Imposing Dirichlet domain condition

Hello all,
I wonder if there is a way to impose a Dirichlet condition not on a boundary but on a domain. Let's say, for instance, that at a time tn of my calculation I want to impose null velocity in my fluid depending on local temperature/concentration/etc. Weak constraints (Lagrange multipliers) are not well suited as the velocity condition is not initially set nor permanent...
So, how can I achieve that ?

Thanks in advance for your help !
Guillaume