After all these years there is still no alternative to rods and brushes and this is how I would sweep your chimney/flue.
Don't let anyone tell you all you need is a hoover! You cannot sweep a chimney with a hoover.
My industrial vacuum is always running beside me to reduce the dust and tidy up at the end, but I use traditional rods and brushes for the sweeping. I always use nylon brushes along with superflex nylon rods in stainless steel flues, in accordance with manufacturers and installers instructions. I would use polypropylene rods and brushes in larger chimneys and a viper rod on an oil liner.
On completion of each sweep I will always do a smoke evacuation test to ensure a good draw or adequate oil flow through chimney/flue.