Inspired by the blood flow through a chamber of a healthy human heart, we present a simplified mathematical and computational model for the flow of an incompressible Newtonian fluid in a fixed domain with volume ruled by an artificial capacitance function. The inclusion of this capacitance function, which is space and time-dependent, gives rise to a differential form resembling the compressible Navier-Stokes equations. The numerical methodology is based on stable Galerkin mixed finite element formulations posed on velocity and pressure. Convergence studies for the two-dimensional steady-state Stokes equation with spatially varying capacitance indicate optimal convergence orders when combining continuous biquadratic velocities with continuous bilinear pressures (Taylor-Hood elements) or discontinuous unmapped linear pressures. On the other hand, we observed a loss of optimality with discontinuous linear mapped approximations for this field, as expected for the classical Stokes problem. Additional convergence studies indicate that the optimal convergence properties are preserved in the resolution of the nonlinear model for smooth solutions. Finally, we present simulations of cardiac cycles in an idealized human left ventricle in two and three dimensions.