Question

Air at ${20}^{\circ }C$ forms a boundary layer near a solid wall in which velocity distribution is given by:
$u=u_0\sin \left(\frac{\pi y}{2\delta }\right)$
The boundary layer thickness is 7 mm and the peak velocity is 9 m/sec. What will be the shear stress in the boundary layer at y = 3.5 mm ? $({\mu }_{air,{20}^{\ast }}=1.81\times {10}^{-5})$

• A. $2.59\times {10}^{-3}Pa$
• B. $25.9\times {10}^{-3}Pa$
• C. $3.66\times {10}^{-3}Pa$
• D. $36.6\times {10}^{-3}Pa$

Answer 1

The correct option is B $25.9\times {10}^{-3}Pa$

$u=u_0\sin \left(\frac{\pi y}{2\delta }\right)$

$\frac{du}{dy}=\frac{\pi u_0}{2\delta }\cos \left(\frac{\pi y}{2\delta }\right)$

at y = 3.5 mm

$\frac{du}{dy}=\frac{\pi \times 9}{2\times 7\times {10}^{-3}}\cos \left(\frac{\pi \times 3.5}{2\times 7}\right)$

$\text{shear stress at y}=3.5,{\tau }_{3.5}=\mu {\left(\frac{du}{dy}\right)}_{y=3.5mm}$

$=1.81\times {10}^{-5}\times \frac{\pi \times 9}{2\times 7\times {10}^{-3}}\cos \frac{\pi }{4}$

$=0.0259Pa$

$=25.9\times {10}^{-3}Pa$