hongiddong said:
If there is no velocity and the fluid is moving still, then there is no kinetic energy, would there be no pressure exerted on the walls?
Or perhaps there is a basal level of kinetic energy of the molecules as they are vibrating?
I think you are misinterpreting what pressure means. Pressure (or rather, static or thermodynamic pressure) is really a measure of the mean kinetic energy of the molecules that make up a gas or liquid. This certainly still exists when a fluid is at rest. The only way to reduce this to zero pressure is to either remove all of the molecules (i.e. create a vacuum) or slow them down to zero velocity (the fluid would change state to a solid long before this would theoretically occur).
The bulk motion of the fluid is represented by the dynamic pressure, which is a continuum quantity and represents the kinetic energy per volume of a flowing fluid. That will certainly go to zero if the fluid slows down to zero flow. If you combine static pressure and dynamic pressure (and sometimes gravity) you get the total pressure for a flow, which pops up a lot in fluid mechanics and is the quantity that is conserved in Bernoulli's equation.
hongiddong said:
Lastly, how can there still be flow if the pressure gradient is constant everywhere? (meaning constant velocity).
You are misusing the term "pressure gradient". If the pressure gradient is constant, that means that the pressure is changing in space (e.g. along a length of pipe) at a constant rate, not that the pressure itself is constant everywhere. With a constant
pressure gradient, there will be acceleration of the flow in the direction of decreasing pressure. If the
pressure itself is constant, then there will be no acceleration (no force), and in the case of no viscosity, it will either remain still or else continue flowing however it already was. If viscosity is involved, then it is similar to friction and no pressure gradient means no flow, and the pressure gradient must be equal or greater to the viscous force in order for flow to be sustained.