Rigorous theory based on transient Eigen expansion rectifying known integral analysis of unsteady viscous encroachment in capillary channels

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2015-08

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Abstract

The primary goal of my thesis is to investigate the well known fluid phenomenon of encroachment of a viscous liquid in a narrow channel due to capillary action. This has been achieved by determining the time-dependent intruded length in a conduit with rectangular cross-section. Although a lot of research has been done on this topic for over a century, all these previous studies resort to a crucial assumption valid only for a mildly transient system. Typically, all of them use an integral formulation where the intrusion is accounted for by equating the total force acting on the fluid to the rate of change of momentum. However, while doing so, the earlier calculations assumed the viscous resistance to be corresponding to a steady state velocity profile. Approximations like this can only be justified when the temporal variation in the system is weak enough to cause negligible transient deviation in the cross-sectional variation of flow. My thesis addresses this issue by proposing a new method where the unsteady field itself is treated as an unknown quantity. Accordingly, a suitable Eigen function expansion has been used to describe the velocity. The time-dependent amplitudes of the expansion and the unsteady penetration length are calculated using a system of ordinary differential equations. Then, a comparative analysis establishes the condition for which the traditional formulations and our rigorous technique differ from each other. It becomes evident that the two methods deviate from each other significantly for shorter time scales, i.e. at the initial stages. In the end, a novel asymptotic perturbation analysis corroborates the convergence of the long-time results by both the approaches while demonstrating how next order behavior differs between the two sets of results. Thus, the accuracy and rigor of the presented work is expected to ensure a more thorough understanding of the dynamics behind this transport phenomenon.

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Keywords

Capillary, Viscous Fluid, Eigen Function, Temporal Variation, Transport Phenomena, Asymptotic Convergence

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