Australia's two largest airlines continue to disagree with each other on how to deal with the safety risk posed by the cloud of ash from a Chilean volcano on Tuesday.

Qantas Airways Ltd. continues to suspend some flights due to ash cloud to the south of Australia but Virgin Australia Ltd. is resuming all services.

The airline suspended flights on Tuesday to and from New Zealand, Adelaide, and Tasmania State. We are putting safety before schedule and won't be resuming services until we are satisfied that we are safe to fly, a Qantas spokesman told WSJ.

We're flying under it and around it, a Virgin Australia spokeswoman said. We have no doubt we're operating safely.

Many Australians who had gone for the Queen's Birthday long-holiday were unable to return home and more than 50,000 passengers were stranded in Australia and New Zealand as flights were canceled on Sunday and Monday.

The ash cloud hovering over southeastern Australia is expected to remain there for another 24 hours, said Gordon Jackson, the supervising meteorologist at the Australian Bureau of Meteorology's volcanic ash unit in Darwin. But we're actually starting to see some of it disperse, so we're keeping a close eye, he said.

Singapore-based budget carrier Tiger Airways Holdings Ltd., canceled flights to and from Adelaide and some between Perth and Melbourne on Tuesday.

Although Puyehue is still erupting, the recent ash clouds are not shooting up high, Jackson said. The chance of the new plumes coming all the way round to Australia is a lot less, he said.