# Homework Help: Boundary layer question

1. Apr 19, 2006

### painfive

Here is the problem: (From sabersky, problem 8.9)

Vapor condenses on a vertical surface to form a liquid film. The film moves under gravity and forms a laminar liquid boundary layer. Derive an expression for the mass flow rate dm/dt as a function of the local film thickness $\delta$. Neglect any velocity components in the y-direction. (the positive x direction is down and y points away from the solid surface)

Answer: dm/dt=$\rho g \delta^3/3 \nu$, where $\rho$ is the density and $\nu$ is the kinematic viscosity.

I'm really stuck on this one. The continuity equation is useless because there must be vapor condensing on to the film (or else dm/dt would be constant). Assuming this vapor has no momentum, I was able to use the momentum equation to get:

$$\frac{\partial}{\partial x} \int_0^\delta \rho u^2 dy =-\tau_0+\rho g$$

where $\tau_0$ is the shear force at the solid surface. Since g and $\nu$ appear not as a sum but as a product in the final answer, I assume there must be another equation relating them. Can anyone help me out here?