The　full　Navier-Stokes-Fourier　system　with　mixed　boundary　condition　that　describes　the　motion　of　shear-thinning　and　incompressible　viscous　fluid　in　a　rotating　multi-screw　extruder　is　investigated.　The　viscosity　is　assumed　to　depend　on　the　shear　rate　and　the　temperature.　The　global　existence　of　suitable　weak　solutions　is　established.　The　fictitious　domain　method　which　consists　in　filling　the　moving　rigid　screws　with　the　surrounding　fluid　and　taking　into　account　the　boundary　conditions　on　these　bodies　by　introducing　a　well-chosen　distribution　of　boundary　forces　is　used.