A flat plate aligned with a uniform stream provides the canonical exact similarity solution of the laminar boundary-layer equations.

Assume

• steady two-dimensional incompressible flow;
• a stationary flat plate beginning at $x=0$;
• uniform outer velocity $U_\infty$;
• zero streamwise pressure gradient;
• a laminar boundary layer.

Then

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

and

$$u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y} =\nu\frac{\partial^2u}{\partial y^2}.$$

The boundary conditions are

$$u(x,0)=0,$$

$$v(x,0)=0,$$

and

$$u(x,y\to\infty)\to U_\infty.$$

Similarity variable

Boundary-layer scaling suggests that the transverse length scale grows like

$$\sqrt{\frac{\nu x}{U_\infty}}.$$

Introduce

$$\boxed{\eta=y\sqrt{\frac{U_\infty}{\nu x}}}.$$

Use a stream function

$$\boxed{\psi(x,y)=\sqrt{\nu U_\infty x},f(\eta)}$$

so that

$$u=\frac{\partial\psi}{\partial y},$$

$$v=-\frac{\partial\psi}{\partial x}.$$

Continuity is then satisfied automatically.

Differentiating gives

$$\boxed{u=U_\infty f'(\eta)},$$

and

$$\boxed{ v=\frac12\sqrt{\frac{\nu U_\infty}{x}} \left[\eta f'(\eta)-f(\eta)\right] }.$$

Blasius equation

Substituting the similarity form into the boundary-layer momentum equation eliminates $x$ and $y$ separately, leaving the nonlinear ordinary differential equation

$$\boxed{f'''+\frac12ff''=0}.$$

The wall and matching conditions become

$$\boxed{f(0)=0},$$

$$\boxed{f'(0)=0},$$

$$\boxed{f'(\infty)=1}.$$

The first two conditions enforce no penetration and no slip at the plate. The last makes the velocity match the uniform outer stream.

There is no elementary closed-form solution. Numerical integration gives

$$\boxed{f''(0)\approx0.3321}.$$

That one numerical constant determines the wall shear.

Velocity-profile similarity

Because

$$\frac{u}{U_\infty}=f'(\eta),$$

profiles at different streamwise positions collapse onto the same curve when plotted against $\eta$.

The $99%$ edge occurs at approximately

$$\eta_{99}\approx4.9,$$

so

$$\boxed{ \delta_{99}(x) \approx4.9\sqrt{\frac{\nu x}{U_\infty}} =\frac{4.9x}{\sqrt{Re_x}} },$$

where

$$\boxed{Re_x=\frac{U_\infty x}{\nu}}.$$

The exact similarity solution therefore confirms the scaling

$$\delta\propto\sqrt{x}.$$

Integral thicknesses

Numerical integration of the Blasius profile gives

$$\boxed{ \delta^*(x) \approx1.72\sqrt{\frac{\nu x}{U_\infty}} },$$

and

$$\boxed{ \theta(x) \approx0.664\sqrt{\frac{\nu x}{U_\infty}} }.$$

Thus the Blasius shape factor is

$$\boxed{H=\frac{\delta^*}{\theta}\approx2.59}.$$

Wall shear and local skin friction

At the wall,

$$\tau_w =\mu\left.\frac{\partial u}{\partial y}\right|_{y=0}.$$

Since

$$u=U_\infty f'(\eta),$$

we obtain

$$\tau_w =\mu U_\infty f''(0) \sqrt{\frac{U_\infty}{\nu x}}.$$

Using $\mu=\rho\nu$ and $f''(0)\approx0.3321$,

$$\boxed{ \tau_w \approx0.332, \frac{\rho U_\infty^2}{\sqrt{Re_x}} }.$$

Define the local skin-friction coefficient

$$c_f(x)=\frac{\tau_w}{\tfrac12\rho U_\infty^2}.$$

Then

$$\boxed{c_f(x)\approx\frac{0.664}{\sqrt{Re_x}}}.$$

The wall shear decreases downstream as $x^{-1/2}$ because the same velocity change is spread across a progressively thicker layer.

Drag of a laminar flat plate

For a plate of length $L$ and width $B$, with one wetted side, the skin-friction drag is

$$D=B\int_0^L\tau_w(x),dx.$$

Integrating the $x^{-1/2}$ shear distribution gives

$$\boxed{ D =\frac12\rho U_\infty^2(BL) \frac{1.328}{\sqrt{Re_L}} },$$

where

$$Re_L=\frac{U_\infty L}{\nu}.$$

Thus the average skin-friction coefficient over the plate is

$$\boxed{\bar C_f=\frac{1.328}{\sqrt{Re_L}}}.$$

This formula applies to the laminar Blasius solution over the stated length. If the boundary layer transitions to turbulence, the laminar relation no longer describes the downstream shear.

Worked example

Air has approximately

$$\rho=1.20,\mathrm{kg/m^3},$$

$$\nu=1.5\times10^{-5},\mathrm{m^2/s}.$$

Let

$$U_\infty=20,\mathrm{m/s},$$

and consider a plate length

$$L=1.0,\mathrm m.$$

Then

$$Re_L =\frac{(20)(1.0)}{1.5\times10^{-5}} \approx1.33\times10^6.$$

If the flow is assumed laminar over the full plate for this ideal calculation, the $99%$ thickness at the trailing edge is

$$\delta_{99}(L) =4.9\sqrt{\frac{(1.5\times10^{-5})(1.0)}{20}} \approx\boxed{4.24,\mathrm{mm}}.$$

The local skin-friction coefficient there is

$$c_f(L) =\frac{0.664}{\sqrt{1.33\times10^6}} \approx\boxed{5.75\times10^{-4}}.$$

The plate-average value is

$$\bar C_f =\frac{1.328}{\sqrt{1.33\times10^6}} \approx\boxed{1.15\times10^{-3}}.$$

For one metre of plate width,

$$D =\frac12(1.20)(20^2)(1.0)(1.0)(1.15\times10^{-3})$$

$$\approx\boxed{0.276,\mathrm N}.$$

The Blasius solution is a foundational benchmark because it connects a reduced boundary-layer PDE to a self-similar velocity profile, exact laminar wall-shear scaling, and a measurable skin-friction drag.