I’m not sure it would be possible to ELI5 this without an awful lot of other stuff.
Equation (4) is simple enough; just Equations (2) + (3).
I think equation (1) is a Newton’s 2nd Law/conservation of momentum thing. Very, very roughly, on the left hand side you have an acceleration-type thing and on the right hand side the first term is pressure * area / mass (so force / mass), then a term for the acceleration due to the viscosity of the material, then the acceleration due to gravity (I think; they don’t seem to have defined G in their table).
And then yes, I think equation (3) is conservation of energy.
But this is a serious astrophysics paper; I think at best you’re looking for ELI5-years-into-(post-)graduate-studies-in-astrophysics.
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