Harry,
Unless the goats are pygmies be prepared to christen them Houdini one & Houdini two .. better escape artists I've yet to come across.
Goats do not eat grass as such, infact they eat almost anything else that they are not obtained for. It's is all down to goats being browsers not grazers .
Though if you really do want a patch to be mowed by them then tethering the goat is the only way I know of it happening .. it gets hungry and bored so eats everything in its reach.
The big down side of tethering is it can be harmful , in that the goat can become injured , chokes or get tangled on the tether or anchor stake.