Relativity & Fast-Moving Objects: Does Time Slow Down?

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This is a video of the 'barking dog' chemistry experiment captured with a high speed camera. When you watch it in real time, the flame appears to move quickly and directly the bottom of the tube. However, when viewed in slow motion, it shows that the flame actually gradually moves down the tube in a series of up and down movements.

As the flame moves extremely fast, (presumably faster than the speed of sound as a sonic boom is released) because of relativity, does this mean time slows down for the flame as it reaches such a high speed? Thus, allowing it to go through more motions than we are perceiving it to?
 
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The moving object in a special relativity situation experiences less elapsed time than an observer in a stationary frame.

But in this situation, the flame is moving far too slow for relativity to be of issue.
 
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