Chapter 1, Problem 1.4CP

Oil of viscosity $\mu$ and density $\rho$ drains steadily down the side of a tall, wide vertical plate, as shown in Fig, C1.4 In the region shown, fully developed conditions exist', that is, the velocity profile shape and the film thickness $\delta$ are independent of distance z along the plate. The vertical velocity w becomes a function only of x, and the shear resistance from the atmosphere is negligible.

(a) Sketch the approximate shape of the velocity profile w(x), considering the boundary conditions at the wall and at the film surface.

(b) Suppose film thickness $\delta$ , and the slope of the velocity profile at the wall, (dw/dx)_wall, are measured by a laser Doppler anemometer (to be discussed in Chap. 6). Find an expression for the viscosity of the oil as a function of $\rho$ , $\delta$(dw/dx)_wall, and the gravitational acceleration g. Note that, for the coordinate system given, both w and (dw/dx)_wallare negative.