In the past 150 years researchers tried to find analytical solutions to simplified hydrodynamic equations describing pressurised flow of liquids in pipelines. The achievements so far are reviewed and categorised. The following transient laminar flows are investigated in detail: (i) accelerating flow, (ii) decelerating flow (iii) reversing flow and (iv) water hammer. The originally derived analytical solutions are in the form of infinite sums, which have to converge and which are truncated in practice. The goal of this work is to investigate underlying assumptions, compare analytical tools and appropriate approximations, and consequently present new approximation of Bessel functions and its roots. Finally, recommendations for usage of these solutions in engineering practice are given.
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