A tractable mathematical model for rectified diffusion

Abstract

A core unresolved challenge in rectified diffusion is the lack of predictive models, which allow accurate modelling of the growth of bubbles over millions of cycles of oscillation. The spherically symmetric problem involves two asymptotic regions in space for the convection and diffusion of the dissolved gas in the liquid, an inner region adjacent to the bubble and an outer region. It also involves three time scales: the shortest time scale associated with the bubble oscillation, the intermediate and the longest time scales for the mass transport across the bubble interface and the outer region, respectively. The problem is analysed systematically using matched asymptotic expansions in space and multi-scales in time, the large parameter being the Péclet number for mass transfer. A tractable mathematical model is formulated for rectified diffusion. It comprises the Rayleigh–Plesset equation for the spherical oscillations of the bubble on the shortest time scale, an ordinary differential equation for the mass of gas in the bubble on the intermediate time scale and a diffusion equation on the longest time scale and the long length scale. Predictions of this simplified model have excellent agreement with experimental results for a single spherical bubble in the bulk over millions of cycles of oscillation. A parametric study is undertaken with several new phenomena/features observed for acoustic frequencies in the range 12 to 62 kHz, acoustic field $0.12$ to $0.22$ bar and the initial gas concentration in the liquid being $0.95$ to $1.0$ of the saturation concentration.

The role of surface tension on the growth of bubbles by rectified diffusion

“…The well-established physical model [8] , [9] includes the Rayleigh–Plesset equation for the dimensional spherical bubble radius , the convection–diffusion equation for the dimensional concentration of gas in the liquid and conservation of mass for the dimensional mass of gas contained in the bubble . In the same order, these equations are (1) , (3) , (4) as follows where the pressure of the liquid at the bubble surface less the external applied pressure is [28] given by and is given by with appropriate boundary and initial conditions.…”

“…Unfortunately, bubble growth through rectification takes millions of oscillation cycles, the numerical simulations of the process have proved to be very challenging with the convection-enhanced diffusion of gas across the bubble interface being the main difficulty. An analytical expression was found for the gas flux at the bubble interface without neglecting convection [8] , [9] which has allowed accurate predictions of the experimental results for rectified diffusion in air–water systems [10] , [11] , [12] , [13] .…”

“…The tractable version of the well-established physical model is outlined in § 2 for completeness. For the detailed model, readers should refer to our previous articles [8] , [9] . In § 3 , a study based on the variation of the surface tension is undertaken.…”

Mass transfer of microbubble in liquid under multifrequency acoustic excitation - A theoretical study

“…Additionally, Zhang [35] , [36] discussed the rectified mass diffusion of non-Newtonian fluids, such as viscoelastic fluids found in various organisms. Recently, Smith and Wang introduced an exceptional model for bubble growth in liquid through rectified diffusion [37] , [38] . Their model offers a straightforward yet precise solution for gas behavior in liquid, enabling accurate predictions of bubble growth over millions of oscillation cycles [37] , [38] .…”

“…Recently, Smith and Wang introduced an exceptional model for bubble growth in liquid through rectified diffusion [37] , [38] . Their model offers a straightforward yet precise solution for gas behavior in liquid, enabling accurate predictions of bubble growth over millions of oscillation cycles [37] , [38] . Notably, their findings highlight the significant influence of shell and area effects on bubble growth in liquid with bulk surfactant concentrations below 2.4 mM, underscoring the importance of surface tension in rectified diffusion for aqueous surfactant solutions [39] .…”