The Soft Point and Its Applications in Body Falling

Abstract

From the mesoscopic point of view, a definition of soft point is introduced by considering the attributes of geometric profile and mass distribution. After that, this concept is used to develop the soft matching technique to simulate the chaotic behaviors of the equations. Especially, a tennis model with deformation factor a(t) is proposed to derive a generalized Newton-Stokes equation v′(t) = λ(v _T − a(t)v(t)). Furthermore, a concept of duality of deformation factor a(t) and velocity v(t) with respect to the generalized Newton-Stokes equation is established. To solve this equation, two data-driven models of a(t) are provided, one is based on the concept of soft matching, while the other is by using the amplitude modulation. Finally, the related iterative algorithm is developed to simulate the motion of the falling body via the duality of a(t) and v(t). Numerical examples successfully demonstrate the phenomenon of chaos, which consists of the continual random oscillations and sudden accelerations. Moreover, the algorithm is tested by using larger coefficients corresponding to the terminal velocity and shows more satisfactory results. It may enable us to characterize the total energy of the dynamical system more accurately.

Keywords

Soft point / Soft matching / Newton-Stokes equation / Duality / Data-driven model

References

[1] Batchelor G K. An Introduction to Fluid Dynamics, 1967, Cambridge: Cambridge University Press

[2] Bird R B, Armstrong R C, Hassager O. Dynamics of Polymetric Liquids, 1987, New York: Wiley Interscience 1

[3] Chen S, Rothstein J P. Flow of wormlike micelle solutions past a confined circular sphere. J. Non-Newton. Fluid., 2004, 116: 205-234

[4] Chhabra R P. Bubbles, Drops and Particles in Non-Newtonian Fluids, 1993, New York: CRC Press

[5] Mckinley G H. Steady and Transient Motion of Spherical Particles in Viscoelastic Liquids, Transport Processes in Bubble, Drops and Particles, 2002, New York: Taylor & Francis

[6] Rajagopalan D, Arigo M T, McKinley G H. The sedimentation of a sphere through an elastic fluid part 2, Transient motion. J. Non-Newton. Fluid., 1996, 65(1): 17-46

[7] Jayaraman A, Belmonte A. Oscillations of a solid sphere falling through a wormlike micellar fluid. Phys. Rev. E., 2003, 67(6): 065301

[8] Lee Y J. Modeling and Simulations of Non-Newtonian Fluid Flows, 2004, University Park, PA: The Pennsylvania State University

[9] Wu Z M, Zhang R. Data-driven modeling for the motion of a sphere falling through a non-Newtonian fluid. Commun. Math. Sci., 2018, 16(2): 425-439

[10] Wu Z M, Zhang R. Learning physics by data for the motion of a sphere falling in a non-Newtonian fluid. Commun. Nonlinear. Sci., 2019, 67: 577-593

[11] Goldman R. On the algebraic and geometric foundations of computer graphics. Acm. T. Graphic, 2002, 21(1): 52-86

[12] Goldman R. The ambient spaces of computer graphics and geometric modeling. IEEE Comput. Graph, 2000, 22(2): 76-84

[13] Goldman R. An Integrated Introduction to Computer Graphics and Geometric Modeling, 2009, New York: CRC Press

[14] Belmonte A. Self-oscillations of a cusped bubble rising through a micellar solution. Rheol. Aata., 2000, 39(6): 554-559

[15] Hassager O. Negative wake behind bubbles in non-Newtonian liquids. Nature, 1979, 279(5712): 402-403