10 – 2D Falling Block (time-dependent)

This exercise extends the falling block problem to a time-dependent case with constant viscosity. The problem couples the solution of the momentum conservation equation (Stokes system) with the advection equation, where density (or phase) is transported through the velocity field.

The main objectives are:

1. Setting up the model domain with a dense block embedded in a viscous medium under gravity,
2. Defining free-slip velocity boundary conditions on all sides,
3. Assembling the coefficient matrix for the Stokes system (constant viscosity → assembled once),
4. Implementing a time loop that in each step:
   • Updates the right-hand side with the current density distribution,
   • Solves the Stokes system for velocity and pressure,
   • Advects density (or phase) using different numerical schemes, including tracers,
   • Recomputes the time step and visualizes the solution,
5. Producing an animation of the sinking block over time.

This problem demonstrates how to couple Stokes flow with advection of material properties, and how the choice of advection scheme affects the evolution of the solution.

Figure 1. Time evolution of the sinking block in a viscous medium using the tracer advection method.