Applied Mathematics and Mechanics >
Similarity transformation-based modeling of the thermally-radiative tetra-hybrid Casson nanofluid flow over a nonlinear stretching sheet using the Clique polynomial collocation method
Received date: 2025-07-24
Revised date: 2025-10-16
Online published: 2025-12-30
Copyright
The flow of a tetra-hybrid Casson nanofluid (Al2O3-CuO-TiO2-Ag/H2O) over a nonlinear stretching sheet is investigated. The Buongiorno model is used to account for thermophoresis and Brownian motion, while thermal radiation is incorporated to examine its influence on the thermal boundary layer. The governing partial differential equations (PDEs) are reduced to a system of nonlinear ordinary differential equations (ODEs) with fully non-dimensional similarity transformations involving all independent variables. To solve the obtained highly nonlinear system of differential equations, a novel Clique polynomial collocation method is applied. The analysis focuses on the effects of the Casson parameter, power index, radiation parameter, thermophoresis parameter, Brownian motion parameter, and Lewis number. The key findings show that thermal radiation intensifies the thermal boundary layer, the Casson parameter reduces the velocity, and the Lewis number suppresses the concentration with direct relevance to polymer processing, coating flows, electronic cooling, and biomedical applications.
U. L. MANIKANTA , K. J. GOWTHAM , B. J. GIREESHA , P. VENKATESH . Similarity transformation-based modeling of the thermally-radiative tetra-hybrid Casson nanofluid flow over a nonlinear stretching sheet using the Clique polynomial collocation method[J]. Applied Mathematics and Mechanics, 2026 , 47(1) : 185 -202 . DOI: 10.1007/s10483-026-3335-6
| [1] | SAKIADIS, B. C. Boundary-layer behavior on continuous solid surfaces: I. boundary-layer equations for two-dimensional and axisymmetric flow. AIChE Journal, 7, 26–28 (1961) |
| [2] | CRANE, L. J. Flow past a stretching plate. Zeitschrift für Angewandte Mathematik und Physik, 21, 645–647 (1970) |
| [3] | BOGNáR, G. On similarity solutions of MHD flow over a nonlinear stretching surface in non-Newtonian power-law fluid. Electronic Journal of Qualitative Theory of Differential Equations, 2016, 1–12 (2016) |
| [4] | GUEDRI, K., KHAN, A., SENE, N., RAIZAH, Z., SAEED, A., and GALAL, A. M. Thermal flow for radiative ternary hybrid nanofluid over nonlinear stretching sheet subject to Darcy-Forchheimer phenomenon. Mathematical Problems in Engineering, 2022, 3429439 (2022) |
| [5] | SAMAT, N. A. A., BACHOK, N., and ARIFIN, N. M. Boundary layer stagnation point flow and heat transfer over a nonlinear stretching/shrinking sheet in hybrid carbon nanotubes: numerical analysis and response surface methodology under the influence of magnetohydrodynamics. Computation, 12, 46 (2024) |
| [6] | TAWADE, J. V., GULED, C. N., NOEIAGHDAM, S., FERNANDEZ-GAMIZ, U., GOVINDAN, V., and BALAMURALITHARAN, S. Effects of thermophoresis and Brownian motion for thermal and chemically reacting Casson nanofluid flow over a linearly stretching sheet. Results in Engineering, 15, 100448 (2022) |
| [7] | SHATANAWI, W., ABBAS, N., SHATNAWI, T. A., and HASAN, F. Heat and mass transfer of generalized Fourier and Fick’s law for second-grade fluid flow at slendering vertical Riga sheet. Heliyon, 9, e14207 (2023) |
| [8] | NAYAK, B., PANDA, M., and MISHRA, S. R. Analysis of Brownian motion and thermophoresis effects with chemical reaction on Williamson nanofluid flow over permeable stretching sheet. Partial Differential Equations in Applied Mathematics, 11, 100908 (2024) |
| [9] | SUDARMOZHI, K., IRANIAN, D., and ALESSA, N. Investigation of melting heat effect on fluid flow with Brownian motion/thermophoresis effects in the occurrence of energy on a stretching sheet. Alexandria Engineering Journal, 94, 366–376 (2024) |
| [10] | GHASEMI, S. E. and HATAMI, M. Solar radiation effects on MHD stagnation point flow and heat transfer of a nanofluid over a stretching sheet. Case Studies in Thermal Engineering, 25, 100898 (2021) |
| [11] | NADEEM, S., ISHTIAQ, B., and ABBAS, N. Impact of thermal radiation on two-dimensional unsteady third-grade fluid flow over a permeable stretching Riga plate. International Journal of Modern Physics B, 37, 2350009 (2023) |
| [12] | ABBAS, N., TUMREEN, M., SHATANAWI, W., QASIM, M., and SHATNAWI, T. A. Thermodynamic properties of second grade nanofluid flow with radiation and chemical reaction over slendering stretching sheet. Alexandria Engineering Journal, 70, 219–230 (2023) |
| [13] | KRISHNA, M. V. and KUMAR, A. V. Chemical reaction, slip effects and non-linear thermal radiation on unsteady MHD Jeffreys nanofluid flow over a stretching sheet. Case Studies in Thermal Engineering, 55, 104129 (2024) |
| [14] | LONE, S. A., ALI, F., SAEED, A., and BOGNáR, G. Irreversibility analysis with hybrid cross nanofluid of stagnation point and radiative flow (TiO2+CuO) based on engine oil past a stretchable sheet. Heliyon, 9, e15015 (2023) |
| [15] | PATGIRI, B., PAUL, A., and SARMA, N. Numerical assessment of viscoelastic tetra hybrid nanofluid flow across a stretchable rotatory disk under the Soret and Dufour aspects. Multidiscipline Modeling in Materials and Structures, 20, 688–706 (2024) |
| [16] | GALAL, A. M., ABBAS, M., ALI, R., SAYDAXMETOVA, S., IBRAHIM, T. K., FAIZ, Z., and ALTHERWI, A. A. Thermal and solutal convective flow of propylene glycol-based tetra-hybrid nanofluid with elastic deformation and Stefan blowing impacts: application to thermal energy. Journal of Thermal Analysis and Calorimetry, 150, 6413–6427 (2025) |
| [17] | BLUMAN, G. W. and COLE, J. D. Similarity Methods for Differential Equations, Springer Science & Business Media, New York (2012) |
| [18] | PAKDEMIRLI, M. and YURUSOY, M. Similarity transformations for partial differential equations. SIAM Review, 40, 96–101 (1998) |
| [19] | CORTELL, R. Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Physics Letters A, 372, 631–636 (2008) |
| [20] | HAMAD, M. A. A. and FERDOWS, M. Similarity solutions to viscous flow and heat transfer of nanofluid over nonlinearly stretching sheet. Applied Mathematics and Mechanics (English Edition), 33(7), 923–930 (2012) https://doi.org/10.1007/s10483-012-1595-7 |
| [21] | MABOOD, F., KHAN, W. A., and ISMAIL, A. M. MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: a numerical study. Journal of Magnetism and Magnetic Materials, 374, 569–576 (2015) |
| [22] | HALDAR, S., MUKHOPADHYAY, S., and LAYEK, G. C. Effects of thermal radiation on Eyring-Powell fluid flow and heat transfer over a power-law stretching permeable surface. International Journal for Computational Methods in Engineering Science and Mechanics, 22, 366–375 (2021) |
| [23] | TRIVENI, B. and RAO, M. V. Chemical reaction and viscous dissipation effects on MHD flow of Powell-Eyring nanomaterial fluid over a nonlinear stretching sheet in a porous medium. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering (2024) https://doi.org/10.1177/09544089241282472 |
| [24] | YOUSIF, M. A., HATAMI, M., and ISMAEL, H. F. Heat transfer analysis of MHD three-dimensional Casson fluid flow over a porous stretching sheet by DTM-Padé. International Journal of Applied and Computational Mathematics, 3, 813–828 (2017) |
| [25] | SHARMA, R. P. and MISHRA, S. Analytical approach on magnetohydrodynamic Casson fluid flow past a stretching sheet via Adomian decomposition method. Heat Transfer, 51, 2155–2164 (2022) |
| [26] | PAVITHRA, C. G., GIREESHA, B. J., and GORLA, R. S. R. Semi-closed solutions of two-dimensional nanofluid flow and heat transfer over a nonlinear stretching sheet embedded with new set of similarity transformations. International Journal of Applied and Computational Mathematics, 10, 37 (2024) |
| [27] | MUKHOPADHYAY, S., DE, P. R., BHATTACHARYYA, K., and LAYEK, G. C. Casson fluid flow over an unsteady stretching surface. Ain Shams Engineering Journal, 4, 933–938 (2013) |
| [28] | SINGH, J., VISHALAKSHI, A. B., MAHABALESHWAR, U. S., and BOGNAR, G. MHD Casson fluid flow with Navier’s and second order slip due to a perforated stretching or shrinking sheet. PLoS One, 17, e0276870 (2022) |
| [29] | AWATI, V. B., GORAVAR, A., KUMAR, M., and BOGNAR, G. Stability analysis of magnetohydrodynamic Casson fluid flow and heat transfer past an exponentially shrinking surface by spectral approach. Case Studies in Thermal Engineering, 60, 104810 (2024) |
| [30] | MKHATSHWA, M. P. Nonlinear radiative mixed convective flow of fourth-grade tetra-hybrid nanomaterial over a horizontal cylindrical surface. Journal of Nanofluids, 13, 1040–1054 (2024) |
| [31] | RANDI? M., HANSEN, P. J., and JURS, P. C. Search for useful graph theoretical invariants of molecular structure. Journal of Chemical Information and Computer Sciences, 28, 60–68 (1988) |
| [32] | NIRMALA, A. N. and KUMBINARASAIAH, S. A novel analytical method for the multi-delay fractional differential equations through the matrix of Clique polynomials of the cocktail party graph. Results in Control and Optimization, 12, 100280 (2023) |
| [33] | HAJIABOLHASSAN, H. and MEHRABADI, M. L. On Clique polynomials. Australasian Journal of Combinatorics, 18, 313–316 (1998) |
| [34] | NIRMALA, A. N. and KUMBINARASAIAH, S. Numerical approach for the Hunter-Saxton equation arising in liquid crystal model through cocktail party graphs Clique polynomial. Journal of Applied Analysis & Computation, 14, 2037–2062 (2024) |
| [35] | HOEDE, C. and LI, X. Clique polynomials and independent set polynomials of graphs. Discrete Mathematics, 125, 219–228 (1994) |
| [36] | JAYANNA, G. K., PRASAD, B. G., and JAYANNA, G. B. Thermal analysis of longitudinal moving fins with different geometries and internal heat generation using the Clique polynomial of the cocktail party graph. Heat Transfer, 54, 3985–4004 (2025) |
| [37] | KUMBINARASAIAH, S., RAMANE, H. S., PISE, K. S., and HARIHARAN, G. Numerical solution for nonlinear Klein-Gordon equation via operational matrix by Clique polynomial of complete graphs. International Journal of Applied and Computational Mathematics, 7, 12 (2021) |
| [38] | TSOU, F. K., SPARROW, E. M., and GOLDSTEIN, R. J. Flow and heat transfer in the boundary layer on a continuous moving surface. International Journal of Heat and Mass Transfer, 10, 219–235 (1967) |
| [39] | ANDERSSON, H. I. and AARSETH, J. B. Sakiadis flow with variable fluid properties revisited. International Journal of Engineering Science, 45, 554–561 (2007) |
| [40] | AHMAD, S., ROHNI, A. M., and POP, I. Blasius and Sakiadis problems in nanofluids. Acta Mechanica, 218, 195–204 (2011) |
/
| 〈 |
|
〉 |
Email Alert
RSS