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First, I had to say I am very confused about fluid mechanics. Here is one question about continuity question.

For the incompressible flow, and density is constant, we have the continuity equation ∇⋅ 𝑣=0. We also have another continuity equation dV/dt= inflow-outflow.

The first equation gives the volume of fluid elements is constant. But the second equation says the volume changes. I think both of them can be used in incompressible flow.

Why are they so different?

I also know there is two specification of the flow field.

One is Lagrangian and second is the Eulerian specification of the flow field

Does it related to this?

Also, in reality, when we fill the balloon water. (suppose ballon has no effect on the water). The balloon expands, it seems the second equation is right and volume is not constant.

Which one is correct?

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The two formulas don't contradict one another.

The first formula says that the total volume of the fluid doesn't change, or in other words that the density doesn't change. That doesn't mean that the volume of fluid in a specific region can't change.

In the example with the balloon, the total volume of water is initially in the plumbing system. When you fill the balloon, the water in the balloon increases, while the water in the plumbing system decreases by the same amount, keeping the total volume fixed.

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