Rolling Spheres: Extending a quest from Galileo’s age

Muhammad Rizwanur Rahman, Prashant R. Waghmare Department of Mechanical Engineering, University of Alberta. mrahman6@ualberta.ca

Keywords: rolling, spheres, droplets, viscous medium.

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The bigger they are the faster they roll, unless they’re too big!

Previously, it was believed that the larger a droplet, the slower it would roll down an incline. Our recent research has demonstrated that, in fact, larger droplets roll faster, unless they are too big. Droplets that are larger than a ‘characteristic size’ will roll more slowly than smaller droplets. Previous researchers could not see this phenomenon because they always experimented in air. Our work shows that the characteristic size, above which droplets begin to roll more slowly, is determined by the viscosity of the surrounding medium. A more viscous medium will increase the characteristic size, above which droplets will roll more slowly. For example, a droplet in water will show a characteristic size, whereas the characteristic size of a droplet in air is so small that researchers never see it. This work elucidates a problem of fundamental physics and has far reaching consequences for research and applications in many fields of engineering and science, including problems of 3D printing, biotechnology, and areas that rely upon micro-fluidics.

Creeping motion of a spherical body immersed in viscous fluids has always attracted scientists and researchers for detailed investigation. Stokes [1], in his pioneering study, discovered that a solid body, falling through a viscous liquid, decelerates quickly prior to attaining the steady terminal velocity. Numerous attempts have been made to investigate the role of different operating parameters while studying such motion in stagnant or moving fluids. In most of the cases, such studies are devoted to analyzing rigid (solid sphere) body motion. When we refer to the classical work of Galileo (~1602) [2] discovering constant acceleration of such a body, a sharp contrast with the observations of Stokes cannot go unnoticed. This is because of the dissipation from the surrounding medium which becomes significant in Stokes flow but negligible in Galileo’s situation. It is well known [1] that in Stokes flow, the resistance force from the surrounding medium results the fall with a diminishing acceleration until it reaches a zero acceleration and attains constant terminal velocity. Contrary to Galileo’s experiment, resistance from the medium becomes substantial in Stoke’s flow. Thus, the continuous diminishing acceleration and the attainment of steady velocity of descent can be attributed to this viscous resistance. In response to Stoke’s observations detailed analyses were performed by numerous researchers to identify the role of the surrounding media in case of rigid body motion [3–5]. Further, Saffman [6] suggested that a rotating and translating sphere subjected to uniform shear experiences a lift force and a side force deflecting the sphere’s trajectory. In a detailed analysis, Bico et al. [7] reported the importance of sliding motion along with the rolling motion of a sphere coated with a viscous layer moving on an incline. In such cases, the sphere size and density as well as the surface tension and thickness of the viscous layer present between the rolling body and the attached substrate affect the steady velocity of descent.

Though the discussion on the solid body motion has long been addressed, the recent development in microfluidic technology demands a detailed understanding of the dynamics of liquid drops in viscous medium for numerous applications. With, the emergence of multiphase microfluidics, the understanding of the fundamental physics of such bodies has become more pronounced. We studied the complex dynamics of compound drops where a smaller inner drop is contained in a bigger outer drop of immiscible liquid. With our effort in investigating the rolling dynamics of a double drop system by varying the volume ratio of the inner to the outer drop, the experiments seemed to exhibit quite unexpected and indecipherable results at first sight with almost no correlation among them. This highlighted a gap in our understanding of the single phase drop motion in a viscous medium. In the early '90s, scaling arguments byof Mahadevan and Pomeau [8] predicted a surprising feature of a single phase droplet rolling on an incline in air medium. According to this argument, the velocity of such a drop should decrease with the increase of its size and this was attributed to the increase in viscous dissipation at the contact area for larger droplets. This claim was later experimentally proveddemonstrated by Richard and Qu´er´e [9]. However, as we see in the historical development of physics (as from Galileo to Stoke), the consideration of a viscous medium significantly alters the outcome.

In our study [10], we examined the scaling analysis [8] for a single-phase drop rolling on an incline in a viscous medium. The theoretical development with this medium consideration resulted in emanation of a velocity behavior that is opposite to the literature [8, 9]. But with increase in drop size, when we reached above a certain threshold, Mahadevan and Pomeau behavior was apparent. However, experimental demonstration of the Mahadevan-Pomeau behavior in the complicated case where the drop is rolling under a viscous medium is not only difficult, rather, to some extent, impossible. Here we exploited the idea of double drop system which allowed us to observe both the behaviors.

Counterintuitively, smaller droplets rolls slower than the bigger ones. This is attributed to the contact resistance. We have observed that the opposite is also true when a critical size is reached. Our study also suggests, by manipulating medium and contact resistance, one can obtain increasing as well as decreasing behaviors for the same drop. The scaling arguments established one additional motion behavior and divided the rolling motion in two different regimes. The study of a drop rolling in air medium seems to contrast that in a viscous medium. But when understandings from the study of the double drop system is blended together with the single drop scenario, the contrast turns into coherence.

References:

[1] G. G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums. Pitt Press Cambridge, 1851, vol. 9.

[2] G. Galili, “Dialogues concerning two new sciences,” New York: Macmillan, vol. 1, p. 105, 1914.

[3] R.Woodward, Transactions of the New York Academy of Sciences, p. 2, 1895.

[4] H. Allen, “L. The motion of a sphere in a viscous fluid,” Philosophical Magazine Series 5, vol. 50, no. 306, pp. 519–534, 1900.

[5] L. Rayleigh, “On the motion of solid bodies through viscous liquid,” Philosophical Magazine Series 6, vol. 21, no. 126, pp. 697–711, 1911.

[6] P. G. Saffman, “The lift on a small sphere in a slow shear flow,” Journal of Fluid Mechanics, vol. 22, no. 02, p. 385, 1965. https://doi.org/10.1017/S0022112065000824

[7] J. Bico, J. Ashmore-Chakrabarty, G. McKinley, and H. Stone, “Rolling stones: The motion of a sphere down an inclined plane coated with a thin liquid film,” Physics of fluids, vol. 21, no. 8, p. 082103, 2009. https://doi.org/10.1063/1.3207884

[8] L. Mahadevan and Y. Pomeau, “Rolling droplets,” Physics of fluids, vol. 11, no. 9, pp. 2449 2453, 1999. https://doi.org/10.1063/1.870107

[9] D. Richard and D. Qu´er´e, “Viscous drops rolling on a tilted non-wettable solid,” Europhysics Letters, vol. 48, no. 3, p. 286, 1999. https://doi.org/10.1209/epl/i1999-00479-1

[10] Rahman MR, Waghmare PR. Influence of outer medium viscosity on the motion of rolling droplets down an incline. Physical Review Fluids. 2018 Feb 2;3(2):023601. https://doi.org/10.1103/PhysRevFluids.3.023601