A cubic approximation is commonly used in conjunction with the von Krmn momentum integral. An alternative form is the sine function: [v_{x}=alpha sin (b y)]

A cubic approximation is commonly used in conjunction with the von Kármán momentum integral. An alternative form is the sine function:

\[v_{x}=\alpha \sin (b y)\]

What should the constants \(\alpha\) and \(b\) be in order to satisfy some conditions on \(v_{x}\) for a flatplate geometry? Repeat the von Kármán analysis using the above expression, and derive a differential equation for \(d \delta / d x\). Integrate this equation to obtain the following expression for \(\delta\) :

\[\delta=\sqrt{\frac{v x}{v_{\infty}} \frac{2 \pi^{2}}{4-\pi}}\]

The mathematics is horrendous except for my students in Lopata.

Also find an expression for the local drag coefficient.