About inverse Laplace transform of a dynamic viscosity function

Urbanowicz, Kamil (author), Bergant, Anton (author), Grzejda, Rafał (author), Stosiak, Michal (author)

A dynamic viscosity function plays an important role in water hammer modeling. It is responsible for dispersion and decay of pressure and velocity histories. In this paper, a novel method for inverse Laplace transform of this complicated function being the square root of the ratio of Bessel functions of zero and second order is presented. The obtained time domain solutions are dependent on infinite exponential series and Calogero–Ahmed summation formulas. Both of these functions are based on zeros of Bessel functions. An analytical inverse will help in the near future to derive a complete analytical solution of this unsolved mathematical problem concerning the water hammer phenomenon. One can next present a simplified approximate form of this solution. It will allow us to correctly simulate water hammer events in large ranges of water hammer number, e.g., in oil–hydraulic systems. A complete analytical solution is essential to prevent pipeline failures while still designing the pipe network, as well as to monitor sensitive sections of hydraulic systems on a continuous basis (e.g., against possible overpressures, cavitation, and leaks that may occur). The presented solution has a high mathematical value because the inverse Laplace transforms of square roots from the ratios of other Bessel functions can be found in a similar way.

• Language: English
• Keywords: inverse Laplace transform, analytical solution, water hammer, dynamic viscosity function, fluid dynamics, pipe flow, Calogero–Ahmed sums
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FS - Faculty of Mechanical Engineering
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.06.2022
• Year: 2022
• Number of pages: 26 str.
• Numbering: Vol. 15, iss. 12, art. 4364
• PID: 20.500.12556/RUL-137576
• UDC: 532.522:517.44
• ISSN on article: 1996-1944
• DOI: 10.3390/ma15124364
• COBISS.SI-ID: 112475651
• Publication date in RUL: 22.06.2022

Record is a part of a journal

• Title: Materials
• Shortened title: Materials
• Publisher: MDPI
• ISSN: 1996-1944
• COBISS.SI-ID: 33588485

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.
• Licensing start date: 20.06.2022

Secondary language

• Language: Slovenian
• Keywords: inverzna Laplaceva transformacija, analitična rešitev, vodni udar, dinamična viskozna funkcija, dinamika tekočin, tok v cevi, Calogero–Ahmedove vrste

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: L2-1825
• Name: Modeliranje zračnih mehurjev ujetih v hidravličnih cevnih sistemih

• Funder: ARRS - Slovenian Research Agency
• Project number: P2-0162
• Name: Večfazni sistemi