Analytical and numerical investigation of viscous fluid-filled spherical slip cavity in a spherical micropolar droplet

Abdulaziz H. Alharbi; Ahmed G. Salem; Abdulaziz H. Alharbi; Ahmed G. Salem

doi:10.3934/math.2024732

AIMS Mathematics

2024, Volume 9, Issue 6: 15097-15118. doi: 10.3934/math.2024732

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Analytical and numerical investigation of viscous fluid-filled spherical slip cavity in a spherical micropolar droplet

Abdulaziz H. Alharbi ^{1
,
,},
Ahmed G. Salem ^{2
,
,}

1.
Department of Mathematics, Jamoum University College, Umm Al-Qura University, Jamoum, 25375, Makkah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, 35516, Mansoura, Egypt

Received: 20 February 2024 Revised: 23 March 2024 Accepted: 19 April 2024 Published: 26 April 2024
MSC : 35Q30, 76B99, 76D05, 76T06

This article presents an analytical and numerical investigation on the quasi-steady, slow flow generated by the movement of a micropolar fluid drop sphere of at a concentrical position within another immiscible viscous fluid inside a spherical slip cavity. Additionally, the effect of a cavity with slip friction along with the change in the micropolarity parameter on the movement of the fluid sphere is introduced. When Reynolds numbers are low, the droplet moves along a diameter that connects their centres. The governing and constitutive differential equations are reduced to a computationally convenient form using appropriate transformations. By using the resulting linear partial differential equations for the stream functions and using the method of separation variables, we can obtain their solutions. General solutions for velocity fields are found using spherical coordinate systems, which are based on the concentric point of the cavity; this allows to obtain solutions to the Navier-Stokes equations internal and external to the spherical droplet. The vorticity-microrotation boundary condition is used in regard to the micropolar droplet case in a viscous fluid. The normalised drag forces acted upon the micropolar drop are illustrated via graphs and tables for diverse values of the viscosity ratio and drop-to-wall radius ratio, with the change of the spin parameter that attaches the microrotation to vorticity. The correction wall factor is shown to increase with an increase in the drop-to-wall radius ratio, when moving from the gas bubble case to the solid sphere case, with an increase in the micropolarity parameter, and with an increase in the slip frictional resistance. This study is relevant due to its potential uses in a variety of biological, natural, and industrial processes, including the creation of raindrops, the investigation of blood flow, fluid-fluid extraction, the forecasting of weather conditions, the rheology of emulsions, and sedimentation phenomena.
- low Reynolds numbers,
- micropolar fluid,
- vorticity-microrotation condition,
- spherical slip cavity
Citation: Abdulaziz H. Alharbi, Ahmed G. Salem. Analytical and numerical investigation of viscous fluid-filled spherical slip cavity in a spherical micropolar droplet[J]. AIMS Mathematics, 2024, 9(6): 15097-15118. doi: 10.3934/math.2024732

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Abstract

This article presents an analytical and numerical investigation on the quasi-steady, slow flow generated by the movement of a micropolar fluid drop sphere of at a concentrical position within another immiscible viscous fluid inside a spherical slip cavity. Additionally, the effect of a cavity with slip friction along with the change in the micropolarity parameter on the movement of the fluid sphere is introduced. When Reynolds numbers are low, the droplet moves along a diameter that connects their centres. The governing and constitutive differential equations are reduced to a computationally convenient form using appropriate transformations. By using the resulting linear partial differential equations for the stream functions and using the method of separation variables, we can obtain their solutions. General solutions for velocity fields are found using spherical coordinate systems, which are based on the concentric point of the cavity; this allows to obtain solutions to the Navier-Stokes equations internal and external to the spherical droplet. The vorticity-microrotation boundary condition is used in regard to the micropolar droplet case in a viscous fluid. The normalised drag forces acted upon the micropolar drop are illustrated via graphs and tables for diverse values of the viscosity ratio and drop-to-wall radius ratio, with the change of the spin parameter that attaches the microrotation to vorticity. The correction wall factor is shown to increase with an increase in the drop-to-wall radius ratio, when moving from the gas bubble case to the solid sphere case, with an increase in the micropolarity parameter, and with an increase in the slip frictional resistance. This study is relevant due to its potential uses in a variety of biological, natural, and industrial processes, including the creation of raindrops, the investigation of blood flow, fluid-fluid extraction, the forecasting of weather conditions, the rheology of emulsions, and sedimentation phenomena.

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