  We study the deformation of drops squeezed between the floor and ceiling of a
microchannel and subjected to a hyperbolic flow. We observe that the maximum
deformation of drops depends on both the drop size and the rate of strain of
the external flow and can be described with power laws with exponents 2.59 +/-
0.28 and 0.91 +/- 0.05 respectively. We develop a theoretical model to describe
the deformation of squeezed drops based on the Darcy approximation for shallow
geometries and the use of complex potentials. The model describes the
steady-state deformation of the drops as a function of a non-dimensional
parameter Ca d^2, where Ca is the capillary number (proportional to the strain
rate and the drop size) and d is a confinement parameter equal to the drop size
divided by the channel height. For small deformations, the theoretical model
predicts a linear relationship between the deformation of drops and this
parameter, in good agreement with the experimental observations.
