# Bernoulli Equation Reference Frames

I'm trying to understand an example on page 107 of "Fundamentals of Fluid Mechanics" seventh edition by Munson, Okiishi, Huebsch, and Rothmayer. The setup of the question is:

Consider a flow of air around a bicyclist moving through still air with velocity $V_0$

In the solution, there is a statement that says that in a coordinate system fixed to the ground, the flow is unsteady as the bicyclist rides by. The Bernoulli equation requires that the flow be steady if it is to be used properly. But in a coordinate system fixed to the bike, it appears as though the air is flowing steadying toward the bike with speed $V_0$. This is the proper reference frame to use.

Question 1: Why does the coordinate system fixed to the ground see an unsteady flow? Maybe I don't understand what steady flow is. From my knowledge, steady flow is where any snapshot of the flow does not change with time. This doesn't mean that the streamlines must be straight, only that each successive particle that passes a given point will follow the same path (i.e., if you are on one streamline, you stay on it and don't divert off of it). Also, does steady flow require that the velocity be constant on each streamline?

## 1 Answer

Steady would mean that flow at a point defined in that coordinate system does not change in time.

If you mark a spot on the ground and look at the air flow above it you'll see it change over time. It will start out being still, it will move as the cyclist passes and then become still again.

If you look at a spot say 1 $m$ ahead of the cyclist (moving with the cyclist). That spot is always 1 $m$ ahead of the cyclist an always has the same properties.

The only funny thing is that, under the moving reference frame, the still air (far ahead and far behind the cyclist) appears to be moving. In the same way that a tree branch is perfectly still but still knocks the cyclist off his bike when he pedals into it.