Question 5.19: The velocity u in a laminar, incompressible, boundary layer ......
The velocity u in a laminar, incompressible, boundary layer over a flat plate is given by
\frac{u}{U_{\infty}}=\frac{3}{2} \frac{y}{\delta}-\frac{1}{2}\left(\frac{y}{\delta}\right)^3
where U_{\infty} is the free stream velocity and 5 is the boundary layer thickness and is given by δ = k√x . Obtain an expression for the velocity component v across the boundary layer.
For a two-dimensional, incompressible flow the continuity equation can be written in differential form as
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0
Given that: \frac{u}{U_{\infty}}=\frac{3}{2} \frac{y}{\delta}-\frac{1}{2}\left(\frac{y}{\delta}\right)^3
Hence, \frac{\partial u}{\partial x}=U_{\infty}\left[-\frac{3}{2} \frac{y}{\delta^2}+\frac{3}{2} \frac{y^3}{\delta^4}\right] \frac{1}{2} k \frac{1}{\sqrt{x}}
Substituting the value of \frac{\partial u}{\partial x} in the continuity equation, one can write
\frac{\partial v}{\partial y}+U_{\infty}\left[-\frac{3}{2} \frac{y}{\delta^2}+\frac{3}{2} \frac{y^3}{\delta^4}\right] \frac{1}{2} k \frac{1}{\sqrt{x}}=0
Integrating with respect to y, we get
v+k U_{\infty} \frac{1}{2 \sqrt{x}}\left[-\frac{3}{2} \frac{y^2}{2 \delta^2}+\frac{3}{2} \frac{y^4}{4 \delta^4}\right]=C
Applying the no-penetration boundary condition, i.e., v = 0, at y = 0 , we obtain
C = 0
Thus, the velocity component across the boundary layer becomes
v=k U_{\infty} \frac{3}{8 \sqrt{x}}\left[\frac{y^2}{\delta^2}-\frac{y^4}{2 \delta^4}\right]