  Due to computational complexity, fluid flow problems are mostly defined on a
bounded domain. Hence, capturing fluid outflow calls for imposing an
appropriate condition on the boundary where the said outflow is prescribed.
Usually, the Neumann-type boundary condition called do-nothing condition is the
go-to description for such outflow phenomenon However, such condition does not
ensure an energy estimate for the Navier--Stokes equations - let alone
establish the existence of solutions. In this paper, we analyze a convective
boundary condition that will capture outflow and establish the existence of
solutions to the governing equation. We shall show existence and uniqueness
results for systems with mixed boundary conditions - Dirichlet condition and
the convective boundary condition. The first system is a stationary equation
where the Dirichlet condition is purely homogeneous, the other is where an
input function is prescribed, and lastly a dynamic system with prescribed input
function. We end by showing numerical examples to illustrate the difference
between the current outflow condition and the usual do-nothing condition.
