The viscous laminar flow of air over a flat plate results in the formation of a boundary layer. The boundary layer thickness at the end of the plate of length L is δ_L. When the plate length is increased to twice its original length, the percentage change in laminar boundary layer thickness at the end of the plate (with respect to δ_L) is ________ (correct to two decimal places).

The correct answer is 41.42%.

Step-by-step Derivation:

For a viscous laminar boundary layer flow of fluid (such as air) over a flat plate, the boundary layer thickness $\delta$ at a distance $x$ from the leading edge is given by the Blasius boundary layer solution:
$\delta =\frac{5x}{\sqrt{{\mathrm{Re}}_x}}$
where ${\mathrm{Re}}_x=\frac{\rho V_{\infty }x}{\mu }$ is the local Reynolds number, $\rho$ is the fluid density, $V_{\infty }$ is the free-stream velocity, and $\mu$ is the dynamic viscosity.

Substituting ${\mathrm{Re}}_x$ into the thickness formula:
$\delta =\frac{5x}{\sqrt{\frac{\rho V_{\infty }x}{\mu }}}=5\sqrt{\frac{\mu x}{\rho V_{\infty }}}$

This shows that the laminar boundary layer thickness is proportional to the square root of the distance from the leading edge:
$\delta \propto \sqrt{x}$

Let ${\delta }_L$ be the boundary layer thickness at the end of the plate of length $L$:
${\delta }_L\propto \sqrt{L}$

When the plate length is increased to twice its original length ($L_2=2L$), the new boundary layer thickness ${\delta }_{2L}$ at the end of this plate is:
${\delta }_{2L}\propto \sqrt{2L}$

The ratio of the new thickness to the original thickness is:
$\frac{{\delta }_{2L}}{{\delta }_L}=\frac{\sqrt{2L}}{\sqrt{L}}=\sqrt{2}\approx 1.4142$

The percentage change in the laminar boundary layer thickness with respect to ${\delta }_L$ is:
$\text{Percentage Change}=\frac{{\delta }_{2L}-{\delta }_L}{{\delta }_L}\times 100$
$\text{Percentage Change}=\left(\sqrt{2}-1\right)\times 100$
$\text{Percentage Change}=\left(1.4142-1\right)\times 100=41.42\%$