Heraclitus is recorded as saying:
Upon those who step into the same river, different and again different waters flow (Arius Didymus, Dox. Gr.)
It is not possible to step twice into the same river...it scatters and again comes together, and approaches and recedes (Plutarch, On the E at Delphi)
We step into and we do not step into the same rivers. We are and we are not (Heraclitus Homericus, Homeric Questions)
These suggest that Heraclitus viewed time as a river that flows; and staying with this metaphor, time moves around us. It's also comparable to Newtons view of absolute time (in fact three views of time - absolute, true and mathematical; which here, he identifies).
Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is known as duration.
However, in physical modelling of time, derived from Newton, it's generally taken that a particle 'moves forward' into time; this being consistent with its motion in space
This difference is taken into account in In the positivist/operational view of science, where it is measurements only that count; and by noting that the motion of a particle relative to absolute time isn't detectable.
Still, there is an ontological difference between the two views; is there an objective view of time that is consistent with Heraclitus's view?
One suggestion might be the 'moving block' view of time; where the present continually adds onto the past.