▁simultane ous ▁solution ▁of ▁a ▁large ▁set ▁of ▁gover ning ▁equations ▁for ▁both ▁sub dom ains ▁and ▁conjug ate ▁conditions . ▁Pat ank ar ▁proposed ▁a ▁method ▁and ▁software ▁for ▁such ▁solutions ▁using ▁one ▁generalized ▁expression ▁for ▁continu ously ▁computing ▁the ▁veloc ities ▁and ▁temperature ▁fields ▁through ▁the ▁whole ▁problem ▁domain ▁while ▁satisfying ▁the ▁conjug ate ▁boundary ▁conditions . ▁ ▁Anal yt ical ▁reducing ▁to ▁condu ction ▁problem ▁ ▁As ▁shown , ▁the ▁well - known ▁Du ham el ' s ▁integral ▁for ▁heat ▁flux ▁on ▁a ▁plate ▁with ▁arbitrary ▁variable ▁temperature ▁is ▁a ▁sum ▁of ▁series ▁of ▁consequ ent ▁temperature ▁derivatives . ▁This ▁series ▁in ▁fact ▁is ▁a ▁general ▁boundary ▁condition ▁which ▁becomes ▁a ▁condition ▁of ▁the ▁third ▁kind ▁in ▁the ▁first ▁approximation . ▁Each ▁of ▁those ▁two ▁expressions ▁in ▁the ▁form ▁of ▁Du ham el ' s ▁integral ▁or ▁in ▁a ▁series ▁of ▁derivatives ▁reduces ▁a ▁conjug ate ▁problem ▁to ▁the ▁solution ▁of ▁only ▁the ▁condu ction ▁equation ▁for ▁the ▁body ▁at ▁given ▁conjug ate ▁conditions . ▁An ▁example ▁of ▁an ▁early ▁conjug ate ▁problem ▁solution ▁using ▁Du ham el ' s ▁integral ▁has ▁been ▁performed . ▁This ▁approach ▁has ▁been ▁applied ▁both ▁in ▁integral ▁and ▁in ▁series ▁forms ▁and ▁is ▁generalized ▁for ▁l amin ar ▁and ▁tur bul ent ▁flows ▁with ▁pressure ▁gradient , ▁for ▁flows ▁at ▁wide ▁range ▁of ▁Pr and tl ▁and ▁Reyn olds ▁numbers , ▁for ▁compress ible ▁flow , ▁for ▁power - law ▁non - New ton ian ▁flu ids , ▁for ▁flows ▁with ▁un st