The sound of tutting in disapproval is sometimes written as 'tsk'. If you wished to indicate that someone had made several such sounds, you might refer to his 'tsktsks'. There are seven consecutive consonants in a row. Any other such words tend to be Germanic-style names such as you find in "Hirschsprung's disease".
Neither of the above possibilities has ten letters, of course, but each does genuinely fit the seven consonants requirement.
To add to Quizmonkey's excellent explanation, the only genuinely vowelless 'words' are those such as
Psst! = Hey, listen!
Sh! = Be quiet!
Zzzz = sleeping (in comics etc)
Brrr! = I'm cold!
Mm-nnh = I see/I agree
Nth as in 'to the nth degree'...it has a vowel sound - 'enth' - but no actual written vowel.
All other words have vowels...including 'Y'.