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I ignored the viscous diffusion term from the NavierStokes formula to encourage better vortex formation (the implicit advection solving I'm using creates enough diffusion on its own for this system).
Click and drag to apply a force to the fluid:


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To learn more about the math involved, check out the following sources:
Fast Fluid Dynamics Simulation on the GPU  a very well written tutorial about programming the NavierStokes equations on a GPU.