Property of reciprocity for stratified laminar flows in channels with arbitrarily shaped cross sections

A stratified laminar flow of several fluids in a channels with an arbitrarily shaped cross section is considered. It is assumed that the hydrostatic problem of finding free boundaries between different fluids is solved and domains of motion of individual fluids are known. Under the assumption that the medium motion arises under the action of an applied pressure gradient and volume gravity forces (or forces of inertia), the property of reciprocity between the applied forces $F_j$ and the flows of different components $Q_i$ , which is manifested as symmetry of the matrix of the flow rate coefficients $L_{ij}$($Q_i=L_{ij}F_j$), is proved in the general form. General symmetric solutions of the problem for a plane channel and a circular tube are presented. Formulas for the coefficient of increasing of the fluid flow rate owing to the presence of a near-wall layer of the gas are derived. It is shown that the flow rate of water in a partly filled channel may exceed the flow rate in a completely filled channel by more than an order of magnitude.