Qualitative analysis on the electrohydrodynamic flow equation

Lazhar Bougoffa; Ammar Khanfer; Smail Bougouffa; Lazhar Bougoffa; Ammar Khanfer; Smail Bougouffa

doi:10.3934/math.2024040

AIMS Mathematics

2024, Volume 9, Issue 1: 775-791. doi: 10.3934/math.2024040

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Research article

Qualitative analysis on the electrohydrodynamic flow equation

1.
Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia
2.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia
3.
Department of Physics, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia

Received: 04 October 2023 Revised: 16 November 2023 Accepted: 23 November 2023 Published: 04 December 2023
MSC : 35R35, 34L15, 34G20

In this paper, we present a comprehensive analysis of the lower and upper bounds of solutions for a nonlinear second-order ordinary differential equation governing the electrohydrodynamic flow of a conducting fluid in cylindrical conduits. The equation describes the radial distribution of the flow velocity in an "ion drag" configuration, which is affected by an externally applied electric field. Our study involves the establishment of rigorous analytical bounds on the radial distribution, taking into account the Hartmann number $ H $ and a parameter $ \alpha. $ An analytic approximate solution is obtained as an improvement of the a priori estimates and it is found to exhibit strong agreement with numerical solutions, particularly when considering small Hartmann numbers. Further, estimates for the central velocity $ w(0) $ of the fluid occurring at the center of the cylindrical conduit were also established, and some interesting relationships were found between $ H $ and $ \alpha. $ These findings establish a framework that illuminates the potential range of values for the physical parameter within the conduit.
- differential equation of electrohydrodynamics,
- approximate solution,
- lower and upper bounds
Citation: Lazhar Bougoffa, Ammar Khanfer, Smail Bougouffa. Qualitative analysis on the electrohydrodynamic flow equation[J]. AIMS Mathematics, 2024, 9(1): 775-791. doi: 10.3934/math.2024040

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Abstract

In this paper, we present a comprehensive analysis of the lower and upper bounds of solutions for a nonlinear second-order ordinary differential equation governing the electrohydrodynamic flow of a conducting fluid in cylindrical conduits. The equation describes the radial distribution of the flow velocity in an "ion drag" configuration, which is affected by an externally applied electric field. Our study involves the establishment of rigorous analytical bounds on the radial distribution, taking into account the Hartmann number $ H $ and a parameter $ \alpha. $ An analytic approximate solution is obtained as an improvement of the a priori estimates and it is found to exhibit strong agreement with numerical solutions, particularly when considering small Hartmann numbers. Further, estimates for the central velocity $ w(0) $ of the fluid occurring at the center of the cylindrical conduit were also established, and some interesting relationships were found between $ H $ and $ \alpha. $ These findings establish a framework that illuminates the potential range of values for the physical parameter within the conduit.

References

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