The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.

The flow of an impinging non-Newtonian jet onto a solid flat plate is examined theoretically in this study. Similarity solutions are sought for both shear-thinning and shear-thickening fluids of the power-law type. The jet is assumed to spread out in a thin layer bounded by a hydraulic jump. In addition to the stagnation-flow region, the flow domain is divided into three main regions: a developing boundary layer, fully viscous boundary layer and hydraulic jump. The anomalous behaviour of power-law fluids at small shear rate is remedied by seeking a two-layer solution in each domain. Such anomalies include the singularity of viscosity for shear-thinning fluids, and the vanishing of viscosity as well the overshoot in velocity for shear-thickening fluids. Although the rate of shear-thinning appears to affect significantly the film profile and velocity, only the overall viscosity influences the position of the hydraulic jump.

To study interfacial particle-to-particle bonding mechanisms, an ultrathin film of pyrrole was deposited on alumina nanoparticles using a plasma polymerization treatment. High resolution transmission electron microscopy experiments showed that an extremely thin film of the pyrrole layer (2 nm) was uniformly deposited on the surfaces of the nanoparticles. In particular, the particles of all sizes (10–150 nm) exhibited equally uniform ultrathin films indicating well-dispersed nanoparticles in the fluidized bed during the plasma treatment. Time-of-flight secondary ion mass spectroscopy experiments confirmed the nano-surface deposition of the pyrrole films on the nanoparticles. The pyrrole-coated nanoparticles were consolidated at a temperature range (approximately 250 °C) much lower than the conventional sintering temperature. The density of consolidated bulk alumina has reached about 95% of the theoretical density of alumina with only a few percent of polymer in the matrix. After low-temperature consolidation, the micro-hardness test was performed on the bulk samples to study the strength that was related to particle-particle adhesion. The underlying adhesion mechanism for bonding of the nanoparticles is discussed.