Viscous flow fields in hyperelastic chambers

Viscous flows in hyperelastic chambers are relevant to many biological phenomena such as inhalation into the lung's acinar region, and medical applications such as the inflation of a small chamber in minimally invasive procedures. In this work, we analytically study the viscous flow and elastic deformation created due to inflation of such spherical chambers from one or two inlets. Our investigation considers the shell's constitutive hyperelastic law coupled with the flow dynamics inside the chamber. For the case of a narrow tube filling a larger chamber, the pressure within the chamber involves a large spatially uniform part, and a small-order correction. We derive a closed-form expression for the inflation dynamics, accounting for the effect of elastic bistability. Interestingly, the obtained pressure distribution shows that the maximal pressure on the chamber's surface is greater than the pressure at the entrance to the chamber. The calculated series solution of the velocity and pressure fields during inflation is verified by using a fully coupled finite element scheme, resulting in excellent agreement. Our results allow the estimation of the chamber's viscous resistance at different pressures, thus enabling us to model the process of inflation and deflation.