  We develop and implement a Bayesian approach for the estimation of the shape
of a two dimensional annular domain enclosing a Stokes flow from sparse and
noisy observations of the enclosed fluid. Our setup includes the case of direct
observations of the flow field as well as the measurement of concentrations of
a solute passively advected by and diffusing within the flow. Adopting a
statistical approach provides estimates of uncertainty in the shape due both to
the non-invertibility of the forward map and to error in the measurements. When
the shape represents a design problem of attempting to match desired target
outcomes, this "uncertainty" can be interpreted as identifying remaining
degrees of freedom available to the designer. We demonstrate the viability of
our framework on three concrete test problems. These problems illustrate the
promise of our framework for applications while providing a collection of test
cases for recently developed Markov Chain Monte Carlo (MCMC) algorithms
designed to resolve infinite dimensional statistical quantities.
