Abstract

A liquid’s mechanical and surface energies are incontrovertibly interconvertible. The changes in internal energy of a fluid in motion that for simplified situations results in Bernoulli’s theorem fails to account for the surface energy (see the physical example below). This is due to the definition of internal energy used to derive the equation which inherently ignores surface energy [1]. This has wide ranging implications.

In this talk firstly, I will establish the deficiency of the fluid’s energy equation using the existing literature [1, 2]. Subsequently, to illustrate the implications, I will examine the exemplar flows of planar (for a simpler geometry) and circular hydraulic jump in micro-gravity. In case of a kitchen sink hydraulic jumps, in Bhagat et al.(2018) I have shown that these jumps occur when the rate of change of the kinetic energy is insufficient to provide the surface energy and the viscous dissipation. Here, I will use the Navier-Stokes equations; I will scale the viscous shallow water equation and show that liquid film’s curvature becomes singular at a finite radius when the local We = 1. I will also present experimental and numerical results, including the experiments conducted in micro-gravity showing kitchen sink hydraulic jumps are created due to surface tension and gravity does not play a significant role. I will show that the deviatoric component of the normal stress which is often ignored is the key to explaining kitchen sink hydraulic jumps. Finally, I will show the consistency of the new energy equation that includes the surface energy with the Navier Stokes equations and it’s application in predicting the disintegration of savart sheets.

Physical example: Consider an inviscid liquid jet impinging onto a flat surface and spreading radially. Ignoring gravity for simplicity, the approximately flat thin film possess minimal curvature, and according to Bernoulli’s principle, the kinetic energy of the fluid parcels remains constant. Now, take a small material volume of the fluid in the jet, and track it at a later time 𝑡, as it spreads to a radius 𝑟 in the film. It can be shown that the ratio of the interfacial area at radius 𝑟 to that in the jet is proportional to 𝑟. Accounting for the total energy of this volume, it now possesses not only its kinetic energy but also additional surface energy due to the increased interfacial area.

References: 1. Batchelor, G. K. An introduction to fluid dynamics. Cambridge university press, 2000. 2. Lighthill, M. J. Waves in fluids. Cambridge university press, 2001. 3. Bhagat RK, Jha NK, Linden PF, Wilson DI. On the origin of the circular hydraulic jump in a thin liquid film. Journal of Fluid Mechanics. 2018 Sep;851:R5.