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Computational model of impacts on viscous surfaces

I started programming computers in 1968 in various languages. I eventually retired from IBM where coding was my main activity.

I am interested in impacts on viscous surfaces and I would like to develop a computational model to simulate the impact and the subsequent viscous relaxation of the impact cavity. In a recent post, a member suggested that a numerical model would be useful for exploring the parameters that lead to different physical behaviors.

I have never coded anything involving the Navier–Stokes equations, and I don't know if that is the most relevant approach for the following problem. Perhaps there are off-the-shelf programs available.

I would like to develop an "excavation" computational model to describe an oblique impact on a viscous non-elastic medium that creates a slanted conical crater, and a second "modification" model to describe the transformation of the slanted conical crater into an elliptical depression as the result of gravity-driven viscous relaxation, as shown in the following image. The models should allow experimentation with different viscosities, impact speeds in the range of 3.0 to 4 km/sec and crater diameters of 60 to 2000 meters.

The models will help to support the hypothesis that the elliptical Carolina Bays can be modeled as conic sections that resulted from secondary impacts of glacier ice boulders which were ejected after an extraterrestrial impact on the Laurentide ice sheet. The following is a LiDAR image of the Carolina Bays.

I would appreciate your suggestions.