Accelerating and Non-accelerating Coordinates - Fluid flow

[Soumalya]

Referring to the problem in the attachment, the author mentions that if we consider the coordinate system attached to the bicycle and the bicycle accelerates or decelerates, the flow past the bicycle becomes unsteady.

For an unsteady flow, we know that nothing changes at a given location on a streamline with time.Thus when the bicycle was moving with a constant velocity, the air was moving relatively past the rider at a constant velocity (steady flow) .Thus all fluid particles were having the same velocity at all locations along the streamline independent of time.There was no convective or local streamwise acceleration.

For the case when the bicycle accelerates or decelerates, the flow of air past the rider accelerates or decelerates relatively at the same rate. Thus, a fluid particle gains velocity (acceleration) as it approaches the rider along the streamline. If we consider another fluid particle behind the fluid particle it has a similar acceleration and say if the first fluid particle moving from one point to another along the streamline gains a certain velocity, the fluid particle behind it would gain a similar velocity in moving between the the same points. Thus, velocity at a given point along the streamline would be invariant of time. What I mean to indicate is presence of convective acceleration along the streamline and absence of local acceleration along the same.

So how is the flow an unsteady one if the bicycle accelerates or decelerates?

 

[Soumalya]

If we consider another fluid particle behind the fluid particle it has a similar acceleration and say if the first fluid particle moving from one point to another along the streamline gains a certain velocity, the fluid particle behind it would gain a similar velocity in moving between the the same points.

I just realized my contemplation was wrong and the velocities achieved by successive fluid particles while passing through a given point along the streamline will increase with time in case the bicycle accelerates.

 

[vanhees71]

The point is that the book assumes the bicyclist moving with constant velocity wrt. an inertial reference frame. Of course, then his reference frame is also an inertial one, and all the laws of fluid dynamics are valid from his point of view too.

It's a good question, whether one has ever worked out fluid-dynamical equations in non-inertial frames of reference. That sounds like an interesting (but also challenging) question!