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For a fluid with viscosity to flow through a pipe that has the same cross-sectional area at both ends, at a constant velocity, there has to be a pressure difference according to Poiseuille's Law. Why exactly is there a change in pressure required to keep the velocity constant?

Is it because according to Bernoulli's principle that Pressure or Pressure-Energy gets converted to Kinetic Energy to speed up the fluid so the mass flow rate at both ends of the pipe stays the same?

So if that's the case in light of Bernoulli's principle, that means the change in pressure in Poiseullie's Law is there so that pressure energy gets converted to Kinetic Energy to fight off the viscosity of the fluid to keep the velocity of the fluid constant?

And

What exactly is Pressure? I know it is the Force divided by the area. I understand that concept, but I've seen the terms Pressure and Pressure Energy used interchangeably when talking about fluids, which creates for some amount of confusion. Aren't Pressure and Pressure Energy different? But when we talk about fluids in light of Bernoulli's principles, it seems as if Pressure and Pressure Energy are the same, which is pretty confusing.

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  • $\begingroup$ This has nothing to do with Bernoulli. In viscous flow, there is a shear force at the wall of the pipe acting on the fluid in the opposite direction of the flow. To balance this force, you need a higher pressure on the cross section upstream than on the cross section downstream. $\endgroup$ – Chet Miller Jan 16 at 23:23
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To answer your first question, I like to think about it in terms of momentum instead of energy—the pressure drop across a channel supplies momentum to the fluid that is then dissipated by the viscous effects. Hence there are no net forces in the flow, and the flow is consequently steady assuming the channel's cross-section is constant. Remember that Poiseuille's law is derived precisely by assuming that the viscous forces exactly compensate for the pressure gradient: $$-\nabla p + \mu \nabla^2 \vec{v} = 0$$ Your second question is much more subtle. In a fluid, I think of pressure as being a molecular energy density, which causes normal forces per unit area on real/imaginary surfaces that have a magnitude equal to this energy density. (These aren't the only forces per unit area a fluid can exert on a surface, but you get my point.) In that sense, this molecular energy density only represents one source of energy for the fluid—other sources include the macroscopic kinetic energy and the gravitational potential energy, both of which Bernoulli's law accounts for.

But this isn't the only way to think about pressure, as there are different definitions of pressure across fields of physics—even within fluid mechanics—that cause any single answer to be suspect. For example, one might consider a thermodynamic pressure that is determined entirely through equations of state akin to what you might learn in a statistical mechanics course. There is also a closely related, but separate, notion of mechanical pressure that is defined as the average of the normal stresses acting on a fluid.

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