Is there a name for the mushy sludge that forms around/beneath meat as it cooks?
The sludge you speak of is actually proteins called myosin (denatures at 120f) and actin (denatures at 150f) going through the stages of denaturing, coagulation and ultimately gelantization.  Protein coagulates when it is denatured, that is destroyed.  Gelatization is a follow on the process of breakdown in connective tissue.