In this paper it is pointed out that the existing solutions for the deflections of thin plates under uniform lateral load and simply supported along all edges can be applied to the determination of the limiting temperature distributions in a fluid flowing in laminar motion in ducts of the same cross sections as those of the plates. The direct application of this solution is permissible only when (a) the axial temperature gradient is constant, with respect to both flow length and cross-section position. This implies uniform heating or cooling. (b) The cross section is a polygon. When the cross-section boundary contains curves, the constants of integration must be adjusted so that the boundary condition of nonslip flow is satisfied. Since the temperature and velocity distributions obtained by this analogy are expressed analytically, values of local heat transfer at any point in the cross-section boundary can be calculated. Examples are given for four cross sections, namely, rectangular, equilateral triangular, right-angled isosceles triangular, and semicircular. The heat-transfer distributions on the boundaries are calculated for square and equilateral triangular cross sections.