \(V\) with \(Re\), and evaluate \(V/U_\infty\) for \(Re = 10^{6}\).
Solution
The condition gives \(\delta/L = Re^{-1/2}\). From continuity (Step 3), \(V = U_\infty\,\delta/L = U_\infty Re^{-1/2}\), so \(V/U_\infty = Re^{-1/2}\). For \(Re = 10^{6}\): \(V/U_\infty = (10^{6})^{-1/2} = 10^{-3}\). \(V/U_\infty = Re^{-1/2} = 10^{-3}\); the wall-normal velocity is a thousand times smaller than the streamwise velocity, confirming \(v\ll u\).
For a Falkner–Skan external flow \(U(x) = C x^{m}\), write the pressure-gradient term \(U\,dU/dx\) and state its sign for \(m>0\), \(m=0\), and \(m<0\). Which case is prone to separation and why?
Solution
\(U\,dU/dx = (Cx^m)(mC x^{m-1}) = mC^2 x^{2m-1}\). For \(m>0\) it is positive (favourable/accelerating), for \(m=0\) it is zero (flat plate, Blasius), for \(m<0\) it is negative (adverse/decelerating). Since \(-\tfrac1\rho\,dp/dx = U\,dU/dx\), \(m<0\) means \(dp/dx>0\): the adverse gradient decelerates near-wall fluid, can drive \(\tau_w\to0\), and is prone to separation (occurring near \(m\approx-0.0904\)). \(U\,dU/dx = mC^2x^{2m-1}\); \(m<0\) separates.
A flat plate in air (\(\rho = 1.2~\mathrm{kg\,m^{-3}}\), \(\nu = 1.5\times10^{-5}~\mathrm{m^2\,s^{-1}}\)) has \(U_\infty = 15~\mathrm{m\,s^{-1}}\) and length \(L = 1.0~\mathrm{m}\), width \(0.5~\mathrm{m}\). Estimate the total laminar friction drag on one side using \(C_D = 1.328\,Re_L^{-1/2}\). Comment on whether the laminar assumption is safe.
Solution
\(Re_L = (15)(1.0)/1.5\times10^{-5} = 1.0\times10^{6}\), so \(Re_L^{-1/2} = 10^{-3}\) and \(C_D = 1.328\times10^{-3}\). Dynamic pressure \(\tfrac12\rho U_\infty^2 = \tfrac12(1.2)(225) = 135~\mathrm{Pa}\); area \(= 1.0\times0.5 = 0.5~\mathrm{m^2}\). \(F_D = C_D\cdot\tfrac12\rho U_\infty^2\cdot A = (1.328\times10^{-3})(135)(0.5) \approx 0.090~\mathrm{N}\). Since \(Re_L = 10^{6}\) is near the flat-plate transition band (\(5\times10^{5}\)–\(3\times10^{6}\)), the rear of the plate may be transitional; the laminar estimate is a lower bound on drag. \(F_D\approx0.090~\mathrm{N}\).
Show, by comparing orders of magnitude, why the streamwise viscous term \(\nu\,\partial_x^2 u\) is dropped while \(\nu\,\partial_y^2 u\) is kept. Estimate their ratio for \(Re = 10^{4}\).
Solution
Using scales \(\partial_x^2 u \sim U_\infty/L^2\) and \(\partial_y^2 u \sim U_\infty/\delta^2\), the ratio is \(\dfrac{\nu\,\partial_x^2 u}{\nu\,\partial_y^2 u} \sim \dfrac{U_\infty/L^2}{U_\infty/\delta^2} = \left(\dfrac{\delta}{L}\right)^2 = Re^{-1}\). For \(Re = 10^{4}\), the ratio is \(10^{-4}\): the streamwise diffusion term is four orders of magnitude smaller than wall-normal diffusion and is negligible at leading order, while wall-normal diffusion is retained because it balances inertia (coefficient held \(O(1)\)). Ratio \(= Re^{-1} = 10^{-4}\).