  The boundary conditions prescribing the constant traction or the so-called
do-nothing conditions are frequently taken on artificial boundaries in the
numerical simulations of steady flow of incompressible fluids, despite the fact
that they do not facilitate a well-posed problem, not allowing to establish the
standard energy estimates. In a pursuit to understand better the possible
consequences of using these conditions, we present a particular set of examples
of flow problems, where we find none or two analytical or numerical solutions.
Namely, we consider problems where the flow connects two such artificial
boundaries. In the simple case of the isotropic radial flows where both steady
and unsteady analytical solutions are derived easily, we demonstrate that while
for some (large) boundary data all unsteady solutions blow up in finite time,
for other data (including small or trivial) the unsteady flows either converge
asymptotically to one of two steady solutions, or blow up in finite time,
depending on the initial state. We then document the very same behavior of the
numerical solutions for planar flow in a diverging channel. Finally, we provide
an illustrative example of two steady numerical solutions to the flow in a
three-dimensional bifurcating tube, where the inflow velocity is prescribed at
the inlet, while the two outlets are treated by the do-nothing boundary
condition.
