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The incompressible Navier-Stokes equations are $u_t+(u\cdot \nabla )u=-\nabla p/\rho +\nu \Delta u$ and $\nabla \cdot u=0$, with no penetration and no slip at a stationary rigid wall. The Euler limit takes $\nu \to 0$ at fixed distance from the wall, leaving the inviscid Euler equations and a no-penetration condition; imposing tangential no slip on that outer problem generally overdetermines it. The Prandtl limit resolves a shrinking wall layer in which transverse viscous diffusion remains comparable to convection and restores no slip.

For the steady flat-plate boundary layer, $\delta /x\ll 1$, continuity makes $v/U=O(\delta /x)$, and $u_{xx}$ is small compared with $u_{yy}$. Normal momentum makes pressure independent of $y$ to leading order. The constant outer speed makes the outer pressure gradient, and hence $p_x$ throughout the layer, zero. The leading equations are $u{u}_x+v{u}_y=\nu u_{yy}$ and $u_x+v_y=0$. With the stream function convention $u={\psi }_y$, $v=-{\psi }_x$, these giveBalancing $U^2/x$ against $\nu U/{\delta }^2$ gives $\delta =\sqrt{\nu x/U}$. There is no independent streamwise scale for the semi-infinite plate, motivating the similarity solutionIt gives $u=U{f}^{\prime }$ and $v=\frac{1}{2}\sqrt{\nu U/x}(\eta f^{\prime }-f)$. Substitution yields $u{u}_x+v{u}_y=-U^{2}f{f}^{\prime \prime }/(2x)$ and $\nu u_{yy}=U^2{f}^{\prime \prime \prime }/x$, so the Blasius equation and conditions areThe first two conditions impose no penetration and no slip; the last matches the outer stream.

The wall shear on one face is $\tau (x)=\rho \nu u_y(x,0)=\rho U\sqrt{\nu U/x}{f}^{\prime \prime }(0)$. Integration from zero to $L$ gives $2\rho U^{2}L{f}^{\prime \prime }(0)/\sqrt{UL/\nu }$ on one face. With the same flow on both faces, the total drag force per unit width isOur normalization has $f^{\prime }(\infty )=1$, so the displayed denominator is one. The factor four refers to both faces; for a plate wetted on only one face the drag is half this value.