A question about the flow delay in response to the pressure change at boundary

zhigang asked me to respond quickly

i think the question is a little too vague.
the first thing that happens in a fluid is that a sound wave propagates and if the pressure change
is maintained then the fluid is set into bulk motion (similar to pressure driven flow in a tube).
but to answer in any more detail requires more information about the geometry, the other boundary conditions, etc.

for example, at early times you can neglect viscosity and the change in pressure accelerates fluid (how much depends on compressibility of the fluid).

some aspects of problems like this may be standard graduate course questions and many details can be worked out
using fourier series ideas. but the basic problem is still a little vague in my view.

the starting point is the navier-stokes and continuity equations and you can assume the flow is only in one direction (unidirectional).