A fluid near a solid wall has an approximated velocity profile given by $u\left( y \right) = \;{U_\infty } \times \sin \left( {\frac{{\pi y}}{{2\delt 0 votes 40 views in General closed A fluid near a solid wall has an approximated velocity profile given by \(u\left( y \right) = \;{U_\infty } \times \sin \left( {\frac{{\pi y}}{{2\delta }}} \right),\;0 \le y \le \delta$ . The walls shear stress is given by:

1. ${\tau _{wall}} = \;\frac{{\pi \mu {U_\infty }}}{{2\delta }}$
2. ${\tau _{wall}} = \;\frac{{3\pi \mu {U_\infty }}}{{\delta }}$
3. ${\tau _{wall}} = \;\frac{{2\pi \mu {U_\infty }}}{{\delta }}$
4. ${\tau _{wall}} = \;\frac{{\pi \mu {U_\infty }}}{{\delta }}$

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Correct Answer - Option 1 : ${\tau _{wall}} = \;\frac{{\pi \mu {U_\infty }}}{{2\delta }}$

Concept:

According to Newton’s law of viscosity, shear stress is given by:

$\tau = \mu \times \frac{{du}}{{dy}} = \;\mu \times \frac{{d\theta }}{{dy}}\;$

τ = shear stress, μ = coefficient of viscosity or absolute viscosity (or dynamic viscosity)

$\frac{{du}}{{dy}} = Velocity\;gradient$

$\frac{{d\theta }}{{dt}}\; = Rate\;of\;angular\;deformation\;or\;Rate\;of\;shear\;strain$

Calculation:

Given:

Velocity profile

$u\left( y \right) = \;{U_\infty } \times \sin \left( {\frac{{\pi y}}{{2\delta }}} \right),\;0 \le y \le \delta$

$\tau = \mu \times \frac{{du}}{{dy}}$

$\tau = \mu \times \frac{{du}}{{dy}} = \mu \times \;{U_\infty } \times\frac{\partial }{{\partial y}}\left( {\sin \left( {\frac{{\pi y}}{{2\delta }}} \right)} \right)$

$= \mu \times {U_\infty } \times \cos \left( {\frac{{\pi y}}{{2\delta }}} \right) \times \frac{\pi }{{2\delta }}$

As we have to find the shear stress at the wall,

y = 0

${\tau _{wall}} = \mu \times {U_\infty } \times \frac{\pi }{{2\delta }} = \frac{{\pi \mu {U_\infty }}}{{2\delta }}$