In this presentation, we address an error analysis of an Eulerian finite element method employed to solve a linearized Navier--Stokes problem posed in a time-dependent domain. In this study, we assume the known evolution of the domain, independent of the solution to the problem at hand. The numerical method under consideration combines a standard BDF-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, we employ Nitsche's method to enforce the boundary conditions. The main focus of this presentation is a convergence analysis of the method for various inf-sup stable velocity--pressure elements. The analysis provides an optimal order convergence estimate in the energy norm for the velocity component and a scaled $L^2(H^1)$-type norm for the pressure component.