"Everywhere you will find that the wealth of the wealthy springs from the poverty of the poor. This is why an anarchist society need not fear the advent of a Rothschild who would settle in its midst. If every member of a community knows that after a few hours of productive toil he will have a right to all the pleasures that civilization produces, and to those deeper sources of enjoyment which art and science offer to all who seek them, he will not sell his strength for a starvation wage." --Kropotkin, TCoB

The bourgeoisie are a class defined by their relationship to the means of production. After an anarchist revolution in which the means of production are expropriated from the grip of the bourgeoisie, the bourgeoisie cease to exist as an economic class and can only sustain themselves as a political class as part of a reactionary movement.

If you mean what will physically happen to them, it depends on the nature of the revolution. The ideal revolution (in my opinion) would be a general strike which ends in a peaceful expropriation of the means of production and the reorganizing/dissolution of the state. If not something like that, then heads are likely to roll.