Unidirectional flow of a thin rivulet on a vertical substrate subject to a prescribed uniform shear stress at its free surface

We use the lubrication approximation to analyze the steady unidirectional flow of a thin rivulet on a vertical substrate subject to a prescribed uniform longitudinal shear stress at its free surface. In particular, we determine the quasisteady stability of a rivulet of prescribed flux, and calculate when it is energetically favorable for a rivulet to split into two narrower rivulets