1 Vol.:(0123456789) Scientifc Reports | (2021) 11:1882 | https://doi.org/10.1038/s41598-021-81417-y www.nature.com/scientificreports Signifcance of nanoparticle’s radius, heat fux due to concentration gradient, and mass fux due to temperature gradient: The case of Water conveying copper nanoparticles Nehad Ali Shah 1,2 , I. L. Animasaun 3* , Jae Dong Chung 4 , Abderrahim Wakif 5 , F. I. Alao 3 & C. S. K. Raju 6 The performance of copper selenide and efectiveness of chemical catalytic reactors are dependent on an inclined magnetic feld, the nature of the chemical reaction, introduction of space heat source, changes in both distributions of temperature and concentration of nanofuids. This report presents the signifcance of increasing radius of nanoparticles, energy fux due to the concentration gradient, and mass fux due to the temperature gradient in the dynamics of the fuid subject to inclined magnetic strength is presented. The non-dimensionalization and parameterization of the dimensional governing equation were obtained by introducing suitable similarity variables. Thereafter, the numerical solutions were obtained through shooting techniques together with 4th order Runge–Kutta Scheme and MATLAB in-built bvp4c package. It was concluded that at all the levels of energy fux due to concentration gradient, reduction in the viscosity of water-based nanofuid due to a higher radius of copper nanoparticles causes an enhancement of the velocity. The emergence of both energy fux and mass fux due to gradients in concentration and temperature afect the distribution of temperature and concentration at the free stream. Te roles of energy and mass fuxes due to temperature and concentration gradients play major roles in electro- chemical processes, dynamics of gases, chemical catalytic reactors, production of copper selenide sequel to the pioneering work by Louis Dufour 1 ; see Ingle and Horne 2 , Hollinger and Lucke 3 , Korzhuev 4 , Kim et al. 5 , Mahdy 6 , Anjum Badruddin 7 . Conservation of energy is capable to unravel the nature of outward fux of energy. Mean- while, Hort et al. 8 once remarked that concentration currents and heat are two driven forces of fuctuations in temperature as in the case of conservation of energy. Te efect of energy fux due to concentration gradient on six diferent gases ( CO 2 + O 2 , N 2 + H 2 , H 2 + CO 2 , N 2 + O 2 , CO 2 + N 2 , H 2 + O 2 ) at various levels of pressure was examined experimentally by Rastogi and Madan 9 . It was remarked that kinetic theory is useful to determine the Dufour coefcient of each gas and temperature diference within the domain (T w - T ∞ ) is dependent on pressure that maintains the motion of each gas. Formation of energy fux due to temperature gradient is highly signifcant in gases than in liquids. In fact, the occurrence in a less viscous fuid is 10,000 larger than in high vis- cous fuid. Tis fact suggests that the magnitude of the Dufour number is larger for gases (Hort et al. 10 ). Dynamics of peristaltic fow through a channel of width 2a subject to energy fux due to concentration gradient and mass fux due to energy gradient was closely examined by Hayat et al. 11 . It was shown that temperature distribution is an increasing property of both Dufour and Soret efects. Although, the increase in the temperature distribution OPEN 1 Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam. 2 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam. 3 Department of Mathematical Sciences, Fluid Dynamics and Survey Research Group, Federal University of Technology, Akure PMB 704, Nigeria. 4 Department of Mechanical Engineering, Sejong University, Seoul 05006, South Korea. 5 Laboratory of Mechanics, Faculty of Sciences Ain Chock, University Hassan II of Casablanca, 20000 Casablanca, Morocco. 6 Department of Mathematics, GITAM Bengaluru, Bangalore, Karnataka 562163, India. * email: ilanimasaun@futa.edu.ng