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We consider a system of two balance laws of Keyfitz–Kranzer type with varying generalized Chaplygin gas, which exhibits negative pressure and is a product of a function of time and the inverse of a power of the density. The Chaplygin gas is a fluid designed to accommodate measurements for the early universe and late-time universal expansion while obeying the pressure–density–time relation. We produce an explanation and description of the non-self-similar Riemann solutions, including the non-classical singular solutions. We also find that due to a direct dependence on time, a change in the regions allowing for combinations of classical and non-classical singular solutions occurs; therefore, a Riemann solution can have different solutions over several time intervals. Our findings are confirmed numerically using the Local Lax–Friedrichs scheme. [ABSTRACT FROM AUTHOR]

Motivated by the industrially important processes of blade coating and cavity filling of liquid crystalline materials, we consider steady, two-dimensional shear-driven (Couette) and pressure-driven (plane Poiseuille) flow of a thin film of a nematic liquid crystal in the slowly varying channel formed between a fixed blade of prescribed shape and a planar substrate. Specifically, blade coating motivates the study of shear-driven flow due to the motion of the substrate parallel to itself with constant velocity, while cavity filling motivates the study of pressure-driven flow due to an imposed pressure drop. We use a combination of analytical and numerical techniques to analyze the Ericksen-Leslie equations governing the fluid velocity and pressure and the director orientation in cases when both the aspect ratio of the channel and the distortion of the director field are small. We demonstrate a variety of flow and director-orientation patterns occurring in different parameter regimes. In the limit of weak flow effects, flow alignment does not occur and the appropriate solution of the governing equations is found explicitly. In the limit of strong flow effects flow alignment occurs and orientational boundary layers exist near the substrate and near the blade, and, in addition, an orientational internal layer may also exist within which the director orientation changes from +θ0 to -θ0, where θ0 is the flow-alignment angle. [ABSTRACT FROM AUTHOR]