About Inverse Laplace Transform of a Dynamic Viscosity Function

Abstract

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.

On the generalization of Calogero-Ahmed summation formulas

“…Finding a generalization of the modified Calogero-Ahmed series began with the need to find the two solutions in the original works of Calogero and Ahmed [21][22][23] that are missing there. These solutions were necessary to develop the formula (4) in the paper [20]. During our research it has been noticed that for the modified Calogero-Ahmed series the following recursive formula has been derived:…”

“…Recent research on the inverse of the function describing the dynamic viscosity function 𝕄 to the time domain [20] revealed that the solution in the time domain can be presented as an infinite series of exponential functions based on zeros of the Bessel function:…”

“…These functions appear in derived solution (4) as the result of multiple convolution calculations. Readers interested in the details are advised to consult the work in [20]. In the works of Calogero [21,22] and Ahmed and Calogero [23], the following solutions can be found for the modified Calogero-Ahmed (5) and the standard Calogero-Ahmed (6) summation formulas, respectively:…”

“…𝕄 2 , can be found in Brereton and Jiang paper [19]. As for the second function 𝕄, a new approach was presented in [20], which can be used to define the final simplified form of this function in the time domain. However, knowledge of the Calogero-Ahmed series is necessary to obtain a complete simplified solution.…”

Load Eccentricity of Compressed Composite Z-Columns in Non-Linear State

“…Commercially available commercial FEM calculation packages allow for simple linear analyses, as well as complex calculations involving geometrically and physically non-linear issues [ 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 ]. It is also worth mentioning that this study involved using the popular finite element method, which is widely used in many fields [ 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 ].…”