The most common and most likely answer to this crossword clue is the 11 letter word EXACERBATES
I've seen this crossword clue in The Daily Telegraph
E
X
A
C
E
R
B
A
T
E
S
Buddy explains!

The straight part
We know that "Fans the flames of" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fans the flames of = exacerbates.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EXACERBATES mean?
EXACERBATES verb- To make worse (a problem, bad situation, negative feeling, etc.); aggravate; exasperate.