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Figure 1

The flow depicted in the figures is irrotational and the fluid is nonviscous, incompressible.

My questions -

  1. Is pressure uniform and constant for the whole cross-section(depicted by $A_1$ and $A_2$)?
  2. Is velocity of every particle constant for a cross-section perpendicular to the tube boundary or there exists a velocity gradient in the pipe perpendicular to the flow?

If the above written were true, can equation of continuity $A_1 v_1 = A_2 v_2$ be applied to finite cross-sectional areas? I don't think so.

Note - I initially asked this question but I feel that was very broad so this new question.

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Given your assumptions, this flow satisfies bernoulli. From the continuity equation, you can see that velocity will vary as the cross sectional area varies. Assuming the vertical distance change is very minor, it can be dropped from the bernoulli equation. From the continuity equation, v2 is less than v1, so that p2 is greater than p1. For a given cross section which is perpendicular to the flow, the magnitude of the velocity will be the same, since there are no friction forces acting on the fluid.

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The equation of continuity is applicable only to narrow tubes of infinitesimally small cross-sectional area.

Consider a tube of above specifications. If the fluid(non-compressible) cannot come out of the side walls of the tube, it can flow in and out only through the cross-sectional area. In this limiting case of infinitesimally small cross-sectional area, we can consider the fluid to have a constant velocity. So, if we want to consider a tube of finite size, we would have to write $$\int \vec{v}_\mathrm{in}\cdot\mathrm{d}\vec{A}_\mathrm{in} = \int \vec{v}_\mathrm{out}\cdot\mathrm{d}\vec{A}_\mathrm{out}$$

Here, $\vec{v}_\rm{in}$ is the velocity of the fluid element going inside the tube and crossing the small area $\mathrm{d}\vec{A}_\mathrm{in}$. Similar notation is used for outgoing flow. So, in general, the value of $\vec{v}_\mathrm{out}$ or $\vec{v}_\rm{in}$ can change from place to place and can be at any angle to the considered area. $\mathrm{d}\vec{A}_\mathrm{in}$ and $\mathrm{d}\vec{A}_\mathrm{out}$ can be any areas bounded by the walls of the tube.

This result can be considered similar to Gauss' Law. If there are no sources or sinks inside the tube, the net incoming fluid flux should be equal to the net outgoing fluid flux.

Similarly, pressure also need not be same for every point of a cross-section. This can be verified by the fact that in a tube of finite size in which the fluid is at rest, the pressure at the bottom is more than up.

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  1. Under steady flow, there is only axial, and NOT radial pressure gradient.
  2. Velocity DOES vary along a cross section, hence that is the MEAN velocity that is used in equation of continuity, or else you need to integrate the velocity distribution over the area of the cross section.
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