Analysis of Meandering Water Rivulets of Finite Amplitude

Abstract

Water rivulets on inclined smooth plate form a stable meandering pattern under certain conditions. The bend theory for describing the meandering water rivulet includes the effect of the surface-tension force. The governing equations are the St. Venant equations of shallow water as shown in curvilinear coordinates. The resultant bend equation is nonlinear. The linear stability analysis of this equation with minute disturbance was done by the senior author. In that analysis, all the small terms (nonlinear terms) were omitted. The stable meander trajectory formed by streams has finite amplitude as derived from Cartesian sinusoidal patterns. These patterns are very similar in shape to those of meandering rivers. The pattern of meandering rivers can be expressed by sine-generated curves. The meandering river trajectory derived from the first-order sine-generated curve contains periodic deviation characteristics. This deviation is caused by either fattening and/or skewing from the first-order sine-generated curve. The approximate solution for the nonlinear analysis of the bend equation (including the surface-tension force) demonstrated that the deviations could be well formulated by the third-order sine-generated curves. Furthermore, the derived water rivulet forms of finite amplitude were verified by laboratory experimental results.