A cylindrical straw $($length L $,$ radius R $)$ is held vertically as shown in the figure, and is fully filled with an incompressible Newtonian fluid with density rho and viscosity mu$.$ The fluid is held in place by sealing the top with a thumb as demonstrated by the instructor in class earlier in the course.

$1.$ In practice a little cavity is created between the thumb and the top surface of the fluid. This overhead pressure p$\_($s$)$ is less than the atmospheric pressure p$\_("$atm $").$ The difference between the two pressures balances the gravitational forces on the fluid pointing downwards. Ignoring surface tension effects, calculate the overhead pressure p$\_($s$)$ for the full straw held static with a thumb blocking the top end. Use g as gravitational acceleration.

$2.$ At t$=0,$ the thumb is removed, and the top end is now exposed to atmospheric pressure, creating a net draining force. Assuming you can ignore transient and surface tension effects and apply the Hagen$-$Poiseuille equation to obtain the expression for the instantaneous volumetric flow rate Q at the bottom.

$3.$ At some time t after, the straw would be partially filled with a column of height h$.$ Assuming you can apply Hagen$-$Poiseuille equation at this precise moment, what is the instantaneous volumetric flow rate at the bottom as a function of height?

$4.$Interpret the equation derived from the previous part with respect to rho,mu$,$h$,$R$.$

$5.$ Convert the previous equation in terms of height h of the column and its derivative with time, with no reference to Q in the final equation.

$6.$ Integrate the previous equation from h$=$L to h$=0$ and t$=0$ to t$=$t$\_("$empty $")$ to obtain an expression for t$\_("$empty $").$

$7.$ Interpret the result for previous equation in terms of rho,mu$,$L$,$R$.$