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

Deformation of an Elastic Beam due to Viscous Flow in an Embedded Parallel Channel Network