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Viscous flows in contact with elastic structures apply both pressure and shear stress at the solid–liquid interface and thus create internal stress and deformation fields within the solid structure. We study the interaction between the deformation of elastic structures, subject to external forces, and an internal viscous liquid. We neglect inertia in the liquid and solid and focus on viscous flow through a thin-walled slender elastic cylindrical shell as a basic model of a soft robot. Our analysis yields an inhomogeneous linear diffusion equation governing the coupled viscous–elastic system. Solutions for the flow and deformation fields are obtained in closed analytical form. The functionality of the viscous–elastic diffusion process is explored within the context of soft-robotic applications, through analysis of selected solutions to the governing equation. Shell material compressibility is shown to have a unique effect in inducing different flow and deformation regimes. This research may prove valuable to applications such as micro-swimmers, micro-autonomous systems and soft robotics by allowing for the design and control of complex time-varying deformation fields.