  We calculated a grid of evolutionary models for white dwarfs with helium
cores (He-WDs) and investigated the occurrence of hydrogen-shell flashes due to
unstable hydrogen burning via CNO cycling. Our calculations show that such
thermal instabilities are restricted to a certain mass range
(M=0.21...0.30Msun), consistent with earlier studies. Models within this mass
range undergo the more hydrogen shell flashes the less massive they are. This
is caused by the strong dependence of the envelope mass on the white dwarf core
mass. The maximum luminosities from hydrogen burning during the flashes are of
the order of 10^5 Lsun. Because of the development of a pulse-driven convection
zone whose upper boundary temporarily reaches the surface layers, the
envelope's hydrogen content decreases by Delta(X)=0.06 per flash. Our study
further shows that an additional high mass-loss episode during a flash-driven
Roche lobe overflow to the white dwarf's companion does not affect the final
cooling behaviour of the models. Independent of hydrogen shell flashes the
evolution along the final white dwarf cooling branch is determined by hydrogen
burning via pp-reactions down to effective temperatures as low as 8000 K.
