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This theoretical and experimental investigation inquires into the various steady and unsteady motions that are possible when a highly viscous liquid partly fills a closed circular cylinder rotated about its horizontal axis at constant angular velocity. Fillings leaving an air bubble in the range roughly 10-20% by volume provide the most lively variety of observable phenomena.The full hydrodynamic problem is too complicated to be amenable to quantitative theoretical treatment, except by numerical analysis which is not yet available; but the abstract qualitative theory developed in § 2 appears to capture all the essentials of experimentally found behaviour. An analogous finite-dimensional system, such as would be presented by a close finite-element approximation, is used to illuminate principles governing the order of multiple solutions and their stability. Then the connection between the full problem and the analogue is demonstrated. Finally a simple argument is outlined confirming the observed stability of the motion at small rates of rotation.The experiments are described in § 3 and their results presented in § 4. For various values of the cylinder's aspect ratio, estimated singularities of the time-independent solution set are recorded as several-branched graphs of ωv/gR versus volume fraction filled by liquid (ω is the angular velocity of the container, R its radius and v the kinematic viscosity of the liquid). The experimental observations are discussed in § 5.

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Abstract. The Upper Ganga River basin is socioeconomically the most important river basin in India and is highly stressed in terms of water resources due to uncontrolled land use and land cover (LULC) activities. This study presents a comprehensive set of analyses to evaluate the population growth, LULC transformations, and water quality nexus for sustainable development in this river basin. The study was conducted at two spatial scales: basin scale and district scale. First, population data were analyzed statistically to study demographic changes, followed by LULC change detection over the period of February–March 2001 to 2012 (Landsat 7 Enhanced Thematic Mapper Plus (ETM+) data) using remote sensing and geographical information system (GIS) techniques. Trends and spatiotemporal variations in monthly water quality parameters, viz. biological oxygen demand (BOD), dissolved oxygen (DO, measured in percentage), fluoride (F), hardness (CaCO3), pH, total coliform bacteria and turbidity, were studied using the Mann–Kendall rank test and an overall index of pollution (OIP) developed specifically for this region, respectively. A relationship was deciphered between LULC classes and OIP using multivariate techniques, viz. Pearson's correlation and multiple linear regression. From the results, it was observed that population has increased in the river basin. Therefore, significant and characteristic LULC changes were observed. The river became polluted in both rural and urban areas. In rural areas, pollution is due to agricultural practices, mainly fertilizers, whereas in urban areas it is mainly contributed from domestic and industrial wastes. Water quality degradation has occurred in the river basin, and consequently the health status of the river has also changed from acceptable to slightly polluted in urban areas. Multiple linear regression models developed for the Upper Ganga River basin could successfully predict status of the water quality, i.e., OIP, using LULC classes.