I was wondering whether there is any equation for viscous liquids (probably derived from energy and mass conservation principles) relating the pressure, rate of volume flow, area of cross section and height of the liquid (or any other parameters if required).
NOTE : I am aware of Poiseuille's law, but that is not what I am looking for as it relates the pressure gradient to the rate of flow.
I think Euler-Bernoulli equation is nothing but the energy equation of a steady-state inviscid flow in Lagrangian form. It means the total non-thermal energy of an imaginary particle doesn't change alongside the stream:
$$P+\rho \frac{v^2}{2}=cte \tag{1}$$
Assuming that body force is negligible. So if you want to extend it to vicious flows you have two possibilities. Of course the obvious way is to use conservative continuum equations for mass, linear momentum and energy for the 3D flow (i.e. Navier-Stokes equations). But if your flow can be reduced to a 1D model, Like a pipe or an orifice, then we can write:
$$\frac{\partial}{\partial x}\left( \bar{P}+ \bar{\rho}\frac{\bar{v}^2}{2}\right)={loss}_x \tag{2a}$$
as the loss of energy at each section of the pipe. If assuming conduction and radiation are negligible (adiabatic) then:
$$\frac{\partial}{\partial x}\left( \bar{h}+ \frac{\bar{v}^2}{2}\right)=0 \tag{2b}$$
Now the Darcy-Weisbach formula can be used to have an estimation of the loss:
$${loss}_x \approx -f \frac{ \bar{\rho} \bar{v}^2}{2 D_H} \tag{3}$$
Where $D_H$ is the hydraulic diameter at that particular section and $f$ is the Darcy friction factor. There are actually a whole bunch of empirical equations for $f$ for different flow regimes. Hagen–Poiseuille equation is the most primitive one for laminar flow. I have explained more here. Anyway combining 2 and 3 :
$$\implies \frac{\partial}{\partial x}\left( \bar{P}+ \bar{\rho}\frac{\bar{v}^2}{2}\right) \approx -f \frac{ \bar{\rho} \bar{v}^2}{2 D_H} \tag{4}$$
Which I think should be a good estimation for the a 1D viscous flow.
