Communications in Mathematical Sciences

Volume 16 (2018)

Number 2

Data-driven modeling for the motion of a sphere falling through a non-Newtonian fluid

Pages: 425 – 439

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n2.a6

Authors

Zongmin Wu (Shanghai Center for Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, China)

Ran Zhang (Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, China)

Abstract

In this paper, we will introduce a mathematical model of the jerk equation to simulate the unstable oscillations of the motion of a falling sphere in the wormlike micellar solution. This differential/algebraic equation (DAE) is established only by learning the experimental data of time vs. velocity with the sparse optimization method. To simulate the solutions of the DAE, four discretization schemes are proposed and compared. Periodic and damped harmonic motion, and nonuniform transient and sustaining oscillations can be observed for the sedimentation of a sphere through the non-Newtonian fluid in the numerical experiments. It successfully presents chaos consistent with the physical behavior, which are highly sensitive to the initial values and experimentally nonreproducible. We can conclude that our model has the ability to capture the primary pattens of the dynamics, which is more meaningful than predict an individual trajectory for the chaotic systems.

Keywords

falling sphere, non-Newtonian fluid, data-driven modeling, sparsity, differential/algebraic equation (DAE), jerk

2010 Mathematics Subject Classification

34B60, 34K28, 41A30, 65P20, 65Z05

Received 21 October 2017

Accepted 22 November 2017

Published 14 May 2018