The energetics of the breakup of a sheet and of a rivulet on a vertical substrate in the presence of a uniform surface shear stress

We use the lubrication approximation to obtain a complete description of the energetics of the breakup both of a fluid sheet of uniform thickness into a periodic array of infinitely many identical thin rivulets and of a single thin rivulet into one or more identical sub-rivulets on a vertical substrate in the presence of a prescribed uniform longitudinal shear stress at the free surface of the fluid by comparing the total energies of the different states. For both problems the situation when the volume flux is positive is relatively straightforward (and, in particular, qualitatively the same as that in the case of no prescribed shear stress), but when the volume flux is negative it is more complicated. However, whatever the value of the prescribed shear stress, there is always a critical thickness below which it is energetically favourable for a sheet to break up into rivulets and a critical semi-width below which it is energetically favourable for a rivulet to remain as a single rivulet, and a critical thickness above which it is energetically favourable for a sheet to remain as a sheet and a critical semi-width above which it is energetically favourable for a rivulet to break up into sub-rivulets.