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About Inverse Laplace Transform of a Dynamic Viscosity Function

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 function...

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Autores principales: Urbanowicz, Kamil, Bergant, Anton, Grzejda, Rafał, Stosiak, Michał
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9231391/
https://www.ncbi.nlm.nih.gov/pubmed/35744421
http://dx.doi.org/10.3390/ma15124364
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author Urbanowicz, Kamil
Bergant, Anton
Grzejda, Rafał
Stosiak, Michał
author_facet Urbanowicz, Kamil
Bergant, Anton
Grzejda, Rafał
Stosiak, Michał
author_sort Urbanowicz, Kamil
collection PubMed
description 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.
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spelling pubmed-92313912022-06-25 About Inverse Laplace Transform of a Dynamic Viscosity Function Urbanowicz, Kamil Bergant, Anton Grzejda, Rafał Stosiak, Michał Materials (Basel) Article 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. MDPI 2022-06-20 /pmc/articles/PMC9231391/ /pubmed/35744421 http://dx.doi.org/10.3390/ma15124364 Text en © 2022 by the authors. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Urbanowicz, Kamil
Bergant, Anton
Grzejda, Rafał
Stosiak, Michał
About Inverse Laplace Transform of a Dynamic Viscosity Function
title About Inverse Laplace Transform of a Dynamic Viscosity Function
title_full About Inverse Laplace Transform of a Dynamic Viscosity Function
title_fullStr About Inverse Laplace Transform of a Dynamic Viscosity Function
title_full_unstemmed About Inverse Laplace Transform of a Dynamic Viscosity Function
title_short About Inverse Laplace Transform of a Dynamic Viscosity Function
title_sort about inverse laplace transform of a dynamic viscosity function
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9231391/
https://www.ncbi.nlm.nih.gov/pubmed/35744421
http://dx.doi.org/10.3390/ma15124364
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