I thought that Bernoulli equation could be used only in the case of non viscous fluid. But doing exercises on *2500 Solved Problems In Fluid Mechanics and Hydraulics (Schaum's Solved Problems Series)* I found that this procedure is followed.

In the case of viscous laminar flow, Bernoulli equation is written as

$$z_1+\frac{v_1^2}{2 g}+\frac{p_1}{\rho g}=z_2+\frac{v_2^2}{2 g}+\frac{p_2}{\rho g}+h_L\tag{1}$$

Where $h_L$ is the head loss (due to viscosity) calculated using Hagen Poiseuille law.

$$h_L=\rho g \frac{8 \eta L \bar{v}}{R^2}\tag{2}$$

Is this a correct way to solve exercises involving viscosity?

Furthermore are there limitation to this use of Bernoulli equation (in case of viscosity)?

In particular

- If the flow is not laminar, I cannot use $(2)$, but can I still write $(1)$ in that way?
- Is $(1)$ valid only along the
**singular**streamline or between different ones (assuming the fluid irrotational)?

## Best Answer

This is a correct way to solve exercises involving viscosity (within certain constraints, e.g., constant viscosity). Eqn. 1 is a version of the Bernoulli equation, modified to include a frictional head loss, and is definitely valid, provided the velocities used are the average velocities. Eqn. 1 without the $h_L$ is valid along a streamline, even for a viscous flow. If Eqn. 1 is being used for a

laminarviscous flow (say in a tube of slowly varying cross section), the kinetic energy terms should not have a 2 in the denominator. See Bird, Stewart, and Lightfoot, Transport Phenomena for details. Here they show that, if the 2 is to be included, then instead of using the average value of v squared, one should use the average value of $v^3$ divided by the average value of v. For laminar flow in a tube, this reduces to twice the square of the average velocity, so, if the average velocity squared is being used in the kinetic energy term, the 2 should not be included in the denominator if the flow is laminar.