  Due to their morphology, the dynamics of bacteria suspended in media can
exhibit complex behaviors. In the presence of a shear, swimming bacteria
experience a drift perpendicular to the shear plane. This drift, termed
rheotaxis, is studied here semi-analytically, numerically and experimentally.
We find the dependency of bacterial orientation and bacterial speed on the
shear rate, in the presence of rotational diffusion. This enables us to show
that the drift speed of bacteria perpendicular to the shear is predominantly
due to bacterial propulsion, and not rheotactic forces. Comparing the drift
speed of bacteria and diffusion leads to the definition of a P\'{e}clet number.
The rheotactic effect increases with the shear, reaching a plateau at very
large P\'{e}clet numbers, in good agreement with experiments of rheotaxis
performed in microfluidic droplets.
