# Applying Bernoulli's equation in fully developed viscous pipe flow

Can we apply the inviscid Bernoulli equation

$$\frac{P}{\rho} + \frac{1}{2}V^2 + gz = constant$$

along the center line of a fully developed pipe flow?

I think we can since the shear stress along the center line is zero ($$\frac{\partial u}{\partial y}=0$$ at the center ), thus resulting in zero shear stress on material elements at the center. There is therefore an inviscid streamline along the center line of the pipe flow.

• After writing my answer - thinking more about what you are asking - you probably want to say something more like - in the neighborhood of the center line. For a viscous fluid there is still a change in the velocity between streamlines at the center - but the gradient is minimum there. Commented Dec 19, 2018 at 21:19
• An ideal gas model is often used to describe the flow of compressible gas in channels of variable cross section, for example, in rocket nozzles. Commented Dec 19, 2018 at 22:56
• Why don't you look at the derivation of Bernoulli from the Euler equation, and see how it changes when you add the viscosity term? Commented Dec 19, 2018 at 23:03