  Exploring fluid-structure interactions is essential for understanding the
physical principle underlying flow control in biological and man-made systems.
Traditionally, we assume that the geometry is known, and from it, the solution
to the coupled elastohydrodynamic problem is determined. Solving the inverse
problem -- finding the geometry that leads to a desired flow -- has received
comparatively less attention. Here, we present a strategy for solving inverse
hydroelastic problems. Specifically, we compute the shape of a soft channel
that yields a desired flow-rate versus pressure-drop relationship. The analysis
is based on low-Reynolds-number hydrodynamics and linear elasticity. We
demonstrate its usefulness in understanding intercellular transport in plants
and the design of check valves. The sensitivity of the algorithm to fabrication
errors and other limitations are discussed.
