Stochastic parametric resonance of the elastic dumbbell in the viscous fluid

The article analyses the occurrence of resonance in the system of two point particles bound with the elastic thread and dipped into the random field of the fluid velocity by means of the numerical simulation technique. The study shows that the considered system of two particles can be the simplest model of the polymeric thread in the turbulent flow. We bring out the dynamic equation system for the relative motion of the particles in the random field of the fluid velocity. The relative fluid velocity dependency on the interparticle distance is taken into account. We use an Euler-Maruyama method for the numerical simulation of the stochastic processes. The dynamics of the relative motion of the particles with regard for the elastic constraint is calculated through the Verlet algorithm. The article analyses the order of the Euler-Maruyama method convergence and the order of Verlet integration approximation. The work presents the results of testing the Verlet integration by means of the correlation with the exact solutions for oscillators at resonance in the viscous fluid. We numerically set limits for resonance excitation in the elastic dumbbell for the periodic and random field of the fluid velocity.

References

[1] Frisch U. Turbulence. The legasy of A.N. Kolmogorov. Cambridge, Cambridge University Press, 1996, 312 p.

[2] Dallas V. Multiscale structure of turbulent channel flow and polymer dynamics in viscoelastic turbulence. Department of Aeronautics & Institute for Mathematical Sciences. Imperial College London, 2010, 148 p.

[3] Graham M.D. Drag reduction in turbulent flow of polymer solutions. Rheology Reviews, 2004, pp. 143–170.

[4] Kivotides D. A method for the computation of turbulent polymeric liquids including hydrodynamic interactions and chain entanglements. Physics Letters A, 2017, vol. 381, no. 6, pp. 629–635.

[5] Kuramoto Y. Rhythms and turbulence in populations of chemical oscillators. Physica A: Statistical Mechanics and its Applications, 1981, vol. 106, no. 1–2, pp. 128–143.

[6] Li F.C, Kawaguchi Y., Yu B., Wei J.J., Hishidae K. Experimental study of drag-reduction mechanism for a dilute surfactant solution flow. International Journal of Heat and Mass Transfer, 2008, vol. 51, no. 3–4, pp. 835–843.

[7] Rubinstein M., Colby R.H. Polymer physics. Oxford University Press, 2003, 443 p.

[8] Odijk T. Turbulent drag reduction in one and two dimensions. Physica A: Statistical Mechanics and its Applications, 2001, vol. 298, no. 1–2, pp. 140–154.

[9] Tsinober A. An informal introduction to turbulence. Springer, 2001, 328 p.

[10] McComb W.D. The physics of fluid turbulence. Clarendon Press, 1992, 602 p.

[11] Landau L.D., Lifshits E.M. Teoreticheskaya fizika. T. VI. Gidrodinamika [Theoretical physics. Vol. VI. Hydrodynamics]. Moscow, Nauka publ., 1988, 736 p.

[12] Hasegawa Y., Arita M. Noise-intensity fluctuation in Langevin model and its higher-order Fokker-Planck equation. Physica A: Statistical Mechanics and its Applications, 2011, vol. 390, no. 6, pp. 1051–1063.

[13] Higham D.J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 2001, vol. 43, no. 3, pp. 525–546.

[14] Liu K., Jin Y. Stochastic resonance in periodic potentials driven by colored noise. Physica A: Statistical Mechanics and its Applications, 2013, vol. 392, no. 21, pp. 5283–5288.

[15] Verlet L. Computer “Experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physical Review, 1967, vol. 159, no. 1, pp. 98–103.