About Inverse Laplace Transform of a Dynamic Viscosity Function

Materials (Basel). 2022 Jun 20;15(12):4364. doi: 10.3390/ma15124364.


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.

Keywords: Calogero–Ahmed sums; analytical solution; dynamic viscosity function; fluid dynamics; inverse Laplace transform; pipe flow; water hammer.

Grant support

A. Bergant gratefully acknowledges the support of Slovenian Research Agency (ARRS) conducted through the research project L2-1825 and the program P2-0162.