Assuming quasi-steady state,

The assumptions we make are
1. The meniscus is part of a sphere, which will happen if we neglect gravity.
2. Shape of meniscus remains same despite movement.
3. Contact angle is independent of velocity.

And the capillary being narrow, the effects due to gravity can be neglected. This leads to Huygen-Poiseuille flow for which the velocity profile is parabolic. The entrance effects have also been neglected. Huygen-Poiseuille flow is valid for low Reynolds number flow in a circular pipe at steady state with a fully developed velocity profile.