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The Navier-Stokes momentum equation is

$$ \rho\frac{\partial \bf{v}}{\partial t}+\rho(\bf{v} \cdot \nabla\bf{v})=-\nabla P + \nabla\cdot \bf{\tau} +\bf f $$

where $\tau$ is the deviatoric stress tensor, $P$ is the pressure, and $\bf f$ is the volumetric density of the body forces acting on the fluid.

I want to add a term to this equation to account of the presence of a piston oscillating harmonically in a fluid-filled cavity. What should be the form of this additional term? enter image description here

I suppose $\tau$ to be the standard one for a newtonian fluid and $\bf f=\rho \bf g$ to account for gravity. So every term of the above equation is known.

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    $\begingroup$ You need to add it as a boundary condition, not as part of the NS equation. $\endgroup$ Commented Sep 5, 2019 at 18:38
  • $\begingroup$ @ChetMiller why should I add it as a boundary condition and not as a force term in the RHT of the NS momentum equation? Is not the momentum equation taking care of all the forces acting on the fluid? And which is the boundary condition I should write? $\endgroup$ Commented Sep 5, 2019 at 21:44
  • $\begingroup$ The NS equations involve forces per unit volume, and the boundary condition involves velocity. So how is it supposed to be included as a force per unit volume.. Plus, the boundary conditions is only present at one location. So what are you going to do, include the boundary condition in the NS equations as a Dirac delta function? It's just not going to work mathematically. $\endgroup$ Commented Sep 5, 2019 at 23:30
  • $\begingroup$ @ChetMiller yes, I understand that mathematically is tricky. This is why I asked this question. My idea was to use a reflecting boundary as a boundary condition (i.e. $\frac{d\rho}{d x}(x=L)=0$) and then add an additional surface force term to describe the additional pressure generated by the piston. It is still not clear to me why this approach is wrong. $\endgroup$ Commented Sep 6, 2019 at 8:25
  • $\begingroup$ The $f$ term in the NS equation represents body forces, as you noted in the question. This means forces that act everywhere on all parts of the fluid, based on the fluid density (such as gravity). A piston is a boundary condition, because it only acts on the boundary of the fluid region. $\endgroup$
    – Time4Tea
    Commented Sep 7, 2019 at 19:57

2 Answers 2

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Although the question is old, the answer may still be useful to someone else.

In my opinion, the pressure imposed by the piston should be included in the equation, not as boundary conditions, rather as a time variable pressure gradient. The function describing the variation of the pressure gradient should be derived according to the armonic oscillation of the piston. So in the case, for example, you know the armonic law of either the movement of the piston or the force applied to the piston, this should be converted in the pressure imposed to the fluid and then included in the equation in the term related to the pressure gradient.

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Navier-Stokes equation is a local equation - each quantity entering it is a function of position and time, e.g., $\rho\rightarrow\rho(\mathbf{x},t)$, and the equation relates quantities and their derivatives at each point $\mathbf{x}$. This is also for the force $\mathbf{f}\rightarrow\mathbf{f}(\mathbf{x},t)$, which is a body force, applied to every point in the liquid (although not necessarily the same for every point.)

On the other hand, a piston would perturbs only a layer at the boundary, which favors its inclusion as a boundary condition. What is problematic here (as pointed in the answer by @Carla) is that it does actually move through space, i.e., we cannot specify it as a condition at a fixed point, e.g., $x=x_0$.

One way of dealing with this is, as the other answer suggest, specifying a (varying in time) pressure gradient at this point - it is an approximate approach, but it is likely to work for many problems.

A more principled approach is to write the equation first in Lagrandian coordinates, where the points adjacent to the piston are either not moving or experience a collision against the boundary of the liquid (i.e., the piston), and then switch to the Eulerian coordinates (for which the equation in teh Q. is written.) This is apparrently a well-known approach, see, e.g.
The exterior non-stationary problem for the Navier-Stokes equations in regions with moving boundaries
MOVING BOUNDARY PROBLEMS (Admittedly, the first links that came up in Google.)

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