Learning physics by data for the motion of a sphere falling in a non-Newtonian fluid

doi:10.1016/j.cnsns.2018.05.007

Communications in Nonlinear Science and Numerical Simulation

Volume 67, February 2019, Pages 577-593

https://doi.org/10.1016/j.cnsns.2018.05.007 Get rights and content

Highlights

•
A nonlinear DAE of the velocity v of a falling sphere in non-Newtonian fluid will be proposed by directly learning the data.
Our model successfully simulates the sustaining oscillations and abrupt increase during the sedimentation of a sphere.
It presents the behavior of a chaotic system which is highly sensitive to initial conditions and experimentally nonreproducible.
The normalized representation covers both the classical physical laws and the nonuniform oscillations.
The data-driven idea will provide scientists with more important tools to support their discovery in the future.

Abstract

In this paper, we will introduce a mathematical model of nonlinear jerk equation of velocity $v = η_{1} \frac{η_{2} v^{″} + 1}{v^{″} + η_{3}} v^{″}$ to simulate the nonuniform oscillations of the motion of a falling sphere in the non-Newtonian fluid. This differential/algebraic equation is established only by learning the experimental data with the generalized Prony method and sparse optimization method. From the numerical results, our model successfully simulates the sustaining oscillations and abrupt increase during the sedimentation of a sphere through a non-Newtonian fluid. It presents the behavior of a chaotic system which is highly sensitive to initial conditions. More statistical and physical discussions about the dynamical features of the model are provided as well. Our model can be interpreted as a nonlinear elastic system, and includes both the uniform and nonuniform oscillatory motion of the falling sphere.

Section snippets

Motivation

Mathematical modeling for the motion of a falling sphere in a bounded cylinder of viscous fluid has been widely studied both in physics and mathematics. Typically a differential equation model is built as a physical law. Back in 1851, Stokes derived that the drag force exerted on a small sphere in viscous Newtonian fluid is proportional to the velocity [3]. This actually describes the motion of a falling sphere with a first-order linear ordinary differential equation (ODE) of velocity. The

Data-driven model of the differential equation

The main process of how to derive the differential equation which describes the motion of the falling sphere in the vicious fluid is investigated in this part. Before that, we briefly recall the physical laws and equations in fluid dynamics used to study the problem.

Numerical results

In this section, a numerical scheme will be provided to deal with the more generalized form of DAE (15), $J_{2} (v^{″}, v) = ξ_{0} v + ξ_{2} v^{″} + ξ_{5} v v^{″} + ξ_{8} v^{″ 2} = 0 .$ One can rewrite it in the implicit ODE form. However, obviously it is an undesirable form since it has two branches. Usually keeping only some branch can destroy the original physical relationship between the variable velocity and its derivatives. (It is verified from the numerical results later that the solutions of Eq. (16) are on both of branches.) This

Statistics for the model

We will firstly study the dynamical features of our model in the perspective of statistics. The results will also be compared with the 5-term model. For more details of the numerical schemes of the 5-term model, one can refer to [54].

The information distance or relative entropy will be considered to measure the divergence between the probability density functions (PDF) from the estimated model, $\hat{ν}$ and the experiment samples ν. It is given by $D (\hat{ν}, ν) = \int ν \log \frac{ν}{\hat{ν}} .$ In Fig. 8(a), we compare PDFs of

Conclusion and discussion

By learning the experimental data, we build the jerk equation (25) to simulate the motion of a falling sphere in the viscoelastic fluids. We believe that this nonlinear elastic system is responsible for the instability. The normalized representation covers both the classical physical laws and the nonuniform oscillations. To identify the underlying features in the DAE, we introduce the generalized Prony method with MQ quasi-interpolation scheme and obtain the parsimony result Eq. (15) with

Acknowledgments

Funding: This work was supported by NSFC Key Project (11631015); NSFC (91330201); Joint Research Fund by NSFC and Research Grants Council of Hong Kong (11461161006).

The authors thank the anonymous referees and the editor for their helpful comments on the manuscript.

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