Question 8.5: A continuous belt (Fig. 8.13) passing upward through a chemi...
A continuous belt (Fig. 8.13) passing upward through a chemical bath at velocity U_{0} , picks up a liquid film of thickness h, density \rho , and viscosity \mu . Gravity tends to make the liquid drain down, but the movement of the belt keep the fluid from running off completely. Assume that the flow is fully developed and that the atmosphere produces no shear at the outer surface of the film. State clearly the boundary conditions to be satisfied by velocity at y = 0 and y = h. Obtain an expression for the velocity profile.
The governing equation is
\mu \frac{\mathrm{d}^{2} u}{\mathrm{~d} y^{2}}=\rho gOr \mu \frac{\mathrm{d} u}{\mathrm{~d} y}=\rho g y+C_{1}
Or \frac{\mathrm{d} u}{\mathrm{~d} y} =\frac{\rho g y}{\mu}+\frac{C_{1}}{\mu}
u =\frac{\rho g y^{2}}{2 \mu}+\frac{C_{1}}{\mu} y+C_{2}
At y=0, u=U_{0}, so C_{2}=U_{0}
At y=h, \tau=0 so \frac{\mathrm{d} u}{\mathrm{~d} y}=0 and C_{1}=-\rho g h
u=\frac{\rho g y^{2}}{2 \mu}-\frac{\rho g h y}{\mu}+U_{0}=\frac{\rho g}{\mu}\left(\frac{y^{2}}{2}-h y\right)+U_{0}