Some simple questions about tweakable ciphers
Suppose the tweakable cipher uses a secret key $k$ and a secret tweak $t$, then syntactically you can regard $K=(k, t)$ as the total secret key.  So, in this case, the 'tweakable cipher' is just an ordinary block cipher, with a longer key.  In order to get a notion different from the usual block cipher, you'll need the tweak to be non-secret.