Hagen-Poiseuille equation

Question

The Hagen-Poiseuille equation states that

$$\Delta p = \frac{8\mu LQ}{\pi R^4}$$

where $\mu$ is the dynamic viscosity of the fluid. I'm just wondering, can the dynamic viscosity be replaced by viscosity to yield the mass flow rate instead of volumetric flow rate as it usually does?

Also, since the viscosity of a fluid is dependent on it temperature, and by extension, it's pressure, and since there is a pressure change ($\Delta p$), under which conditions should the viscosity value be provided? Thanks a lot.

Answer 1

The dynamic viscosity $\mu$ and the kinematic viscosity $\eta$ are related by:

$$\eta=\frac{\mu}{\rho}$$

where $\rho$ is the fluid's specific mass.

Since as volumetric flow rate $Q$ and mass flow rate $Q_M$ are related by:

$$Q=\frac{Q_M}{\rho}$$

we can now write:

$$\Delta p = \frac{8\mu LQ}{\pi R^4}=\frac{8\eta LQ_M}{\pi R^4}$$

This is of course what $\eta$ is intended for: treat fluid dynamic problems with mass flow rate, rather than volumetric flow rate.

Also, since the viscosity of a fluid is dependent on it temperature, and by extension, it's pressure, and since there is a pressure change (Δp), under which conditions should the viscosity value be provided?

Hagen-Poiseuille is intended for use with incompressible fluids (such as most common liquids):

$$\frac{\partial V}{\partial p} \approx 0$$

This means that the liquid's bulk properties, like viscosity, is more or less invariant to pressure $p$. Also, the range of pressures encountered in fluid dynamics tends to be modest.