Bill, a young woman who makes chips in a shop, attends lectures given by the Doctor, who has been serving as a university professor for a number of years (accompanied by Nardole).
After the Doctor meets her personally, he offers to become her personal tutor
One day, she meets a young woman, Heather, who points out a strange puddle. The puddle sucks Heather in as the new "pilot." Taking on her form, it then proceeds to pursue Bill.
Bill, alarmed, runs to the Doctor, who ushers her into the TARDIS. However the water-Heather manages to follow them wherever they go throughout time and space.
The Doctor determines that the "water" is actually what amounts to futuristic motor oil that leaked from a space craft, and it merged with the unfortunate Heather.
It also took on her final thoughts and desires, including an offer for Bill to come away with her. Bill convinces what used to be Heather to say goodbye, and the form dissolves back into a puddle.
The Doctor takes Bill back to the university and is going to wipe her memory, but she convinces him to let her remember. He is further convinced to allow her to travel with him as a proper companion.