Most equations don't know of the other ones unless you specify the coupling in some way, often as a r.h.s. term. You can always iterate any loose coupling and most often obtain convergence.
In this case the elasticity equation does not know about the additional pressure caused by the resistance of the fluid. This should be seen as a additional boundary force equivalent to pressure. That coupling should work well when the effect of fluid is small but requires more advanced methods when not.