- 1Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, Thailand
- 2Department of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Pakistan
- 3Department of Basic Sciences, College of Sciences and Theoretical Studies, Dammam-branch, Saudi Electronic University, Riyad, Saudi Arabia
- 4Department of Basic Sciences, College of Sciences and Theoretical Studies, Medinah-branch, Saudi Electronic University, Riyad, Saudi Arabia
- 5Mechanical Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Wadi addawaser, Saudi Arabia
- 6Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt
- 7Center of Research, Faculty of Engineering, Future University in Egypt New Cairo, New Cairo, Egypt
This research highlights the utilization of two viscosity models to study the involvement of variable properties in heat and momentum transport in a rotating Carreau fluid past over a cone. The rheology of the Carreau material is assessed by the variable dynamic viscosity over the heating cone. The transport of momentum phenomenon is modeled by considering generalized Ohm’s law in Carreau liquid and thermal transport in derived by considering variable thermal conductivity, heat flux model. The considered model is derived in the form of nonlinear PDEs with boundary layer analysis. The nonlinear PDEs are converted into coupled ODEs by using approximate transformation and converted equations are solved numerically by finite element methodology. The impact of numerous parameters is displayed graphically, and their behavior is discussed in detail.
Introduction
The features of heat transfer are applicable in several industrial, chemical, and thermal processes. This type of phenomenon is applicable in polymers, gas engineering, cooling process, solar cells, water, thermal process, and oils. Recent work on heat transfer in several rheological fluids should be mentioned. Nazeer et al. (2022) estimated the thermal aspects of multiphase flow taking into account magnetohydrodynamic and thermal radiation over a stretching frame. Luan et al. (2022) discussed turbulator effects in heat transfer phenomena in an energy transfer mechanism. Imran et al. (2022) discussed enhancement in thermal energy using the concept of several shapes of nanoparticles involving hybrid nanoparticles over a horizontal plate. The characteristics of fluids exhibit a range of rheological behavior. Therefore, it is very complex to study rheological behaviors, taking into account rheological stress (strain relation). The Newtonian model contains only viscosity rheology and its utilization in non-Newtonian contexts provides inaccurate information with respect to mass diffusion, flow, and heat energy. Carreau fluid is type of non-Newtonian fluid. Mathematical model related to Carreau fluid was investigated by Carreau (1972). The stress tensor regarding Carreau fluid is defined as
where
Recent investigations have shown that mass diffusion and thermal conductivity during transport mass species and heat cannot increase and remain constant. During changes of thermal energy, viscosity and thermal conductivity are functions of heat energy during the transport of particles. Alhussain and Tassaddiq (2022) proposed a model of variable viscosity in the presence of hybrid nanomaterial in Casson fluid, considering thin film considering magnetic field in channel. They have utilized analytical approach to simulate numerical consequences. Kumawat et al. (2022) performed numerical results of entropy generation in MHD flow considering variable viscosity in the presence of variable viscosity. Nazir et al. (2020) simulated the numerical study of a Carreau liquid using the concept of variable properties in mass diffusion and heat energy involving non-Fourier’s law over a frame using finite element scheme. It was found that the highest heat energy was obtained for variable viscosity than for constant viscosity. Chaurasiya et al. (2022) discussed the influence of variable thermal conductivity taking into account semi-conductor sources using a finite element scheme. Wang et al. (2022) investigated the impacts of tri-hybridized nanofluid taking into account variable properties of fluid in the presence of non-Fourier’s law computed by the FE-method. Naseem et al. (2021) discussed the numerical consequences of Soret and Dufour effects involving the role of thermal radiation over a stretching frame, including variable thermal conductivity. Sohail et al. (2021) investigated the performance of entropy generation in a Casson fluid inserting the impact of Lorentz force. They discussed features of variable properties including various effects past a stretching frame. Akbar et al. (2022) performed investigations of unsteady flow in peristaltic transport using variable viscosity (function of temperature) simulating an exact solution approach. Akram et al. (2022a) estimated the thermal features of peristaltic flow involving electroosmotically adding a suspension of nanomaterial in a curved microchannel. Maraj et al. (2022) developed mathematically modeling of rotational MHD flow in the presence of Hall force inserting two types of nanoparticles using slip conditions in a vertical channel. Habib and Akbar (2021) investigated a new to nanofluids using sensitive Staphlococcus aureus and Staphlococcus aureus. They concluded that drug resistance can be reduced and overcome using drug conjugate and gold nanoparticles. Akram et al. (2021) introduced thermal transfer characterizations inserting hybrid nanofluid (Ag–Au) considering electroosmotic pumping in microchannel. Akram et al. (2022b) analyzed electroosmotic flow based on water-silver nanofluids considering peristalsis flow using two various approaches of nanofluid. Akram et al. (2022c) introduced investigation of electroosmosis in the presence of MHD peristaltic flow containing suspension of SWCNTs nanofluid filled in aqueous media. Multiple aspects of Carreau liquid applied to a magnetic field in a suspension of a nanofluid using slip effects across a slandering surface were studied by Raju et al. (2019). Khan and Sultan (2015) discussed thermal features of Dufour and Soret effects in Eyring-Powell liquid using a porous heated cone. Dawar et al. (2021) investigated flow configurations in Williamson fluid, taking into account non-isosolutal conditions inserting nanofluid over wedge and cone.
The utilization of variable properties to study momentum and thermal transport in Carreau model past over a cone has not been sufficiently thoroughly studied. This contribution covers this question, and the modeled equations are solved numerically via finite element approach, and the impact of different involved parameters on velocity and temperature is discussed.
Mathematically development and physical consequences
zThree-dimensional thermal features in terms of variable properties in a Carreau liquid in rotating cone are visualized. Two kinds of forces, namely, ion slip and Hall forces, are considered. The rheology of a Carreau liquid in the presence of variable viscosity is imposed into fluidic motion. Heat energy assessed using the non-Fourier approach. Lorentz force is inserted along the z-direction of a heated cone. Additionally, motion associated with particles is generated due to rotational movement of cone with angle
Conservation laws (Khan et al., 2014; Malik et al., 2016) are defined as follows:
The reduced form of the PD equations (Khan et al., 2014; Malik et al., 2016) is given below:
BCs (boundary conditions) (Malik et al., 2016) are
Viscosity and thermal conductivity (Nazir et al., 2020) are functions of heat energy, as follows:
The similarity variables (Malik et al., 2016) of current the analysis are
Eq. 12 is known as a transformation, and the selection of the transformation is adopted to satisfy the continuity equation. Transformations are used in Eqs. 7–9. The ODEs are derived as
The Prandtl fluid at infinity is delivered as
Substituting Eq. (16) into Eq. (15), we have
Wall stresses (skin friction coefficients) are formulated as
The heat transfer rate in terms of variable thermal conductivity is
Numerical methodology
The formulated ODEs are numerically simulated by a numerical scheme (FEM) [14, 16, and 31]. The description associated with FEM is mentioned below.
Discretization of domain
In this step, the desired ODEs are transformed into numbers of elements (300 elements). Residuals of the current problem are achieved. Eqs. 13–15 are strong forms with boundary conditions. A strong form can be converted into a weak form by collecting all terms on one side and multiplying it by weight functions and integrated it on whole domain. The weighted residuals of the problem implementation due to
Shape functions
Various shape functions are utilized in the current problem. In this approach, however, a linear shape function is implemented to obtain an approximate solution. Several shape functions are used in FEM. However, the linear type of shape function is utilized in this numerical approach. Shape functions are based on a linear type of polynomial. The values variables (
Shape functions are
Developments of stiffness matrices using assemble process
The Galerkin approach is utilized to derive the stiffness matrices. The concept of the assembly process is implemented to find stiffness matrices and global stiffness matrix. The sizes of the boundary vectors, stiffness matrices, and source vectors are
Assembly method
The assembly method is used to assemble all elements. The residual is defined as
The convergence of the problem must existed under
Code development and validation
Code based on FEM is developed on MAPLE. The Galerkin finite element method is used to simulate the problem on MAPLE 18. The FEM code is verified in a published study (Malik et al., 2016), which is recorded in Table 1. The grid-independent analysis is shown in Table 2.
Outcomes and discussion
The three-dimensional thermal aspects of a Carreau liquid in the presence of variable viscosity in a rotating cone are developed. Features related to ion slip and Hall theory are added in momentum equations. The concept associated with Cattaneo-Christov model (CCM) is utilized, involving heat generation/heat absorption. Moreover, variable fluidic properties are utilized in terms of variable thermal conductivity and viscosity dependent heat energy. The developed model is numerically solved using the finite element approach. The graphical results in view of thermal and velocity field are shown below.
Contrast among constant and variable viscosities in velocity field
In this section, Figures 2–9 are plotted reading secondary and primary velocity fields against several parameters. It is noted that dot curves are associated with constant viscosity for
Contrast among constant and variable viscosities in temperature field
This subsection contains graphical explanation associated with thermal impacts for two cases (variable viscosity and constant viscosity) against change in heat source,
Measurement of shear stresses and temperature gradient
Table 3 demonstrates the impacts of shear stresses and Nusselt numbers against variation in
Prime consequences of performed research
The thermal features of 3D-Carreau model are developed under considerations of ion slip and Hall forces in rotating cone. Variable thermal properties in term of viscosity, mass diffusion, and thermal conductivity are addressed. Chemical species and external heat sources are utilized in the presence of a non-Fourier law. The following are the most important outcomes from this investigation:
The computations of the problem become grid independent at 300 elements;
Heat energy diffuses faster into non-Newtonian liquids for temperature-dependent viscosity than for the diffusion of heat energy for constant viscosity;
Motion into fluidic particles is enhanced against higher values of ion slip and Hall currents parameters;
Heat energy is boosted versus higher values of
An enhancement in thermal energy occurs according to
The thermal transfer rate is applicable in metal grinding, wire paintings, glasses melting, chemical products, solar collectors, machine cutting, food processing, electronic components, nuclear reaction, solar systems, glass fiber, and so on.
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
UN: Conceptualization; Investigation; Software; Validation; Writing—review and editing. MS: Data curation; Writing—original draft; Writing—review and editing; Visualization; Methodology. KM: Formal analysis; Funding acquisition; Software. AS: Methodology; Project administration; Resources; Funding acquisition. RA: Project administration; Formal analysis. AG: Visualization; Supervision; Funding acquisition. MS: Software; Validation; Visualization; Supervision; Funding acquisition.
Funding
This work was partially funded by the research center of the Future University in Egypt, 2022.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
Akbar, N. S., Maraj, E. N., Noor, N. F. M., and Habib, M. B. (2022). Exact solutions of an unsteady thermal conductive pressure driven peristaltic transport with temperature-dependent nanofluid viscosity. Case Stud. Therm. Eng. 35, 102124. doi:10.1016/j.csite.2022.102124
Akram, J., Akbar, N. S., Alansari, M., and Tripathi, D. (2022). Electroosmotically modulated peristaltic propulsion of TiO2/10W40 nanofluid in curved microchannel. Int. Commun. Heat Mass Transf. 136, 106208. doi:10.1016/j.icheatmasstransfer.2022.106208
Akram, J., Akbar, N. S., and Tripathi, D. (2021). A theoretical investigation on the heat transfer ability of water-based hybrid (Ag–Au) nanofluids and Ag nanofluids flow driven by electroosmotic pumping through a microchannel. Arab. J. Sci. Eng. 46 (3), 2911–2927. doi:10.1007/s13369-020-05265-0
Akram, J., Akbar, N. S., and Tripathi, D. (2022). Analysis of electroosmotic flow of silver-water nanofluid regulated by peristalsis using two different approaches for nanofluid. J. Comput. Sci. 62, 101696. doi:10.1016/j.jocs.2022.101696
Akram, J., Akbar, N. S., and Tripathi, D. (2022). Electroosmosis augmented MHD peristaltic transport of SWCNTs suspension in aqueous media. J. Therm. Anal. Calorim. 147 (3), 2509–2526. doi:10.1007/s10973-021-10562-3
Alhussain, Z. A., and Tassaddiq, A. (2022). Thin film blood based Casson hybrid nanofluid flow with variable viscosity. Arab. J. Sci. Eng. 47 (1), 1087–1094. doi:10.1007/s13369-021-06067-8
Carreau, P. J. (1972). Rheological equations from molecular network theories. Trans. Soc. Rheology 16 (1), 99–127. doi:10.1122/1.549276
Chaurasiya, V., Chaudhary, R. K., Awad, M. M., and Singh, J. (2022). A numerical study of a moving boundary problem with variable thermal conductivity and temperature-dependent moving PCM under periodic boundary condition. Eur. Phys. J. Plus 137 (6), 714. doi:10.1140/epjp/s13360-022-02927-w
Dawar, A., Shah, Z., Tassaddiq, A., Kumam, P., Islam, S., and Khan, W. (2021). A convective flow of Williamson nanofluid through cone and wedge with non-isothermal and non-isosolutal conditions: A revised buongiorno model. Case Stud. Therm. Eng. 24, 100869. doi:10.1016/j.csite.2021.100869
Farooq, U., Waqas, H., Khan, M. I., Khan, S. U., Chu, Y. M., and Kadry, S. (2021). Thermally radioactive bioconvection flow of Carreau nanofluid with modified Cattaneo-Christov expressions and exponential space-based heat source. Alexandria Eng. J. 60 (3), 3073–3086. doi:10.1016/j.aej.2021.01.050
Habib, M. B., and Akbar, N. S. (2021). New trends of nanofluids to combat Staphylococcus aureus in clinical isolates. J. Therm. Anal. Calorim. 143 (3), 1893–1899. doi:10.1007/s10973-020-09502-4
Imran, M., Farooq, U., Muhammad, T., Khan, S. U., and Waqas, H. (2021). Bioconvection transport of Carreau nanofluid with magnetic dipole and nonlinear thermal radiation. Case Stud. Therm. Eng. 26, 101129. doi:10.1016/j.csite.2021.101129
Imran, M., Yasmin, S., Waqas, H., Khan, S. A., Muhammad, T., Alshammari, N., et al. (2022). Computational analysis of nanoparticle shapes on hybrid nanofluid flow due to flat horizontal plate via solar collector. Nanomaterials 12 (4), 663. doi:10.3390/nano12040663
Khan, N. A., Aziz, S., and Khan, N. A. (2014). Numerical simulation for the unsteady MHD flow and heat transfer of couple stress fluid over a rotating disk. PLoS One 9 (5), e95423. doi:10.1371/journal.pone.0095423
Khan, N. A., and Sultan, F. (2015). On the double diffusive convection flow of Eyring-Powell fluid due to cone through a porous medium with Soret and Dufour effects. AIP Adv. 5 (5), 057140. doi:10.1063/1.4921488
Kumawat, C., Sharma, B. K., Al-Mdallal, Q. M., and Rahimi-Gorji, M. (2022). Entropy generation for MHD two phase blood flow through a curved permeable artery having variable viscosity with heat and mass transfer. Int. Commun. Heat Mass Transf. 133, 105954. doi:10.1016/j.icheatmasstransfer.2022.105954
Luan, X. D., Xu, Y. P., Ayed, H., and Selim, M. M. (2022). Heat transfer treatment of nanomaterial with considering turbulator effects. Int. Commun. Heat Mass Transf. 131, 105787. doi:10.1016/j.icheatmasstransfer.2021.105787
Malik, M. Y., Jamil, H., Salahuddin, T., Bilal, S., Rehman, K. U., and Mustafa, Z. (2016). Mixed convection dissipative viscous fluid flow over a rotating cone by way of variable viscosity and thermal conductivity. Results Phys. 6, 1126–1135. doi:10.1016/j.rinp.2016.11.027
Maraj, E. N., Zehra, I., and SherAkbar, N. (2022). Rotatory flow of MHD (MoS2-SiO2)/H2O hybrid nanofluid in a vertical channel owing to velocity slip and thermal periodic conditions. Colloids Surfaces A Physicochem. Eng. Aspects 639, 128383. doi:10.1016/j.colsurfa.2022.128383
Nabwey, H. A., Alshber, S. I., Rashad, A. M., and Mahdy, A. E. N. (2022). Influence of bioconvection and chemical reaction on magneto-Carreau nanofluid flow through an inclined cylinder. Mathematics 10 (3), 504. doi:10.3390/math10030504
Naseem, T., Nazir, U., El-Zahar, E. R., Algelany, A. M., and Sohail, M. (2021). Numerical computation of Dufour and Soret effects on radiated material on a porous stretching surface with temperature-dependent thermal conductivity. Fluids 6 (6), 196. doi:10.3390/fluids6060196
Nazeer, M., Saleem, S., Hussain, F., Zia, Z., Khalid, K., and Feroz, N. (2022). Heat transmission in a magnetohydrodynamic multiphase flow induced by metachronal propulsion through porous media with thermal radiation. Proc. Institution Mech. Eng. Part E J. Process Mech. Eng., 095440892210752. doi:10.1177/09544089221075299
Nazir, U., Saleem, S., Nawaz, M., Sadiq, M. A., and Alderremy, A. A. (2020). Study of transport phenomenon in Carreau fluid using Cattaneo–Christov heat flux model with temperature dependent diffusion coefficients. Phys. A Stat. Mech. its Appl. 554, 123921. doi:10.1016/j.physa.2019.123921
Nazir, U., Sohail, M., Selim, M. M., Alrabaiah, H., and Kumam, P. (2021). Finite element simulations of hybrid nano-Carreau Yasuda fluid with hall and ion slip forces over rotating heated porous cone. Sci. Rep. 11 (1), 19604–19615. doi:10.1038/s41598-021-99116-z
Raju, C. S. K., Hoque, M. M., Khan, N. A., Islam, M., and Kumar, S. (2019). Multiple slip effects on magnetic-Carreau fluid in a suspension of gyrotactic microorganisms over a slendering sheet. Proc. Institution Mech. Eng. Part E J. Process Mech. Eng. 233 (2), 254–266. doi:10.1177/0954408918776723
Reedy, S., Srihari, P., Ali, F., and Naikoti, K. (2022). Numerical analysis of Carreau fluid flow over a vertical porous microchannel with entropy generation. Partial Differ. Equations Appl. Math. 5, 100304. doi:10.1016/j.padiff.2022.100304
Sohail, M., Chu, Y. M., El-Zahar, E. R., Nazir, U., and Naseem, T. (2021). Contribution of joule heating and viscous dissipation on three dimensional flow of Casson model comprising temperature dependent conductance utilizing shooting method. Phys. Scr. 96 (8), 085208. doi:10.1088/1402-4896/ac00e5
Sohail, M., Nazir, U., El-Zahar, E. R., Alrabaiah, H., Kumam, P., Mousa, A. A. A., et al. (2022). A study of triple-mass diffusion species and energy transfer in Carreau-Yasuda material influenced by activation energy and heat source. Sci. Rep. 12 (1), 10219–10317. doi:10.1038/s41598-022-13890-y
Song, Y. Q., Waqas, H., Al-Khaled, K., Farooq, U., Gouadria, S., Imran, M., et al. (2022). Aspects of thermal diffusivity and melting phenomenon in Carreau nanofluid flow confined by nonlinear stretching cylinder with convective Marangoni boundary constraints. Math. Comput. Simul. 195, 138–150. doi:10.1016/j.matcom.2022.01.001
Wang, F., Nazir, U., Sohail, M., El-Zahar, E. R., Park, C., and Thounthong, P. (2022). A Galerkin strategy for tri-hybridized mixture in ethylene glycol comprising variable diffusion and thermal conductivity using non-Fourier’s theory. Nanotechnol. Rev. 11 (1), 834–845. doi:10.1515/ntrev-2022-0050
Nomenclature
PDEs partial differential equations
ODEs ordinary differential equations
FEM finite element method
Keywords: finite element method, nonlinear ODEs, heat transfer, boundary layer theory, variable properties
Citation: Nazir U, Sohail M, Mukdasai K, Singh A, Alahmadi RA, Galal AM and Eldin SM (2022) Applications of variable thermal properties in Carreau material with ion slip and Hall forces towards cone using a non-Fourier approach via FE-method and mesh-free study. Front. Mater. 9:1054138. doi: 10.3389/fmats.2022.1054138
Received: 26 September 2022; Accepted: 04 November 2022;
Published: 23 December 2022.
Edited by:
Ali Saleh Alshomrani, King Abdulaziz University, Saudi ArabiaReviewed by:
Noreen Akbar, National University of Sciences and Technology (NUST), PakistanNajeeb Alam Khan, University of Karachi, Pakistan
Copyright © 2022 Nazir, Sohail, Mukdasai, Singh, Alahmadi, Galal and Eldin. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Muhammad Sohail, muhammad_sohail111@yahoo.com; Sayed M. Eldin, sayed.eldin22@fue.edu.eg