Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet

Bosire Nyabate Dimnah; Johana Kibet Sigey; Viona Ojiambo Nakhulo

doi:doi:10.11648/j.acm.20251405.14

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Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet

Bosire Nyabate Dimnah^*

, Johana Kibet Sigey

, Viona Ojiambo Nakhulo

Published in Applied and Computational Mathematics (Volume 14, Issue 5)

Received: 30 September 2025 Accepted: 18 October 2025 Published: 31 October 2025

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Abstract

The rapid expansion of industrial and engineering activities has introduced new and complex heat transfer challenges. Many modern systems operate under conditions that require high thermal loads, thereby demanding effective heat management strategies. To address this, advancements in nanotechnology have been employed to improve the thermal performance of conventional fluids. In this study, nanoparticles were uniformly dispersed within a base fluid to enhance its thermal conductivity, leading to a significant improvement in convective heat transfer rates. The mathematical formulation of the problem involved the continuity, momentum, energy, and concentration equations, which were expressed as partial differential equations (PDEs). Through the application of similarity transformations, these equations were reduced to a system of nonlinear ordinary differential equations (ODEs). The transformed equations were then solved numerically using the collocation method (BVP4C) implemented in MATLAB, a robust solver known for its stability and accuracy in boundary value problems. The obtained results demonstrated that the presence of nanoparticles considerably improves the heat transfer characteristics of the fluid by enhancing both the temperature and velocity distributions. These findings provide valuable insights for the development of advanced nanofluids with optimized thermal properties. Such fluids hold great potential for use as efficient coolants in a wide range of applications, including electronic device cooling, automotive thermal systems, air conditioning units, and power generation equipment, where enhanced thermal management is essential for operational reliability and energy efficiency.

Published in	Applied and Computational Mathematics (Volume 14, Issue 5)
DOI	10.11648/j.acm.20251405.14
Page(s)	277-292
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Brownian Motion, Thermophoretic Effect, Copper Nanofluid, Stretching Sheet

References

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[2]	Bhatti, M. M., Lu, D., Shahid, M., & Alsharif, A. M. (2023). Combined effects of Brownian motion, thermophoresis and Joule heating in nanofluid thin film flow. International Communications in Heat and Mass Transfer, 141, 106622. https://doi.org/10.1016/j.icheatmasstransfer.2022.106622
[3]	Singh, R., Kumar, D., & Sharma, P. (2023). Nanofluid flow over a porous stretching sheet with suction and injection effects. Case Studies in Thermal Engineering, 42, 102651. https://doi.org/10.1016/j.csite.2023.102651
[4]	Khan, A., Hussain, S., Ullah, A., & Ali, I. (2023). Slip effects and thermal stratification in nanofluid flow over a stretching cylinder. Chinese Journal of Physics, 85, 109–121. https://doi.org/10.1016/j.cjph.2023.02.019
[5]	Azeem, M., Ayub, M., & Khan, N. S. (2022). Thin film flow of copper nanofluid with particle shape effects over a stretching sheet under slip and convective boundary conditions. Heat Transfer, 51(8), 7103–7120. https://doi.org/10.1002/htj.22536
[6]	Tawade, J. V., Guled, C. N., Noeiaghdam, S., Fernandez-Gamiz, U., Govindan, V., & Balamuralitharan, S. (2022). Effects of thermophoresis and Brownian motion for thermal and chemically reacting Casson nanofluid flow over a linearly stretching sheet. Results in Engineering, 15, 100448. https://doi.org/10.1016/j.rineng.2022.100448
[7]	Kumar, A., Reddy, C. V., & Shehzad, S. A. (2022). Computational study of unsteady thin film nanofluid flow with viscous dissipation. Journal of Thermal Analysis and Calorimetry, 147, 10197–10214. https://doi.org/10.1007/s10973-021-10724-0
[8]	Ramzan, M., Farooq, U., & Chung, J. D. (2022). Chemical reaction and Joule heating effects on hybrid nanofluid thin film flow with viscous dissipation. Mathematical Problems in Engineering, 2022, 9856234. https://doi.org/10.1155/2022/9856234
[9]	Waqas, M., Farooq, M. U., Hayat, T., & Alsaedi, A. (2022). Stratified nanofluid thin film flow with Brownian motion and thermophoresis over a nonlinear stretching sheet. Physics of Fluids, 34(9), 093303. https://doi.org/10.1063/5.0101200
[10]	Ahmed, S., Khan, M. I., & Sherif, E. S. M. (2022). Entropy generation in hybrid nanofluid thin film flow with chemical reaction and thermal radiation. Journal of Molecular Liquids, 360, 119391. https://doi.org/10.1016/j.molliq.2022.119391
[11]	Salih, A. (2013). Stream function-vorticity formulation. Department of Aerospace Engineering, Indian Institute of Space Science and Technology, Thiruvananthapuram-Mach, 10.
[12]	Abu-Hamdeh, N. H., Aljinaidi, A. A., Eltaher, M. A., Almitani, K. H., Alnefaie, K. A., Abusorrah, A. M., \& Safaei, M. R. (2021). Implicit finite difference simulation of Prandtl-Eyring nanofluid over a flat plate with variable thermal conductivity: a Tiwari and Das model. Mathematics, 9(24), 3153.
[13]	Jamshed, W., et al. (2021). Single-phase modeling of Cu–engine oil nanofluid flow using VIM. Alexandria Engineering Journal, 60(6), 5559–5571.
[14]	Hayat, T., et al. (2020). Darcy–Forchheimer nanofluid flow over a nonlinear stretching sur-face with radiation. International Journal of Numerical Methods for Heat & Fluid Flow, 30(7), 3723–3743.
[15]	Ullah, M. Z. (2022). Radiative and Darcy-Forchheimer hybrid nanofluid flow over an inclined stretching surface due to nonlinear convection and homogeneous heterogeneous reactions. Waves in Random and Complex Media, 1-17.
[16]	Singh, S. P., Upreti, H., & Kumar, M. (2024). Flow and heat transfer assessment in magnetized Darcy-Forchheimer flow of Casson hybrid nanofluid through cone, wedge, and plate. BioNanoScience, 14(1), 395-408.

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APA Style

Dimnah, B. N., Sigey, J. K., Nakhulo, V. O. (2025). Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet. Applied and Computational Mathematics, 14(5), 277-292. https://doi.org/10.11648/j.acm.20251405.14

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ACS Style

Dimnah, B. N.; Sigey, J. K.; Nakhulo, V. O. Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet. Appl. Comput. Math. 2025, 14(5), 277-292. doi: 10.11648/j.acm.20251405.14

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AMA Style

Dimnah BN, Sigey JK, Nakhulo VO. Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet. Appl Comput Math. 2025;14(5):277-292. doi: 10.11648/j.acm.20251405.14

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@article{10.11648/j.acm.20251405.14,
  author = {Bosire Nyabate Dimnah and Johana Kibet Sigey and Viona Ojiambo Nakhulo},
  title = {Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {5},
  pages = {277-292},
  doi = {10.11648/j.acm.20251405.14},
  url = {https://doi.org/10.11648/j.acm.20251405.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251405.14},
  abstract = {The rapid expansion of industrial and engineering activities has introduced new and complex heat transfer challenges. Many modern systems operate under conditions that require high thermal loads, thereby demanding effective heat management strategies. To address this, advancements in nanotechnology have been employed to improve the thermal performance of conventional fluids. In this study, nanoparticles were uniformly dispersed within a base fluid to enhance its thermal conductivity, leading to a significant improvement in convective heat transfer rates. The mathematical formulation of the problem involved the continuity, momentum, energy, and concentration equations, which were expressed as partial differential equations (PDEs). Through the application of similarity transformations, these equations were reduced to a system of nonlinear ordinary differential equations (ODEs). The transformed equations were then solved numerically using the collocation method (BVP4C) implemented in MATLAB, a robust solver known for its stability and accuracy in boundary value problems. The obtained results demonstrated that the presence of nanoparticles considerably improves the heat transfer characteristics of the fluid by enhancing both the temperature and velocity distributions. These findings provide valuable insights for the development of advanced nanofluids with optimized thermal properties. Such fluids hold great potential for use as efficient coolants in a wide range of applications, including electronic device cooling, automotive thermal systems, air conditioning units, and power generation equipment, where enhanced thermal management is essential for operational reliability and energy efficiency.
},
 year = {2025}
}

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TY  - JOUR
T1  - Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet

AU  - Bosire Nyabate Dimnah
AU  - Johana Kibet Sigey
AU  - Viona Ojiambo Nakhulo
Y1  - 2025/10/31
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251405.14
DO  - 10.11648/j.acm.20251405.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 277
EP  - 292
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251405.14
AB  - The rapid expansion of industrial and engineering activities has introduced new and complex heat transfer challenges. Many modern systems operate under conditions that require high thermal loads, thereby demanding effective heat management strategies. To address this, advancements in nanotechnology have been employed to improve the thermal performance of conventional fluids. In this study, nanoparticles were uniformly dispersed within a base fluid to enhance its thermal conductivity, leading to a significant improvement in convective heat transfer rates. The mathematical formulation of the problem involved the continuity, momentum, energy, and concentration equations, which were expressed as partial differential equations (PDEs). Through the application of similarity transformations, these equations were reduced to a system of nonlinear ordinary differential equations (ODEs). The transformed equations were then solved numerically using the collocation method (BVP4C) implemented in MATLAB, a robust solver known for its stability and accuracy in boundary value problems. The obtained results demonstrated that the presence of nanoparticles considerably improves the heat transfer characteristics of the fluid by enhancing both the temperature and velocity distributions. These findings provide valuable insights for the development of advanced nanofluids with optimized thermal properties. Such fluids hold great potential for use as efficient coolants in a wide range of applications, including electronic device cooling, automotive thermal systems, air conditioning units, and power generation equipment, where enhanced thermal management is essential for operational reliability and energy efficiency.

VL  - 14
IS  - 5
ER  -

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Author Information

Bosire Nyabate Dimnah

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology (JKUAT), Juja, Kenya

Contact Email

http://orcid.org/0009-0001-2639-1113
Johana Kibet Sigey

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology (JKUAT), Juja, Kenya

Contact Email

http://orcid.org/0009-0004-1790-7351
Viona Ojiambo Nakhulo

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology (JKUAT), Juja, Kenya

Contact Email

http://orcid.org/0000-0002-3311-5539

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APA Style

Dimnah, B. N., Sigey, J. K., Nakhulo, V. O. (2025). Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet. Applied and Computational Mathematics, 14(5), 277-292. https://doi.org/10.11648/j.acm.20251405.14

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ACS Style

Dimnah, B. N.; Sigey, J. K.; Nakhulo, V. O. Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet. Appl. Comput. Math. 2025, 14(5), 277-292. doi: 10.11648/j.acm.20251405.14

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AMA Style

Dimnah BN, Sigey JK, Nakhulo VO. Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet. Appl Comput Math. 2025;14(5):277-292. doi: 10.11648/j.acm.20251405.14

Copy | Download

@article{10.11648/j.acm.20251405.14,
  author = {Bosire Nyabate Dimnah and Johana Kibet Sigey and Viona Ojiambo Nakhulo},
  title = {Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {5},
  pages = {277-292},
  doi = {10.11648/j.acm.20251405.14},
  url = {https://doi.org/10.11648/j.acm.20251405.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251405.14},
  abstract = {The rapid expansion of industrial and engineering activities has introduced new and complex heat transfer challenges. Many modern systems operate under conditions that require high thermal loads, thereby demanding effective heat management strategies. To address this, advancements in nanotechnology have been employed to improve the thermal performance of conventional fluids. In this study, nanoparticles were uniformly dispersed within a base fluid to enhance its thermal conductivity, leading to a significant improvement in convective heat transfer rates. The mathematical formulation of the problem involved the continuity, momentum, energy, and concentration equations, which were expressed as partial differential equations (PDEs). Through the application of similarity transformations, these equations were reduced to a system of nonlinear ordinary differential equations (ODEs). The transformed equations were then solved numerically using the collocation method (BVP4C) implemented in MATLAB, a robust solver known for its stability and accuracy in boundary value problems. The obtained results demonstrated that the presence of nanoparticles considerably improves the heat transfer characteristics of the fluid by enhancing both the temperature and velocity distributions. These findings provide valuable insights for the development of advanced nanofluids with optimized thermal properties. Such fluids hold great potential for use as efficient coolants in a wide range of applications, including electronic device cooling, automotive thermal systems, air conditioning units, and power generation equipment, where enhanced thermal management is essential for operational reliability and energy efficiency.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Brownian Motion and Thermophoretic Effect on Thin-Film Copper Nanofluid Flow over a Stretching Sheet

AU  - Bosire Nyabate Dimnah
AU  - Johana Kibet Sigey
AU  - Viona Ojiambo Nakhulo
Y1  - 2025/10/31
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251405.14
DO  - 10.11648/j.acm.20251405.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 277
EP  - 292
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251405.14
AB  - The rapid expansion of industrial and engineering activities has introduced new and complex heat transfer challenges. Many modern systems operate under conditions that require high thermal loads, thereby demanding effective heat management strategies. To address this, advancements in nanotechnology have been employed to improve the thermal performance of conventional fluids. In this study, nanoparticles were uniformly dispersed within a base fluid to enhance its thermal conductivity, leading to a significant improvement in convective heat transfer rates. The mathematical formulation of the problem involved the continuity, momentum, energy, and concentration equations, which were expressed as partial differential equations (PDEs). Through the application of similarity transformations, these equations were reduced to a system of nonlinear ordinary differential equations (ODEs). The transformed equations were then solved numerically using the collocation method (BVP4C) implemented in MATLAB, a robust solver known for its stability and accuracy in boundary value problems. The obtained results demonstrated that the presence of nanoparticles considerably improves the heat transfer characteristics of the fluid by enhancing both the temperature and velocity distributions. These findings provide valuable insights for the development of advanced nanofluids with optimized thermal properties. Such fluids hold great potential for use as efficient coolants in a wide range of applications, including electronic device cooling, automotive thermal systems, air conditioning units, and power generation equipment, where enhanced thermal management is essential for operational reliability and energy efficiency.

VL  - 14
IS  - 5
ER  -

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