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The overarching objective of the present study is to apply a quasi-two-dimensional approach to analyze the laminar flow of lubricating oil. Lubricating oils are non-Newtonian by nature. For these types of oils, the Sisko fluid model is the most suitable model of the nonlinear stress–strain relationship for these types of oils. It is hoped that by omitting the dependence of flow quantities in one direction, more qualitative information can be obtained on the characteristics of the purely three-dimensional boundary layer flow of lubricating oils. Some of the most familiar flow geometries discussed are steady flow over a flat plate, a corner of a wedge, and a stagnation region; steady flow in a convergent and divergent channel; and impulsively started flow over an infinite flat plate and semi-infinite flat plate. The governing equations of all flow geometries are transformed into nonlinear ordinary differential equations (ODE) using the free parameter transformation. The results are discussed briefly in the graphical presentation.

The time development of the symmetrical standing zones of recirculation, which is formed in the early stages of the impulsively started laminar flow over the square cylinder, have been studied numerically. The Reynolds number considered ranges from 25 to 1,000. Main flow characteristics of the developing recirculation region aft of the square cylinder and its interaction with the separating shear layer from the leading edges are studied through the developing streamlines. Other flow characteristics are analysed in terms of pressure contours, surface pressure coefficient, wake length and drag coefficient. Four main-flow types and three subflow types of regimes are identified through a detailed analysis of the evolution of the flow characteristics. Typically, for a given Reynolds number, it is noted that flow starts with no separation (type I main-flow). As time advances, symmetrical standing zone of recirculation develops aft of the square cylinder (type II main-flow). The rate of growth in width, length and structure of the aft end eddies (sub-flow (a)) depends on the Reynolds number. In time, separated flow from the leading edges of the square cylinder also develops (type III main-flow) and forms growing separation bubbles (sub-flow (b)) on the upper and lower surfaces of the square cylinder. As time advances, the separation bubbles on the upper and lower surfaces of the cylinder grow towards downstream regions and eventually merge with the swelling symmetrical eddies aft of the cylinder. This merging of the type II and type III flows created a complex type IV main-flow regime with a disturbed tertiary flow zone (sub-flow (c)) near the merging junction. Eventually, depending on the Reynolds number, the flow develops into a particular category of symmetrical standing recirculatory flow of specific characteristics.