Air is flowing at a velocity of $5 \mathrm{~m} / \mathrm{s}$ over a plane surface. Derive an expression for the thickness of the laminar sub-layer and calculate its value at a distance of 1 m from the leading edge of the surface.
Assume that within the boundary layer outside the laminar sub-layer, the velocity of flow is proportional to the one-seventh power of the distance from the surface and that the shear stress $R$ at the surface is given by:
$$
\left(R / \rho u_s^2\right)=0.03\left(u_s \rho x / \mu\right)^{-0.2}
$$
where $\rho$ is the density of the fluid ( $1.3 \mathrm{~kg} / \mathrm{m}^3$ for air), $\mu$ is the viscosity of the fluid $\left(17 \times 10^{-6} \mathrm{~N} \mathrm{~s} / \mathrm{m}^2\right.$ for air), $u_s$ is the stream velocity $(\mathrm{m} / \mathrm{s})$, and $x$ is the distance from the leading edge $(\mathrm{m})$.