How do we sense the flow of time? According to the Abhidhamma, at each given instant there is a consciousness that arises and ceases completely before the next consciousness arises. Each consciousness is only aware of the present moment (just an instant and only that instant) and therefore cannot know about previous instants. 

So, how is it possible that we have a sense of the flow of time? How do we know that there was a past?
Past is a conceptual construct that only seems solid in the absence of careful examination. But if you look closely you will see that Past is assembled by the mind from multiple cues (as is Present by the way).

The experience of second-to-second flow of time is a byproduct of chitta-vrtti, the associative cycle, when each subsequent dharma (~thought) comes by association with the previous one. The sense of relatedness of the two dharmas is what creates the illusion of flow.

Disclaimer: this answer represents my own experience and understanding and is not meant to reflect an official position or to be consistent with views of any Buddhist school.