Growth of lipid-coated multi-microbubbles in viscoelastic tissues
Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, 32511, Shebin El-Koom, Egypt
2 Department of Theoretical and Mathematical Physics, Ural Federal University, 620083, Ekaterinburg, Russia
3 Fluid Dynamics and Seismics Lab, Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow, Russia
Accepted: 27 March 2022
Published online: 27 April 2022
In this work, we propose the theoretical and mathematical approaches of lipid-coated multi-microbubbles in viscoelastic tissues during the growth process of microscopic bubbles. The mathematical approaches have two systems of microbubbles. The first one is a system of a lipid-coated single microbubble, and the second one is the system of multi-microbubbles. The dynamics of the lipid-coated single and multi-microbubbles occur under the effect of constant and variable surface tension. The mathematical models are formulated based on the mass equation, modified Keller–Miksis equation, and concentration equation. The system of nonlinear modified Keller–Miksis equations is solved analytically by using the modification of the Plesset–Zwick method during the lipid-coated microbubbles growth process. We consider that the lipid shells and viscoelastic medium are taken to build and study the growth of lipid-coated multi-microbubbles. The number of microbubbles “n” and some physical parameters as effects of bubble–bubble interactions, Van der Waals hard-core radius, elasticity modulus, polytropic exponent “κ” and others are investigated and studied on the growing of lipid-coated single and multi-microbubbles. Our results demonstrate that the appearance of the lipid-coated multi-microbubbles can significantly decrease the radii of microbubbles during the growth process. Moreover, the lipid shells of single microbubbles are more significant than in the case of multi-microbubbles at through considering the effect of surface tension.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2022