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The Navier-Stokes system for incompressible fluid says $$ \begin{aligned} \dot{\mathbf{u}} + (\mathbf{u}\cdot\nabla)\mathbf{u}-\nu\Delta\mathbf{u}+\nabla{p} &= 0,\\ \nabla\cdot\mathbf{u}&=0. \end{aligned} $$ The non-slip boundary condition for the velocity $\mathbf{u}$ says $\mathbf{u}|_{\partial\Omega}=0.$ But the pressure boundary condition is unclear. Most references I found use Neumann boundary condition which is $\nabla{p}\cdot\mathbf{n}=0$ on $\partial\Omega$ ($\mathbf{n}$ is normal vector of $\partial\Omega$). As consequence, $p$ can be determined up to an arbitrary additive constant! Does this make any sense in physics? How can I determine the constant?

There is also an initial value condition for $p$. We can add an arbitrary function $q(t)$ satisfying $q(0)=0$ to $p(\mathbf{x},t).$ That is, if $p(\mathbf{x},t)$ is a solution, so is $p(\mathbf{x},t)+q(t).$

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The motion of the fluid parcel only depends on the gradient of the pressure, not the pressure itself. Think about it this way: we know that pressure increases the deeper you go into water. But if I place a cylinder in the water, the flow will always look like (the colors represent pressure with red being high pressure and blue low pressure; the arrows are velocity vectors)

enter image description here

no matter how deep it is in the water because it is the change in the pressure that determines the flow. So yes, $p$ being defined to an additive constant makes complete sense.

I don't know that you can solve for the constant because there are a multitude of gauges, $\phi$, that can satisfy the $\nabla\phi=0$ requirement in the NS equations.

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As long as we are far from compressible effects (low Mach) and the 0 pressure under which cavitation may appear, yes, pressure is defined up to a constant if boundary conditions are given in terms of velocity. If you provide boundary condition in terms of normal stress (e.g. at an inlet), then you provide a pressure value with this boundary condition. But you can add any constant to this boundary pressure and you will obtain the same flow.

Note that the B.C. in $\nabla p\cdot n$ you mention is only an artifact of some numerical techniques where the pressure equation is not solved directly (Chorin method).

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