A steady fully developed film of water 2mm thick flows down

A steady, fully developed film of water 2mm thick flows down a vertical plate of glass as shown in the diagram. Neglect friction with the air around it. What is the surface velocity of the water film?

Solution

Solution:

For given conditions navier stokes equation will reduced to

stress is proportional to velocity gradient

stress = viscosity x velocity gradient

= 0.89 mpa s x (v-0)/0.002

stress will be equal to force / area

force acting will be due to gravity that is density x volume x g

= density x area x height x g

stress become = density x area x height x g/area

= density x height x g

plugging back to equation

density x height x g = 0.89 x v/0.002

v = density x height x g x 0.002 /0.89

= 22.022 h m/s

(here h is not given)