# On the boundary condition of solid boundaries for an enclosed, non moving fluid

First and foremost, I might say horribly wrong things. Feel free to correct any inconsistencies in the following post.

Let's assume an incompressible, viscous, newtonian fluid that is not moving and enclosed by a solid surface.

The total stress tensor of an incompressible, viscous, newtonian fluid is defined by :

$\sigma^f(u,p) = -p.Id + \mu (\nabla u + \nabla u ^T)$

As the fluid is static, we have $\nabla u = 0$, hence the total stress tensor is reduced to its hydrostatic part $\sigma^f(u,p) = -p.Id$

In Fluid/Structure interaction settings, we define a fluid problem ($\mathcal{F}$), a solid problem ($\mathcal{S}$), and we define a system of coupling condition ($\mathcal{C}$) which are usually defined by a kynematic condition ($\mathcal{C}_1$) and a dynamic condition ($\mathcal{C}_2$), with $\mathcal{C}_1$ and $\mathcal{C}_2$ defined by :

• $(\mathcal{C}_1) : u_f = u_s$

• $(\mathcal{C}_2) : \sigma^f(u,p).n^f = \sigma^s(d).n^s$

Up until here, I do think that I understand what is happening.

In partitioned solvers, I read numerous times that the fluid solver is transfering forces to the solid solver, and that the solid solver is transfering displacement to the fluid solver. I am rather unsatisfied by this because of $\mathcal{C_2}$. Going back to the case of my incompressible enclosed fluid stated above (again, stop me if my interpretation is wrong), if one of the sides tries to compress the fluid (like a piston), the fluid will not move nor will the side of the solid. However, the intracavity pressure will rise, as will the stress inside the solid.

My question are the following :

• Was I wrongly understanding how and what data had to be transfered during iterations of a partitioned FSI solver ?

• Assuming a fluid problem where you would not like to compute the solid behavior. What type of boundary condition would be used to impose both the velocity of the fluid at the solid boundaries AND the stress ?

Thanks in advance for anything that could be useful to my understanding

• "... if one of the sides tries to compress the fluid... the fluid will not move nor will the side of the solid..." Can you explain what you are trying to say here? If a piston pushes on the fluid, the solid would definitely be moving because it's a piston. And the fluid would be moving out of the way of the new position of the solid. This would send compression waves through the fluid towards the side opposite the chamber. This all serves to increase the pressure in the chamber. – tpg2114 Jun 1 '16 at 15:29
• In the original setting, I assume the fluid to be incompressible. Under this assumption, the fluid cannot be compressed by the piston. I would assume that the piston force is transmitted through the interface and, following Pascal's law, the pressure increases everywhere, right ? – Al_th Jun 1 '16 at 15:46