(Ignoring your last paragraph)
Well, you need a material that sinks when frozen (most substances do) and a situation of thermal equilibrium. The dark side needs to cool enough to form solids, which sink. The bright side needs to pick up enough heat to melt the same amount of material.
Now why doesn't the back side eventually collect all the solid, building up to a mountain? I expect the top layer to be hottest, so the solids that sink won’t melt there. Why does it freeze over at the top, at all? The cooling material will sink while still liquid.
I imagine a planet wide convection flow, hot on top and cool returning at depth.
Perhaps the surface crust is a different material that floats on the main fluid. Whether solid or liquid it still floats above the inner fluid. But, only in the main downdraft region does it get pulled down since everything is going down. It will float back up once it clears the downdraft. So that might work: you have the appearance of a constant conveyor belt being pulled under, at the darkest antisolar point. The solid rains back up a few miles away, to join the bottom of the material headed toward the sinking zone.
No whirlpool though. The planet doesn’t rotate very fast, so the convection currents will be rather straight.
No eruption either. The hottest spot is a constant upwelling, but it's all liquid. The best you could figure might be to have a thin floating crust made out of a different material, but stays solid even under the hottest point.