Impact of a spherical interface on a concentrical spherical droplet

Ahmed G. Salem; Turki D. Alharbi; Abdulaziz H. Alharbi; Anwar Ali Aldhafeeri; Ahmed G. Salem; Turki D. Alharbi; Abdulaziz H. Alharbi; Anwar Ali Aldhafeeri

doi:10.3934/math.20241378

AIMS Mathematics

2024, Volume 9, Issue 10: 28400-28420. doi: 10.3934/math.20241378

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Impact of a spherical interface on a concentrical spherical droplet

1.
Department of Mathematics, Faculty of Science, Mansoura University, 35516, Mansoura, Egypt
2.
Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Makkah, Saudi Arabia
3.
Department of Mathematics, Jamoum University College, Umm Al-Qura University, Jamoum, 25375, Makkah, Saudi Arabia
4.
Department of Mathematics and Statistics, Faculty of Science, King Faisal University, P.O. Box 400, Al Ahsa, 31982, Saudi Arabia

Received: 15 July 2024 Revised: 29 August 2024 Accepted: 06 September 2024 Published: 08 October 2024
MSC : 35Q30, 76B99, 76D05, 76T06

In this paper, an analytical and numerical technique are examined in order to analyse the Stokes flow determination problem due to a viscous sphere droplet moving at a concentric instantaneous position inside a spherical interface separating finite and semi-infinite immiscible fluid phases. Here, when only one of the three phases of the fluid (micropolar fluid) has a microstructure, attention is focused on this case. The motion is considered when Reynolds- and capillary-numbers are low, and the droplet surface and the fluid-fluid interface have insignificant deformation. A general solution is obtained in a spherical coordinate system based on a concentric position to analyse the slow axisymmetric movement of the micropolar fluid, considering microrotation and velocity components. Boundary conditions are initially fulfilled at the fluid-fluid interface and subsequently at the droplet surface. The normalised hydrodynamic drag force applying to a moving viscous droplet appears to be a function of the droplet-to-interface radius ratio, which increases monotonically and becomes unbounded when the droplet surface touches the fluid-fluid interface. The numerical outcomes of the normalised drag force acting on the viscous droplet are derived for different values of the parameters, and are presented in a tabular and graphical framework. A comparison was made between our numerical outcomes for the drag force and the pertinent data for the special cases found in the literature.
- micropolar fluid,
- axisymmetric motion,
- low Reynolds numbers,
- normalised hydrodynamic drag force,
- fluid-fluid interface effect
Citation: Ahmed G. Salem, Turki D. Alharbi, Abdulaziz H. Alharbi, Anwar Ali Aldhafeeri. Impact of a spherical interface on a concentrical spherical droplet[J]. AIMS Mathematics, 2024, 9(10): 28400-28420. doi: 10.3934/math.20241378

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Abstract

In this paper, an analytical and numerical technique are examined in order to analyse the Stokes flow determination problem due to a viscous sphere droplet moving at a concentric instantaneous position inside a spherical interface separating finite and semi-infinite immiscible fluid phases. Here, when only one of the three phases of the fluid (micropolar fluid) has a microstructure, attention is focused on this case. The motion is considered when Reynolds- and capillary-numbers are low, and the droplet surface and the fluid-fluid interface have insignificant deformation. A general solution is obtained in a spherical coordinate system based on a concentric position to analyse the slow axisymmetric movement of the micropolar fluid, considering microrotation and velocity components. Boundary conditions are initially fulfilled at the fluid-fluid interface and subsequently at the droplet surface. The normalised hydrodynamic drag force applying to a moving viscous droplet appears to be a function of the droplet-to-interface radius ratio, which increases monotonically and becomes unbounded when the droplet surface touches the fluid-fluid interface. The numerical outcomes of the normalised drag force acting on the viscous droplet are derived for different values of the parameters, and are presented in a tabular and graphical framework. A comparison was made between our numerical outcomes for the drag force and the pertinent data for the special cases found in the literature.

References

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