$"$

Please solve step by step using paper and without using neumerical methods

$"$

Two identical vertical cylindrical tanks $D[m]$ in diameter and $L[m]$ high are placed side by side as shown in the illustration below. They are connected at the bottom by a tube $L_1[m]$ long and $d[m]$ in diameter. At time $t=0,$ tank B is full of oil and tank $A$ is empty. A short tube $L_2[m]$ long and $d[m]$ in diameter is also an outlet from tank B$.$ Both outlet tube and connecting tube are horizontal and at the bottom of the tanks. Both tubes are opened simultaneously at $t=0.$

a$)$ Formulate a mathematical model that describe the change in the levels of $tankA$ and $tankB$ with respect to time.

b$)$ Solve the models and determine the maximum level in tank A$.$

Assume viscous flow through the tubes so that:

$v_2=\frac{d^{2}g{h}_B}{32L_2}$

$v_1=\frac{d^{2}g(h_B-h_A)}{32L_1}$

Data:

$D=2[m]$ tank diameter

$L=3[m]$ Initial height of oil in tank B

$d=0\mathrm{.}01[m]$ tube diameter

$L_1=0\mathrm{.}50[m]$

$L_2=0\mathrm{.}25[m]$

Density $=999$

Viscosity $=1$x${10}^{-3}$

C$)$ find an equation for hA & hB

Note : without using nuemerical methods $($Euler $,$ RK $,$ etc $)$ and without using $2$nd order DE$.$