  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
bi-stability. 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 estimating
the chamber's viscous resistance at different pressures, thus enabling us to
model the process of inflation and deflation.
