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

In this paper, we will introduce a mathematical model of nonlinear jerk equation of velocity v=η1η2v″+1v″+η3v″ 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.