# External force in the Navier-Stokes momentum equation

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?

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.

• You need to add it as a boundary condition, not as part of the NS equation. – Chet Miller Sep 5 at 18:38
• @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? – Alessandro Zunino Sep 5 at 21:44
• 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. – Chet Miller Sep 5 at 23:30
• @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. – Alessandro Zunino Sep 6 at 8:25
• 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. – Time4Tea Sep 7 at 19:57