  The paper develops a finite element method for the Navier-Stokes equations of
incompressible viscous fluid in a time-dependent domain. The method builds on a
quasi-Lagrangian formulation of the problem. The paper provides stability and
convergence analysis of the fully discrete (finite-difference in time and
finite-element in space) method. The analysis does not assume any CFL time-step
restriction, it rather needs mild conditions of the form $\Delta t\le C$, where
$C$ depends only on problem data, and $h^{2m_u+2}\le c\,\Delta t$, $m_u$ is
polynomial degree of velocity finite element space. Both conditions result from
a numerical treatment of practically important non-homogeneous boundary
conditions. The theoretically predicted convergence rate is confirmed by a set
of numerical experiments. Further we apply the method to simulate a flow in a
simplified model of the left ventricle of a human heart, where the ventricle
wall dynamics is reconstructed from a sequence of contrast enhanced Computed
Tomography images.
