Exercise Laminar Flow:

Model the flow of a glass flowing out a reservoir.

The aim of this problem is to model an industrial process in order to generate a glass flowing out a two dimensional reservoir. For this purpose, the glass, assumed to be a Newtonian incompressible liquid of viscosity $$ and density $,$ fall by its own weight from a tank of section A and height $H(t).$ This glass passes between two walls separated by a distance $h,$ with a length $L,$ to finally exit to the atmosphere. It is requested:

Determine an equation relating the flow rate per unit length $q,$ exiting the two$-$dimensional conduit to the height of liquid in the reservoir.

Determine an equation in the reservoir that relates the height of the liquid and the flow rate entering the conduit.

Using the above equations, determine a differential equation for the height $H(t)$ and solve it$.$