Similarity solutions for unsteady shear-stress-driven flow of Newtonian and power-law fluids : slender rivulets and dry patches

Unsteady flow of a thin film of a Newtonian fluid or a non-Newtonian power-law fluid with power-law index N driven by a constant shear stress applied at the free surface, on a plane inclined at an angle α to the horizontal, is considered. Unsteady similarity solutions representing flow of slender rivulets and flow around slender dry patches are obtained. Specifically, solutions are obtained for converging sessile rivulets (0 < α < π/2) and converging dry patches in a pendent film (π/2 < α < π), as well as for diverging pendent rivulets and diverging dry patches in a sessile film. These solutions predict that at any time t, the rivulet and dry patch widen or narrow according to |x|3/2, and the film thickens or thins according to |x|, where x denotes distance down the plane, and that at any station x, the rivulet and dry patch widen or narrow like |t|−1, and the film thickens or thins like |t|−1, independent of N

Shakedown Limits of Slab Track Substructures and Their Implications for Design

This paper presents an approach to shakedown of slab track substructures subjected to train loads. The train load is converted into a distributed moving load on the substructure surface using a simplified track analysis. Based on the lower-bound dynamic shakedown theorem, shakedown solutions for the slab track substructures are obtained over a range of train speeds between zero and the critical speed of the track. It is found the shakedown limit is largely influenced by the ratio of layer elastic moduli and the ratio of train speed to critical speed rather than their absolute values. An attenuation factor, as a function of the critical speed and the friction angle of subsoil, is proposed to effectively obtain the shakedown limit of the slab track substructure at any train speed. In light of the shakedown solutions, improvements to the existing design and analysis approaches are also suggested