How does flow develop in an initially empty pipe under the no-slip condition?
Let us consider a circular pipe, initially empty, and an external body of water moving at a constant velocity $v$. At a certain instant $t$, this body of water enters the pipe through its inlet cross-section, which we will denote as $S_{0}$.
According to the no-slip boundary condition, the velocity of the outermost annular layer of fluid, which is in contact with the pipe wall, must be zero. This outer layer, in turn, slows down the adjacent layer. However, it does not have sufficient time to transmit this deceleration to the innermost layers.
In other words, at cross-section $S_{0}$ and at time $t$, the fluid layer in contact with the wall has zero velocity, the adjacent layer has a slightly reduced velocity, while the remaining inner layers still move at the original velocity $v$.
This implies that, over a time interval $dt$, the inner layers travel a distance $v\,dt$, which is greater than the distance covered by the outer layers (zero for the layer immediately adjacent to the wall). It would then seem that, at time $t + dt$, a gap should appear near the wall at the next cross-section $S_{1}$.
What exactly happens at this point? Do the fluid particles from the inner region move radially outward to fill this gap, somewhat like the flow in a fountain? If so, they would have to come to rest upon reaching the wall. Meanwhile, the particles passing above them are slowed down, but this effect still has not propagated to the innermost layers within such a short time interval.
Applying the same reasoning to the subsequent cross-sections $S_{2}$, $S_{3}$, $\ldots$, $S_{n}$ would seemingly imply that a boundary layer never forms.
So where is the flaw in this reasoning? How is this apparent paradox resolved? What is the actual physical mechanism by which an initially empty pipe becomes filled with fluid?