Static Liquid Bridges in Axisymmetric Elastic Tubes
The figure below illustrates the formation of an occluding liquid
bridge from an initially axisymmetric liquid film of uniform thickness
H, inside an elastic cylindrical tube of wall thickness h and
radius R0.
The liquid film is unstable to axisymmetric disturbances (indicated by
the dashed red line): The initially uniformly
distributed fluid drains into the regions of larger film thickness.
Since the fluid pressure is lower in the region where the film
thickness grows, the tube wall is pulled radially
inwards. This moves the air-liquid
interface closer to the tube's centreline. An occluding liquid bridge forms
when opposite points on the air-liquid interfaces come into contact.
Following the formation of the liquid bridge, fluid continues to drain
from the liquid lining into the liquid bridge. In the final equilibrium
state, the liquid bridge is enclosed
by two air-liquid interfaces which have the shape of spherical caps
and meet the tube wall at a certain contact angle. The pressure jump
over the air-liquid interfaces compresses the tube wall in the region that is
wetted by the liquid bridge.
This problem was investigated in detail by Halpern & Grotberg
[`Fluid-Elastic Instabilities of Liquid-Lined Flexible tubes.' Journal
of Fluid Mechanics 244, 615-632 (1992)]. Their study predicted large axisymmetric deformations of the tube wall when the liquid
bridge is being formed. This raises the question of the system's
stability to non-axisymmetric perturbations since cylindrical tubes
tend to buckle when they are compressed too strongly.
