  The shape of a microchannel during flow through it is instrumental to
understanding the physics that govern various phenomena ranging from
rheological measurements of fluids to separation of particles and cells. Two
commonly used approaches for obtaining a desired channel shape (for a given
application) are (i) fabricating the microchannel in the requisite shape and
(ii) actuating the microchannel walls during flow to obtain the requisite
shape. However, these approaches are not always viable. We propose an
alternative, passive approach to {\it a priori} tune the elastohydrodynamics in
a microsystem, towards achieving a pre-determined (but not pre-fabricated) flow
geometry when the microchannel is subjected to flow. That is to say, we use the
interaction between a soft solid layer, the viscous flow beneath it and the
shaped rigid wall above it, to tune the fluid domain's shape. Specifically, we
study a parallel-wall microchannel whose top wall is a slender soft coating of
arbitrary thickness attached to a rigid platform. We derive a nonlinear
differential equation for the soft coating's fluid--solid interface, which we
use to infer how to achieve specific conduit shapes during flow. Using this
theory, we demonstrate the tuning of four categories of microchannel
geometries, which establishes, via a proof-of-concept, the viability of our
modeling framework. We also explore slip length patterning on the rigid bottom
wall of the microchannel, a common technique in microfluidics, as an addition
`handle' for microchannel shape control. However, we show that this effect is
much weaker in practice.
