Replenishing hydrogen in the core of the sun How is hydrogen supplied to the core of the sun? Shouldn't the radiative zone prevent this? Shouldn't heavier helium fill the core? How much of the hydrogen can the sun fuse?
There is very little mixing in the core of the Sun, where the stratification is fixed by radiative (rather than convective) heat transfer. The heavier helium does "fill the core", but takes about 12 billion years to do so, during which time, the concentration of helium gradually increases.

During its main sequence lifetime, most of the energy generation takes place in the central 20% of the Sun, where approximately 600 million tonnes of H is turned into He every second. (You get this by dividing the solar luminosity by $c^2$ and note that the pp H-burning chain is 0.7% efficient.) If we take this as a good average value over the main sequence lifetime (the Sun was fainter in the past, but will be brighter in the future), we calculate that $2\times 10^{29}$ kg of H is turned into He, which is 10% of the solar mass, and about 13% of its initial H.

Once the Sun evolves away from the main sequence, it burns more hydrogen in a shell around the core, and the convective envelope can mix fresh material into this shell. Given that the Sun is expected to leave behind a $\sim 0.5 M_{\odot}$ C/O white dwarf at the end of its life, then at least half of its mass will have been processed by nuclear reactions (and probably more, since some processed material is lost in the solar wind during the red giant phase).