In a hourglass, does the sand flow through the neck depend on the amount of sand in the upper glass? If we consider a sand flow analogous to fluid flow, then it should depend linearly, but in that case amount of sand to represent the given time would rise squared depending on time?
Fluid approximations do not work well at the scale of sand in hourglasses for most hourglasses. Almost all of the sand is statically braced against the walls and floor of the hourglass. Instead, you have a small region of instability above the hole where it is not possible for the sand to be braced; as that unstable portion falls through the hole, more falls in from above. There is slow flow of sand downwards, but it's still mostly statically braced. Thus, to a first approximation, flow rates are determined by grain size and shape (and material), and the immediate geometry around the hole (predominantly the size).