I have a model about a fluid being extruded on a moving bed as in the 3D printing process. The model is a boundary-value problem where the right boundary is the point where the fluid attaches to the bed while at the left boundary new fluid is continuously provided. The reference frame is the fluid filament centerline with the origin at the extruder nozzle. As such, the right boundary is unknown, although its Cartesian coordinates are known as the height is constant and the horizontal Cartesian coordinate is linked to the moving speed of the bed.
I thought of representing the moving boundary as a collection of discrete time steps and to use the solution of the previous iteration as the initial guess for the current time. The simulation for a single time snapshot is run by using the wrapping strategy suggested here.
My problem is how to include the freshly extruded fluid at the left boundary considering that, with the above wrapping strategy, my simulating interval is always 1.