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We're being told there's no need to panic buy fuel, but queues have been reported across the South Coast, with some petrol stations having to shut.
It's after tanker drivers at five of the seven main supply companies have voted in favour of industrial action over terms and conditions, as well as safety standards.
Unite, which represents around 2,000 drivers who deliver to Shell, Esso and major supermarkets, is demanding minimum standards for pay, hours, holiday and redundancy.
The Army could be brought in to help drive supplies to petrol stations.
David Cameron's chaired a special COBRA meeting to discuss contingency plans, but the government says there is no need to panic buy.
Some drivers appear not to be taking any chances. There have been queues reported at stations in Hampshire.
Hampshire police said that "spurious rumours'' online about shortages were fuelling panic buying.
"It is important for members of the public to bear in mind that members of the Unite union are still in talks with their employers, dedicated to finding a solution which will make strike action unnecessary,'' a spokeswoman said.
"There is currently no problem with fuel supplies in Hampshire and therefore we would urge all motorists to continue as normal with their fuel purchases - there is no need to stock up. Petrol stations in Hampshire and the Isle of Wight are not being closed."
Dorset Police have urged the public not to panic buy fuel.
Chief Inspector Nick Maton said: "There is no disruption to the fuel supply in the UK and members of the public should not panic buy.
"The actions of some motorists in queuing irresponsibly at petrol stations is causing danger to other road users.
"Police are taking action, requesting petrol stations to close temporarily in order to keep traffic flowing.
"Once the queues have dispersed, the petrol stations may reopen for short periods.''
The force said it was not closing all petrol stations in the county. A spokeswoman said:
"Dorset Police will only be requesting the closure of the petrol stations worst affected by congestion in order to keep traffic flowing, requests of this nature will be made direct from Dorset Police officers to designated petrol stations where necessary."
Thanks to Helena Mace for the photo.