The production rate, useful life and operating efficiency of electric submergible pumping systems (ESP) is improved by operating the system so that the motor (220, 320) is cooled by two passes of production. Production fluids remove waste heat from the motor (220, 320) by passing the fluid in contact with the exterior of the motor (220, 320). The improved cooling permits the motor to be operated at a lower temperature improving life and efficiency, and/or to be operated at higher power at a similar temperature. Additionally, oversized and excess equipment is not required, further improving performance and economics for the user. In another aspect, the invention is a method of pumping fluids using a motor having vortex generators (366) on its exterior surfaces or on the interior surfaces of a surrounding shroud.