Purchase a cooler to hold as much dry ice as you want to use. This will be the low-lying fog that sinks to the floor.
Place the dry ice into the cooler, wearing gloves for protection. Dry ice is solid carbon dioxide and can burn your skin due to its extremely low temperature.
Pour hot water over the dry ice to produce the fog/smoke. Use a fan to disperse it over a prescribed area.
Use a submersible water heater to produce continual smoke by keeping the water heated in the cooler. The hotter the water, the more quickly the smoke will dispense. Use your own judgment on how hot the water should be, based on how long you wish the fog to flow.