  Elastic deformation due to embedded fluidic networks is currently studied in
the context of soft-actuators and soft-robotic applications. In this work, we
analyze interaction between the elastic deflection of a slender beam and
viscous flow within a long serpentine channel, embedded in the elastic beam.
The channel is positioned asymmetrically with regard to the midplane of the
beam, and thus pressure within the channel creates a local moment deforming the
beam. We focus on creeping flows and small deflections of the elastic beam and
obtain, in leading order, a fourth-order partial integro-differential equation
governing the time-dependent deflection field. This relation enables the design
of complex time-dependent deformation patterns of beams with embedded channel
networks, including inertia-like standing and moving wave solutions in
configurations with negligible inertia.
