Consider the fluid system shown in Fig. $1.$ The fluid system consists two tanks with capacitance $c_1$ and $c_2$ and a valve with resistance $R.$ In addition, the fluid system is driven by a pump with prescribed flow rate $Q_s(t).$

$($a$)$ Draw the linear graph.

$($b$)$ Write down the elemental equation of each element.

$($c$)$ Write down the continuity equation at the bottom of the first tank $c_1.$

$($d$)$ A fluid particle migrates from tank $c_1$ to tank $c_2.$ Write down the compatibility equation along the path.

Figure $1$: A fluid system

Figure $2$: An electrical system