Conditions of validity of basic fluid equations

Question

I easily get confused about the conditions of validity of basic fluid equations. I list the conditions for these equations to hold true, as I could understand them.

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1. $\rho v S=\mathrm{constant}$: steady flow
2. $ v S=\mathrm{constant}$: steady flow, $\rho=\mathrm{constant}$ (incompressible fluid)
3. $ \frac{1}{2}\rho v^2+\rho g h+ p=\mathrm{constant}$: steady flow, $\rho=\mathrm{constant}$ (incompressible fluid), no viscosity $(\eta=0)$
4. $Q=\frac{\pi R^4 \Delta p}{8 \eta L}$: steady flow, laminar motion

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Firstly I'm not sure if 2. really holds without the condition of $\eta=0$. That means that anytime 4. is valid (for a fluid with $\eta \neq 0$) then 2. is valid too (and so is 1.). Is this true?

Secondly, is there a condition on $\rho$ in 4.? Must $\rho$ be constant or not?

If I forgot any condition please suggest it, as I can understand better when I can use one or another equation.

Answer 1

1. Steady flow in a channel of constant cross sectional area. Flat velocity profile, unless v is the average axial velocity.

2. Same as 1, except incompressible.

3. Also valid along streamline of steady viscous flow

4. Steady incompressible laminar flow of viscous fluid in a straight circular cylindrical horizontal pipe. Constant temperature.

Answer 2

Equation 2 is the equation of continuity which does not depend on the viscosity of the fluid.

Equation 4 is the Poiseuille equation which assumes that the fluid is incompressible (density = constant) and the flow is laminar.