(Solved): Derive an analytical expression for the laminar boundary layer thickness () on a cylindrical wire ...

Derive an analytical expression for the laminar boundary layer thickness $(\delta )$ on a cylindrical wire with radius $a$, as function of the distance $\mathbf{z}$ that the flow travel along the wire. The flow is parallel to the wire, of which the velocity far from the wire is given as $u_{\infty }$. The density and viscosity of the fluid are $\rho$ and $\mu$, respectively. Assume that $\delta \gg a$ Hint 1: Start with deriving the mass and momentum balances for a fluid element in CCS. Hint 2: Use the following ansatz for differential momentum balances: $u=u_{\infty }\sin \left(\frac{\pi }{2}\zeta \right)$ where $\zeta =\frac{r}{\delta }$ and $r$ is the radial distance from the surface of the wire. Hint 3: Use the trigonometric integration tables (삼각함수 적분표) for the calculation.