 turbul lubric turbul lubric paper concern hydrodynam turbul motion lubric layer proceed reynold equat introduc approxim current lubric problem lubric film thick gener motion equat turbul lubric written prandtl mix length hypothesi exact approxim solut veloc distribut lubric layer discuss point pressur gradient reynold number influenc veloc distribut well differ respect laminar flow order simpl formula exact depend rate flow pressur gradient dimensionless form replac linear relat slope depend reynold number approxim allow obtain pressur differenti equat simpl form pressur equat integr case journal bear assum constant variabl viscos lubric compar experiment data smith d fuller good qualit agreement point 