  Photoexcited electron-hole pairs (excitons) in transition metal
dichalcogenides (TMDC) experience an effective force when these materials are
non-uniformly strained. In the case of strain produced by a sharp tip pressing
at the center of a suspended TMDC membrane, the excitons are transported to the
point of the highest strain at the center of the membrane. This effect, exciton
funneling, can be used to increase photoconversion efficiency in TMDC, to
explore exciton transport, and to study correlated states of excitons arising
at their high densities. Here, we analyze the limits of funneling efficiency in
realistic device geometries. The funneling efficiency in realistic monolayer
TMDCs is found to be low, $ <5 \;\%$ both at room and low temperatures. This
results from dominant diffusion at room temperature and short exciton lifetimes
at low temperatures. On the other hand, in TMDC heterostructures with long
exciton lifetimes the funneling efficiency reaches $\sim 50\;\%$ at room
temperature, as the exciton density reaches thermal equilibrium in the funnel.
Finally, we show that Auger recombination limits funneling efficiency for
intense illumination sources.
