The Numerical Scheme for the History Force Integrals in Hydrodynamics

A high-order precision numerical scheme was proposed for the Basset type kernel integrals and the Basset type integro-differential equations that often arise in hydrodynamics. The scheme was tested on the model integrals and on the problems of particle and bubble focusing in a standing acoustic wave in liquid. The scheme showed around 2 orders higher speed of integral estimation compared with the existing analogs. The obtained results show that the integral can be approximated with high order of precision and the real problem is simulated well by the proposed scheme. A variable step technique was used to increase the precision of integro-differential equation simulation even more by eliminating the discrepancy at the very start of the simulation. The proposed numerical scheme may find its applications in many biological and medical problems related to acoustophoresis or to particle and bubble focusing by ultrasound.