Here in th eU.S., there's a product readily available from DIY or even many grocery stores called ZUD... it's rust remover containing a mild version of Oxalic acid. It comes in a powder and one floods the rust area with water and then sprinkle on the ZUD. I'd try to keep it off any lawn area and try the process on a good warm afternoon since it appears to work better warm. A good stiff brush to scrub it before rinsing helps. Barring that one can always use Muriatic acid... again available from DIY stores. It is a semi-dilute hydrochloric acid, so use appropriate care. In fact, I would cut it by half with water and then use it full strength if required. If you use water, pour the acid into the water, never the reverse... It will remove the rust but it usually leaves a lighter color on the concrete... again, be sure to take appropriate precautions if you use it... gloves, old leather shoes, and especially safety glasses...
Best luck!