$\text {If the straight-line approximation $u = U _ { \infty } y / \delta$ to the velocity profile is used, the boundary-}$$\text {layer thickness for a zero-pressure gradient flow on a flat plate would be approximated to be:}$
A) $15 \mathrm {~cm}$
B) $20 \mathrm {~cm}$
C) $25 \mathrm {~cm}$
D) $30 \mathrm {~cm}$

Question

Quizplus · Accepted Answer

$\delta = 3 \sqrt { v x / U _ { \infty } }$
$\text {Substitute into the von Karman integral equation:}$
\[\begin{array} { c } 
\tau _ { 0 } = \frac { d } { d x } \int _ { 0 } ^ { \delta } \rho U _ { \infty } \frac { y } { \delta } \left( U _ { \infty } - U _ { \infty } \frac { y } { \delta } \right) d y \
= \rho U _ { \infty } ^ { 2 } \frac { d } { d x } \int _ { 0 } ^ { \delta } y \left( 1 - \frac { y } { \delta } \right) d y = \rho U _ { \infty } ^ { 2 } \frac { d } { d x } \left( \frac { \delta ^ { 2 } } { 2 } - \frac { \delta ^ { 2 } } { 3 } \right) = \rho U _ { \infty } ^ { 2 } \frac { \delta } { 3 } \frac { d \delta } { d x } \
\text { But, } \quad \tau _ { 0 } = \left. \mu \frac { \partial u } { \partial y } \right| _ { y = 0 } = \mu \frac { U _ { \infty } } { \delta } . \quad \text { So, } \mu \frac { U _ { \infty } } { \delta } = \rho U _ { \infty } ^ { 2 } \frac { \delta } { 3 } \frac { d \delta } { d x } \
\therefore \delta ^ { 2 } d \delta = \frac { 3 v } { U _ { \infty } } d x . \text { Integrate, using } \delta = 0 \text { at } x = 0 \text { and obtain } \delta = 3 \sqrt { v x / U _ { \infty } } .
\end{array}\]