Liquid Flow in Prismatic Channels Resting on a Parabolic Contour

A set of exact solutions is suggested for the Poisson equation in a plane region of a certain form for describing a flow of Newtonian incompressible liquid in a channel at low Reynolds numbers. The motion of an ideal liquid in a region bounded by a rotating contour, torsion of a rod, deflection of a membrane at zero displacement on the boundary of its region, and heating of a rod under certain conditions are equivalent to a mathematically formulated problem of such a flow. A possibility is demonstrated to construct approximate solutions to the said problem for certain sections of a channel, having a parabolic segment of the contour, using exact solutions to this problem. Examples are given for the construction of flow regions in a prismatic channel resting on a parabolic contour.