  The paper addresses an error analysis of an Eulerian finite element method
used for solving a linearized Navier--Stokes problem in a time-dependent
domain. In this study, the domain's evolution is assumed to be known and
independent of the solution to the problem at hand. The numerical method
employed in the study combines a standard Backward Differentiation Formula
(BDF)-type time-stepping procedure with a geometrically unfitted finite element
discretization technique. Additionally, Nitsche's method is utilized to enforce
the boundary conditions. The paper presents a convergence estimate for several
velocity--pressure elements that are inf-sup stable. The estimate demonstrates
optimal order convergence in the energy norm for the velocity component and a
scaled $L^2(H^1)$-type norm for the pressure component.
