In this work the Navier-Stokes equations for non-stationary incompressible flow of the Newtonian fluid in time dependent domain are studied. The geometry of the flow domain changes in time according to fluid properties such as stress tensor. The motivation for our study comes from medicine—the simulation of blood flow in arteries and veins. After choosing an appropriate mathematical model of the flow in a domain with viscoelastic compliant walls, we deal with its theoretical analysis. We prove the existence of a weak solution using the weak compressible approximation in a moving domain with given deformation function. In our approach the fluid-structure interface condition is treated using a permeable-wall approach decoupling the fluid and the deformable structure. Finally we present some numerical experiments illustrating the convergence of the iteration with respect to the domain deformation function as well as the behavior of the moving wall for decreased permeability.