How can I justify the formula for orifice flow rate? I am writing my first math paper, and I am using a formula for the mass flow rate of a liquid through an orifice. I found this formula mentioned in some online videos (like this one) but I could not find the source of the formula. The formula is
$$m' = \frac{(\Delta p)^{\frac{1}{\alpha}}}{R}$$
where $m'$ is the mass flow rate, $\Delta p$ is the pressure difference between the two sides of the orifice, $\alpha$ is 1 for laminar flow and 2 for turbulent flow, and $R$ is the resistance of the orifice.
How is this formula justified? I'm looking for a derivation with accompanying citations: either a paper in which the formula is derived (it need not be a main result), or citations for the assumptions of the derivation.
Some searching hints at the formula being derived from Bernoulli's Equation, or from the Poiseuille Equation (maybe Hagen-Poiseuille?), but I cannot find a clear match. Also, all the equations seem to deal specifically with laminar or turbulent flow.
I also found this question which seems to have some connection to my question (it mentions that the degree-one term dominates laminar flow, while the degree-two term dominates turbulent flow) but I cannot get a concrete reference from it.
 A: The laminar case with $\alpha=1$ (that is, $\frac{dm}{dt} \propto \Delta p$) does indeed follow directly from Pouseille's equation.
The turbulent case, $\frac{dm}{dt} \propto \sqrt{\Delta p}$, seems to be an upper limit on flow rate for incompressible fluids, derived from the Bernoulli equation $p + \frac12 \rho v^2 = \text{constant}$.
Note that the constant of proportionality (your $R$) for a given geometry will generally not be the same for laminar and for turbulent flow; in the two cases the constants $R_\text{laminar}, R_\text{turbulent}$ must even have different units.
The boundary between laminar flow and turbulent flow is messy; a good starting place for analysis is to pretend that flow is either laminar or turbulent and prepare to be unsurprised by subtleties.
A: Looking at the flow in a narrow pipe (laminar flow in pipe) we have the following relations
$$ \begin{array}{ll}
\text{Mass flow rate} & m' = \rho A v \\
\text{Pressure Resistance} & \frac{\Delta p}{\rho g} = f \frac{\ell}{d} \frac{v^2}{2 g} \\
\text{Friction Factor} & f = \frac{64}{\rm Re} \\
\text{Reynold's number} & {\rm Re} = \frac{\rho d v}{\mu}
\end{array}$$
Combined together yield
$$ m' = \frac{\rho A d^2}{32 \mu \ell} \Delta p $$
which matches the given equation for laminar flow if $R = \frac{32 \mu \ell}{\rho A d^2}$.
For turbulent flow the friction factor is more complex, and so they must have done some curve fitting to get to the $\sqrt{\Delta p}$ factor (probably a velocity related term since $\Delta p \propto \frac{\ell}{d} v^2$).
