- 1School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, China
- 2Department of Mathematics, University of Management and Technology, Lahore, Pakistan
- 3School of Computer Science and Technology, Zhejiang Normal University, Jinhua, Zhejiang, China
- 4Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, Saudi Arabia
- 5Research Centre, Future University in Egypt, New Cairo, Egypt
The study of hybrid nanoliquids can aid in developing numerous advanced features that facilitate heat transmission, such as pharmaceutical processes, hybrid-powered engines, microelectronics, engine cooling, and domestic refrigerators. In the current study, a mathematical model is designed to elaborate the physical inception of an unsteady second-grade hybrid nanofluid with
1 Introduction
Investigations into non-Newtonian materials have been ongoing since the past century due to their unique characteristics and fascinating rheological properties. These materials are widely used across various industries, including chemical engineering, metal processing, food, and plastics. Non-Newtonian fluids have a range of applications, including biofluids, glassblowing, synthetic fibers, cosmetics, food, pharmaceuticals, shampoo, and metal spinning. These fluids exhibit different behaviors and can be classified as dilatant, shear-thickening, thixotropic, or shear-thinning. Rheologists have identified various fluid models, such as Casson, Maxwell, Burgers, Williamson, Oldroyd-B, third-grade, Jeffrey, micropolar, Sisko, and Sutterby Cross. However, second-grade fluids behave differently under different conditions, which explains the characteristics of shear-thickening, shear-thinning, and Newtonian effects. Second-grade fluids have gained the attention and devotion of intellectuals due to their dynamic properties [1–8]. Stretching a plastic sheet, on the other hand, is not always linear. An exponentially stretched sheet’s heat transport characteristics have a broader range of technical applicability. The heat transfer ratio of the continuously expanded surface increases rapidly with the expansion rate and temperature variations, which regulates the outcome when the copper wire is thinned and diluted. The techniques involved in these methods significantly impact the final product quality due to the effect of stretching kinematics and concurrent heating or cooling. Khan and Sanjayanand [9] analyzed a second-grade fluid’s steady flow and heat conductivity with an exponentially extending surface using the Runge–Kutta fourth-order (RK4) method. Rehman et al. [10] investigated the steady flow of a second-grade fluid over an exponentially stretching sheet using the Keller box and homotopy analysis approaches. Nadeem et al. [11] explored the flow and heat transfer of second-grade (viscoelastic) liquids in thermal radiation. Ramzan and Bilal [12] calculated the mixed convection of a second-grade nanofluid caused by time-dependent MHD, thermal radiation, and diffuse surfaces. Pakdemirli et al. [13] used perturbation analysis to examine the properties of a second-grade fluid. Recently, many researchers have studied second-grade nanofluids over an exponentially stretching surface [14–23].
Professionals like unsteady flow in several engineering organizations since it contributes to better mechanisms over their deeds [24, 25]. Moreover, even in ideal flow conditions, unnecessary destabilizing effects can occur around the system. The behavior of unstable boundary layer (BL) flow is unique compared to steady-state flow because the control equation has additional time-dependent conditions that degrade the structure of BL separation and fluid motion. However, through a healthier consideration of unstable fluid flow presentations in manufacturing dealings, contemporary enterprise techniques that permit improved structure dependability, productivity, and cost saving of multiple dynamical devices are possible [26]. Zaib et al. [27] discussed the computational exploration of a time-dependent flow with heat flux past an exponentially contracting surface.
The spectacle of heat transfer in electromagnetic waves is called thermal radiation. It happens because the two mediums have a significant temperature difference. In manufacturing and physical science, radiative influences are a crucial part. In the polymer manufacturing sectors, where heat-controlling variables influence the ultimate product quality to some extent, thermal radiation impacts are essential in controlling heat transfer. In addition, the radiation effects of missiles, aircraft, solar radiation, gas turbines, liquid metal fluids, spacecraft, nuclear power plants, and MHD accelerators are also prominent. Pantokratoras and Fang [28] were pioneers in examining the effect of nonlinear thermal radiation on Sakiadis flow. Dogonchi and Ganji [29] evaluated the impact of radiant heat on the MHD flow of a water-based nanofluid in a channel that can shrink, stretch, and diverge or converge. Khan et al. [30] studied the radiation flow of hybrid nanofluids through porous surfaces [30]. Many researchers [31–36] are involved in nonlinear thermal radiation.
Recognizing the need for improved thermal conductivity in traditional fluids, a new type of nanofluid called “hybrid nanofluid” is presented to provide highly industrialized heat conductivity. Two or more semiconductor materials are mixed with a base fluid to make a hybrid nanofluid. Different nanomaterials include carbon nanotubes [37], metals, metal oxides, and carbides. Numerous investigators are now interested in hybrid nanofluid due to its significance for the betterment of thermodynamic characteristics in real-world applications [38, 39], as a result of Choi and Eastman’s [40] outstanding findings that gave the unique notion of nanoliquid. Hybrid nanofluids are also used in various applications, including electrical gadget cooling [41], cooling of domestic refrigerators [42], automobile braking fluid, transformers, heat exchangers, and solar water heating [43]. Suresh et al. [44] explored the effects of a hybrid nanofluid
The fuzzy set theory (FST) [54] has proved to be a valuable technique for modeling uncertainties in recent decades, providing models with a more accurate view of reality and allowing them to express themselves with a broader perspective [55–59]. After modeling real-world problems, they convert into partial differential equations (PDEs) or ordinary differential equations (ODEs). Uncertainty issues may arise during the development of a dynamic model. Researchers must deal with inaccurate data, parameters, dynamical variability, and complex relationships. As a result, many scientists use fuzzy models to depict dynamical systems to prevent artificial data accuracy and produce more realistic results. The fuzzy differential equation (FDE) is critical in overcoming these challenges. Initially, Chang and Zadeh [60] proposed the basic idea of fuzzy derivatives. Dubois and Prade [61] proposed the idea of fuzzy numbers (FNs) for solving an FDE. Kaleva [62] introduced the concept of FDEs in a fuzzy environment. Recently, FDEs played a significant role in fluid dynamics, such as the effects of MHD and gravitation on the third-grade fluid through an inclined channel in a fuzzy atmosphere, which were quantitatively explored by Nadeem et al. [63]. They used the triangular fuzzy numbers to analyze ambiguity. The heat transmission of SWCNTs MWCNTS on a third-grade nanofluid along an inclined channel in a fuzzy atmosphere was explored by Siddiqui et al. [64]. For comparison and uncertainty, they used nanoparticle volume fraction as TFN.
A careful review of the previously cited literature reveals several breaks and confines. No preceding studies have examined the unsteady MHD flow of the second-grade hybrid
The motivations for performing this analysis inspire the following research questions:
1) How do the thermal characteristics of nanoparticles vary when nonlinear thermal radiation features are used?
2) How do different developing parameters affect heat transfer and flow rates?
3) How does heat transfer improve in heat source/sink and magnetic force implications?
4) Why is the homotopy analysis method (HAM) preferred over the other methods?
5) Ho\w does the Lorentz force affect the velocity of the second-grade hybrid nanofluid by applying the magnetic field?
2 Mathematical formulation
The time-dependent, 2D incompressible, and unsteady flow of the MHD viscoelastic (second-grade) hybrid
When using the BL approximation, the governing equations for continuity, momentum, and heat are established on all of the preceding assumptions [12, 51]:
and the boundary conditions are
where
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TABLE 1.
The thermophysical properties of hybrid nanofluids are as follows [51]:
The following similarity transformations are presented in [10] to simplify the governing Eqs 1–3 along with the boundary conditions (4). The stream function
Using Eq. 6, Eqs (2), (3) can be condensed to the following set of nonlinear ODEs in the context of the abovementioned relations [36]:
with the constraints
where the unsteadiness parameter is
The stretching/shrinking parameter is
Using Eq. 6 in Eq. 10 and Eq. (11) yields the following relationship:
where
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FIGURE 2. Membership function of TFN [54].
2.1 Homotopy analysis method
The HAM is a multifaceted investigative system that solves nonlinear equations with several variables. Based on Eq. 9, the HAM computes consequential Eqs 7, 8. Linear operators and preliminary approximations are mandatory to surprise the process through this technique. Consequently, we used them as
The properties of the operator described above are as follows:
where
where
The boundary conditions become [12]
Equations (7)–(9) convert into nonlinear operators like Eqs 18–21, and then, the series solution becomes
2.2 Fuzzification
Using fuzzy concepts, comparing nanofluid and hybrid nanofluid is also explored in this study. The nonlinear ODEs convert into FDEs, and the nanoparticle volume percentage is taken as a TFN. The governing FDE is converted into a double parametric form. In this case, Eq. 8 can be converted into an interval form using the
Let
The
To handle this scenario, the FDEs are renewed into lower
3 Results and discussion
An unsteady flow analysis was performed on a second-grade hybrid
The influence of the magnetic (M) parameter on the velocity field is depicted in Figure 3. For higher values of M, the velocity dropped in both cases. Lorentz pressure is responsible for this phenomenon, which arises from the cooperation of electric and magnetic fields during an electrically conducted fluid flow. So, the fluid velocity in the BL is controlled by the generated Lorentz force. As a result, as M rises, the velocity of the fluid and hybrid nanofluids falls. The interaction of magnetic fields is significant in different technical and industrial applications, such as crude oil extraction, geothermal systems, and groundwater hydrology. The change of the second-grade parameter (
As shown in Figure 12, f ′′(0) increases with M and decreases as
3.1 Fuzzy results and discussion
Figure 16 portrays the calculated fuzzy temperature using volume fractions of
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TABLE 3. Comparison of current results of
4 Conclusion
This study analyzed the unsteady MHD second-grade hybrid
• The fluid velocity is dropped with the magnetic parameter, while the fluid velocity is boosted with the second-grade fluid parameter.
• The fluid temperature increases while the fluid velocity declines with the improvement of
• The fluid temperature boosts against higher values of
• The fluid velocity grows versus the stretching/shirking parameter while the fluid temperature declines.
• The skin friction coefficient is reduced with a rise in unsteady and second-grade parameters while growing with magnetic parameters.
• For higher values of Nr, H,
• The maximum width of the fuzzy fluid temperature of the hybrid nanofluid was observed during a fuzzy analysis using a triangular MF, indicating that the fuzziness level is higher than that of regular nanofluids.
• The
The findings of this study can be used to drive future progress in which the heating system’s heat outcome is analyzed with nanofluids or hybrid nanofluids of various kinds (Maxwell, third-grade, Casson, Carreau, micropolar fluids, etc).
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding authors.
Author contributions
RZ, MN: conceptualization, methodology, and writing–original draft. MN, MS: data curation, investigation, resources, software, and writing–original draft. IS: formal analysis, supervision, validation, and writing–original draft. IK, AM: funding acquisition, project administration, resources, visualization, and data curation, writing.
Funding
The authors declare financial support was received for the research, authorship, and/or publication of this article. This project was funded by Zhejiang Normal University, Jinhua, Zhejiang, China.
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.
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Nomenclature
Keywords: second-grade fluid, exponential stretching surface, thermal radiation, hybrid nanofluid, triangular fuzzy number (TFN)
Citation: Zulqarnain RM, Nadeem M, Siddique I, Samar M, Khan I and Mohamed A (2023) Numerical study of second-grade fuzzy hybrid nanofluid flow over the exponentially permeable stretching/shrinking surface. Front. Phys. 11:1301453. doi: 10.3389/fphy.2023.1301453
Received: 25 September 2023; Accepted: 17 October 2023;
Published: 09 November 2023.
Edited by:
Felix Sharipov, Federal University of Paraná, BrazilReviewed by:
B. Venkateswarlu, Yeungnam University, Republic of KoreaAndaç Batur Çolak, Istanbul Commerce University, Türkiye
Wasfi Shatanawi, Hashemite University, Jordan
Copyright © 2023 Zulqarnain, Nadeem, Siddique, Samar, Khan and Mohamed. 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: Mahvish Samar, mahvishsamar@hotmail.com