MatLab Bvp4c Technique to Compute Thermophoresis and Brownian Motion in Nanofluid Flow Over a Transient Stretching Sheet

Atifa Latif; Saman Iftikhar; Hadia Ali; Syed S. Naqvi

Authors

Atifa Latif Department of Mathematics, GC University Faisalabad Pakistan
Saman Iftikhar Faculty of Computer Studies, Saudi Arabia, Arab Open University, Jeddah, Saudi Arabia
Hadia Ali Department of Mathematics, GC University Faisalabad Pakistan
Syed Sheraz Naqvi Department of Mathematics, GC University Faisalabad Pakistan

Keywords:

Bvp4c approach, Lobato-IIIA, Computer Applications, MatLab Solver, Mathematica NDSolve, Nano fluid, Heat transfer, Boundary layer, Unsteady Stretching sheet, Mass transfer

Abstract

This physical phenomenon examined the transport mechanisms of heat and mass within a nanofluid thin film. The nanofluid thin film is situated over an unsteady stretching sheet, which is one of the pioneering contributions to the field, focusing specifically on the flow dynamics of nanofluid thin films. This foundational framework is established by Buongiorno’s fluid model. The mathematical model is applied for the evaluation of the nanofluid film, which adeptly weaves in significant phenomena, including Brownian motion as well as thermophoresis. The mathematical model is achieved in the form of non-linear partial differential equations (PDEs) for computation with the help of computer applications. Firstly, the analytical framework of similarity transformations is applied to non-linear PDEs to convert them into ordinary differential equations (ODEs). Secondly, these ODEs have been critically examined and prepared for coding in MatLab by reducing their high order into first order. The software Mathematica and MatLab have been employed to solve the boundary value problem (BVP). The built-in BVP4c solver is applied to obtain accurate solutions in the form of graphs and numerical values. The current analysis yields significant results revealing that both the free surface temperature and the volume fraction of nanoparticles tend to increase in response to variations in both unsteady conditions and magnetic parameters. Furthermore, the outcomes demonstrate that the interaction among diverse nanofluid variables with the phenomenon of viscous energy loss contributes to a reduction in the overall heat transfer rate. The potential effect of these proficient thermal management techniques is crucial, especially in microelectronics and energy systems.

References

iaeme iaeme, “COMBINED HEAT AND MASS TRANSFER IN MHD THREE-DIMENSIONAL POROUS FLOW WITH PERIODIC PERMEABILITY & HEAT ABSORPTION,” Int. J. Mech. Eng. Technol. , Accessed: Sep. 16, 2025. [Online]. Available: https://www.academia.edu/2464148/COMBINED_HEAT_AND_MASS_TRANSFER_IN_MHD_THREE_DIMENSIONAL_POROUS_FLOW_WITH_PERIODIC_PERMEABILITY_and_HEAT_ABSORPTION

M. I. Khan, S. Qayyum, S. Kadry, W. A. Khan, and S. Z. Abbas, “Theoretical investigations of entropy optimization in electro-magneto nonlinear mixed convective second order slip flow,” J. Magn., vol. 25, no. 1, pp. 8–14, 2020, doi: 10.4283/JMAG.2020.25.1.008,.

A. Latif, M. A. Rana, and M. Hussan, “Periodic permeable free convective 3-dimensional flow of a second grade fluid with slip effect,” Phys. Scr., vol. 96, no. 8, p. 085207, May 2021, doi: 10.1088/1402-4896/ABC282.

F. A. M. Ijaz Khan, “Heat transport and nonlinear mixed convective nanomaterial slip flow of Walter-B fluid containing gyrotactic microorganisms,” Alexandria Eng. J., vol. 59, no. 3, 2020, doi: 10.1016/j.aej.2020.04.042.

D. H. Bagh Ali, Sajjad Hussain, Yufeng Nie, Ahmed Kadhim Hussein, “Finite element investigation of Dufour and Soret impacts on MHD rotating flow of Oldroyd-B nanofluid over a stretching sheet with double diffusion Cattaneo Christov heat flux model,” Powder Technol., vol. 377, pp. 439–452, 2021, doi: https://doi.org/10.1016/j.powtec.2020.09.008.

L. A. Jawad Ahmed, Masood Khan, “Transient thin film flow of nonlinear radiative Maxwell nanofluid over a rotating disk,” Phys. Lett. A, vol. 383, no. 12, pp. 1300–1305, 2019, doi: https://doi.org/10.1016/j.physleta.2019.01.024.

S. N. Ali Sulaiman Alsagri, “MHD Thin Film Flow and Thermal Analysis of Blood with CNTs Nanofluid,” Coatings, vol. 9, no. 3, p. 175, 2019, doi: https://doi.org/10.3390/coatings9030175.

N. A. & N. H. Zeeshan, Haroon Ur Rasheed, Waris Khan, Ilyas Khan, “Numerical computation of 3D Brownian motion of thin film nanofluid flow of convective heat transfer over a stretchable rotating surface,” Sci. Rep., 2002, [Online]. Available: https://www.nature.com/articles/s41598-022-06622-9

C. Sulochana and S. R. Aparna, “Unsteady magnetohydrodynamic radiative liquid thin film flow of hybrid nanofluid with thermophoresis and Brownian motion,” Multidiscip. Model. Mater. Struct., vol. 16, no. 4, pp. 811–834, Dec. 2019, doi: 10.1108/MMMS-08-2019-0160.

Z. S. Saleem Nasir, “Darcy Forchheimer nanofluid thin film flow of SWCNTs and heat transfer analysis over an unsteady stretching sheet,” AIP Adv., vol. 9, p. 015223, 2019, doi: https://doi.org/10.1063/1.5083972.

G. C. Yanhai Lin, Liancun Zheng, Xinxin Zhang, Lianxi Ma, “MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generation,” Int. J. Heat Mass Transf., vol. 84, pp. 903–911, 2015, doi: https://doi.org/10.1016/j.ijheatmasstransfer.2015.01.099.

A. D. Zahir Shah, “Impact of Nonlinear Thermal Radiation on MHD Nanofluid Thin Film Flow over a Horizontally Rotating Disk,” Appl. Sci., vol. 9, no. 8, p. 1533, 2019, doi: https://doi.org/10.3390/app9081533.

S. A. R.C. AZIZ, I. HASHIM, “Flow and Heat Transfer in a Nanofluid Thin Film over an Unsteady Stretching Sheet,” Sains Malaysiana, vol. 47, no. 7, pp. 1599–1605, 2018, [Online]. Available: https://www.ukm.my/jsm/pdf_files/SM-PDF-47-7-2018/31 R.C. Aziz.pdf

J. J. Zhao, Y. Y. Duan, X. D. Wang, and B. X. Wang, “Effect of nanofluids on thin film evaporation in microchannels,” J. Nanoparticle Res., vol. 13, no. 10, pp. 5033–5047, Oct. 2011, doi: 10.1007/S11051-011-0484-Y/METRICS.

J. Ahmed, M. Khan, and L. Ahmad, “Transient thin-film spin-coating flow of chemically reactive and radiative Maxwell nanofluid over a rotating disk,” Appl. Phys. A Mater. Sci. Process., vol. 125, no. 3, pp. 1–17, Mar. 2019, doi: 10.1007/S00339-019-2424-0/METRICS.

Z. A. Alhussain and A. Tassaddiq, “Thin Film Blood Based Casson Hybrid Nanofluid Flow with Variable Viscosity,” Arab. J. Sci. Eng., vol. 47, no. 1, pp. 1087–1094, Jan. 2022, doi: 10.1007/S13369-021-06067-8/METRICS.

I. Asad Ullah, “A Magnetite–Water-Based Nanofluid Three-Dimensional Thin Film Flow on an Inclined Rotating Surface with Non-Linear Thermal Radiations and Couple Stress Effects,” Energies, vol. 14, no. 17, p. 5531, 2021, doi: https://doi.org/10.3390/en14175531.

J. Buongiorno, “Convective Transport in Nanofluids,” J. Heat Transfer, vol. 128, no. 3, pp. 240–250, Mar. 2006, doi: 10.1115/1.2150834.

M. E. A. Faroogh Garoosi, Leila Jahanshaloo, Mohammad Mehdi Rashidi c, Arash Badakhsh, “Numerical simulation of natural convection of the nanofluid in heat exchangers using a Buongiorno model,” Appl. Math. Comput., vol. 554, pp. 183–203, 2015, doi: https://doi.org/10.1016/j.amc.2014.12.116.

M. M. R. M. Sheikholeslami, D.D. Ganji, “Magnetic field effect on unsteady nanofluid flow and heat transfer using Buongiorno model,” J. Magn. Magn. Mater., vol. 416, pp. 164–173, 2016, doi: https://doi.org/10.1016/j.jmmm.2016.05.026.

N. B. Nor Ashikin Abu Bakar, “Nanofluid Flow using Buongiorno Model over a Stretching Sheet and Thermophysical Properties of Nanoliquids,” Indian J. Sci. Technol., vol. 9, 2016, [Online]. Available: https://indjst.org/articles/nanofluid-flow-using-buongiorno-model-over-a-stretching-sheet-and-thermophysical-properties-of-nanoliquids

M. Qasim, Z. H. Khan, R. J. Lopez, and W. A. Khan, “Heat and mass transfer in nanofluid thin film over an unsteady stretching sheet using Buongiorno’s model,” Eur. Phys. J. Plus 2016 1311, vol. 131, no. 1, pp. 1–11, Jan. 2016, doi: 10.1140/EPJP/I2016-16016-8.

S. T. Mohyud-Din, U. Khan, N. Ahmed, and B. Bin-Mohsin, “Heat and mass transfer analysis for MHD flow of nanofluid inconvergent/divergent channels with stretchable walls using Buongiorno’s model,” Neural Comput. Appl., vol. 28, no. 12, pp. 4079–4092, Dec. 2017, doi: 10.1007/S00521-016-2289-5/METRICS.

A. Mishra and M. Kumar, “Numerical analysis of MHD nanofluid flow over a wedge, including effects of viscous dissipation and heat generation/absorption, using Buongiorno model,” Heat Transf., vol. 50, no. 8, pp. 8453–8474, Dec. 2021, doi: 10.1002/HTJ.22284.

M. Mustafa, “MHD nanofluid flow over a rotating disk with partial slip effects: Buongiorno model,” Int. J. Heat Mass Transf., vol. 108, pp. 1910–1916, 2017, doi: https://doi.org/10.1016/j.ijheatmasstransfer.2017.01.064.

A. V. R. Natalia C. Roşca, “Flow and Heat Transfer Past a Stretching/Shrinking Sheet Using Modified Buongiorno Nanoliquid Model,” Mathematics, vol. 9, no. 23, p. 3047, 2021, doi: https://doi.org/10.3390/math9233047.

F. S. A.-M. Sohail Nadeem, Wang Fuzhang, Fahad M. Alharbi, Farrah Sajid, Nadeem Abbas, A.S. El-Shafay, “Numerical computations for Buongiorno nano fluid model on the boundary layer flow of viscoelastic fluid towards a nonlinear stretching sheet,” Alexandria Eng. J., vol. 61, no. 2, pp. 1769–1778, 2022, doi: https://doi.org/10.1016/j.aej.2021.11.013.

M. Sarfraz, M. Khan, M. Z. Ullah, and D. Abuzaid, “Significance of Buongiorno’s model on viscoelastic MHD flow over a heated lubricated surface subject to Joule heating,” IJMPB, vol. 37, no. 18, p. 2350171, Jul. 2023, doi: 10.1142/S0217979223501710.

J. Raza, S. Dero, L. A. Lund, and Z. Omar, “Duality and stability of MHD Darcy–Forchheimer porous medium flow of rotating nanofluid on a linear shrinking/stretching sheet: Buongiorno model,” Int. J. Numer. Methods Heat Fluid Flow, vol. 32, no. 5, pp. 1517–1539, Apr. 2022, doi: 10.1108/HFF-01-2021-0054.

B. Nayak, S. Acharya, and S. R. Mishra, “Elastico-viscous Buongiorno model nanofluid flow over a stretching sheet with radiative heat transfer phenomena,” Int. J. Ambient Energy, vol. 43, no. 1, pp. 8894–8906, Dec. 2022, doi: 10.1080/01430750.2022.2111357.