D
$h_0$
h(t)
Stopper
Water flow
d
Figure 1: Water flow from the tank (with discharge coefficient).
The initial water level in the tank is $h_0$, and the water level drops continuously after the stopper is opened. Accounting for discharge losses via a coefficient $0 < C_d \leq 1$, the draining process is modeled by the separable differential equation:
$$\frac{dh}{\sqrt{h}} + C_d \sqrt{2g} \left(\frac{d^2}{D^2}\right) dt = 0,$$
where $h(t)$ is the water height at time t, D is the diameter of the tank, d is the diameter of the outflow orifice, g is the gravitational constant, and $C_d$ is the discharge coefficient.
(a)
Solve the differential equation (8) and express the general solution in terms of $h(t)$.
(b)
Determine the constant of integration from part (a) using the initial condition $h(0) = h_0$, and hence express $h(t)$ explicitly.
