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The pressure at the bottom of the salty side of the partition is higher than at the bottom of the fresh side. This can be readily seen by integrating up hydrostatic balance from a height z above the bottom to the top, where we assume that p = 0 — see the figure.
where the limits on the integration are from z to the top.
Thus the pressure at the bottom on the salty side is ρsaltgh and the pressure at the bottom on the fresh side is ρfreshgh. Thus there is a pressure gradient force in the direction x, toward the fresh reservoir of magnitude:
where
is the reduced gravity, h is the depth of the container and L is the lateral dimension of the container. Given that water tends to flow down the pressure gradient from high to low pressure we expect, and indeed observe, the water to flow from the salty reservoir to the fresh reservoir. We can estimate the magnitude of the expected flow speeds by assuming a balance between the acceleration terms in the pressure gradient force in the momentum equation thus:
implying a flow speed of, using Eqs.(1) and (2)
If Δρ/ρ is only 0.1%, the above implies a speed of u ~ 1cm/s if h = 10 cm, roughly in accord with observed flow speeds.