Osmosis (excluded volume dynamics)
Salt molecules stay in upstream compartment and do not throw the membrane. Water molecules throw the membrane from downstream to upstream because the concentration gradient of water molecules (opposite to salt concentration gradient) is then permanent. The quantity throwed Q(t) is related to the salt concentration gradient delta C(t) by the relation Q(t) = Pe * s / E * delta C(t) where Pe is the membrane permeability to the salt and E is the thickness of the membrane. The integration of this relation with time should not lead to a linear function of time in the present case because the upstream concentration is varying as water throwed to upstream involves the dilution of the solution.
Example 2, case of osmosis on a totaly selective membrane :