The Milk Bottle Problem

A milk bottle is allowed to stand so that the cream rises to the top. This occurs without any change in total volume. Does the pressure near the base of the bottle change ? Surely, since the total weight W of the liquid remains constant, the pressure on the base, of area A, being W/A remains constant. After separation, the pressure immediately below the cream is less than what it was before separation. So maybe . . . A mathematician uses a regular milk bottle of volume V, containing liquid of density d. After settling, the liquid is separated into two components of density d_1 and d_2, which occupy volumes of V_1 and V_2 respectively. As per the diagram for the mathematician's milk bottle, V_1 = h_1*A_1 and V_2 = h_2*A_2 , where h_1 and h_2 are the heights of the respective component volumes. The mathematician starting from this basis calculates a change in pressure after separation -- or is there one ? A would-be-physicist (wbp) challenges the mathematician.

Graphic reproduced from
Harvey A. Cohen, A Dragon Hunter's Box", with illustrations by Jeni Rawson, Hanging Lake Books, Warrandyte, Victoria, Australia, 1974.
This graphic is (C) Harvey A. Cohen, 1974, 1975, 2002 and its reproduction for any commercial purpose is forbidden.
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