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The history effect in bubble growth and dissolution. Part 1. Theory

Published online by Cambridge University Press:  30 June 2016

Pablo Peñas-López ,
Miguel A. Parrales ,
Javier Rodríguez-Rodríguez  and
Devaraj van der Meer
Pablo Peñas-López*
Affiliation:
Fluid Mechanics Group, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés (Madrid), Spain
Miguel A. Parrales
Affiliation:
Fluid Mechanics Group, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés (Madrid), Spain
Javier Rodríguez-Rodríguez
Affiliation:
Fluid Mechanics Group, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés (Madrid), Spain
Devaraj van der Meer
Affiliation:
Physics of Fluids Group, Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
*
†Email address for correspondence: papenasl@ing.uc3m.es
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Abstract

The term ‘history effect’ refers to the contribution of any past mass transfer events between a gas bubble and its liquid surroundings towards the current diffusion-driven growth or dissolution dynamics of that same bubble. The history effect arises from the (non-instantaneous) development of the dissolved gas concentration boundary layer in the liquid in response to changes in the concentration at the bubble interface caused, for instance, by variations of the ambient pressure in time. Essentially, the history effect amounts to the acknowledgement that at any given time the mass flux across the bubble is conditioned by the preceding time history of the concentration at the bubble boundary. Considering the canonical problem of an isolated spherical bubble at rest, we show that the contribution of the history effect in the current interfacial concentration gradient is fully contained within a memory integral of the interface concentration. Retaining this integral term, we formulate a governing differential equation for the bubble dynamics, analogous to the well-known Epstein–Plesset solution. Our equation does not make use of the quasi-static radius approximation. An analytical solution is presented for the case of multiple step-like jumps in pressure. The nature and relevance of the history effect is then assessed through illustrative examples. Finally, we investigate the role of the history effect in rectified diffusion for a bubble that pulsates under harmonic pressure forcing in the non-inertial, isothermal regime.

JFM classification

Drops and Bubbles: Bubble dynamics Drops and Bubbles: Drops and Bubbles

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Papers
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Journal of Fluid Mechanics , Volume 800 , 10 August 2016 , pp. 180 - 212
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© 2016 Cambridge University Press 

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