- 1Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Terengganu, Malaysia
- 2Department of Mathematics and Statistics, Hazara University Mansehra, Mansehra, Pakistan
- 3Mechanical Engineering, Faculty of Engineering & Technology, Future University in Egypt, New Cairo, Egypt
- 4Department of Mathematics, University of Swabi, Swabi, Pakistan
- 5School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu, China
- 6Mechanical Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Wadi addawaser, Saudi Arabia
- 7Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt
The heat transfer ratio plays an important role in the industrial and engineering sectors; in this model, the authors used the hybrid nanofluid because the heat transfer ratio of the hybrid nanofluid is more than that of the base fluid. The key objective of this research work is to boost up the heat transfer ratio, for example, not only the accomplishment of energy is enough but is also expected to regulate the feeding of energy, and this is possible only to approve the development of heat transmission liquids to the mechanism of the expenditures of energy and improvement. The current research study investigates the influence of Marangoni convection, solar radiation, and viscous dissipation on the bioconvection couple stress flow of the hybrid nanofluid over a shrinking surface. This type of flow has some important application in the industrial and engineering sectors for the purpose of cooling and heating effect. To transform the non-dimensionless form of the differential equation to the dimensionless form, the authors used the defined similarity transformation. The transformed dimensionless form of the differential equation is solved by the homotopic analysis method. The obtained important result is determined with the help of graphs which is obtained from velocity and temperature equations. The impression of different parameters such as couple stress parameter, Marangoni convection parameter, nanoparticle volume fraction, solar radiation parameter, magnetic field parameter, thermophoresis parameter, Eckert number, and Prandtl number is taken over graphs. The skin friction coefficient and Nusselt number are described in the form of tables.
Introduction
Nanotechnology has produced massive consciousness amongst the researchers due to its wide possibility of applications in different branches of science and technology. The important use of the research on nanotechnology is to boost the heat transfer ratio. To enhance the heat transfer ratio, different approaches are used for the regular fluid. Several researchers work on the combination of solid and liquid for the improvement of ratio. The importance of magneto-Marangoni convection due to the immoderate uses precisely, sprinkling of thin film flow, use in the important apparatus in atom, and helpful in making semiconductors, played a very enormous role in the process of welding and showed good performance in the crystal growth. Marangoni convection is also used in material science, varnish, and silicon melts and plays an important role in factories and industries. Moreover, to color the ground, the Marangoni phenomenon is commonly used. The pigment is hanged on the exterior surface of the essential medium like water or other thickness fluids in this procedure. Das et al. (2008) for the first time introduced the concept of nanofluids. Buongiorno (2006) studied the convective heat transport in nanofluids. Due to the important uses of nuclear reactors, plasma studies, wire drawing, hot rolling, and manufacturing of glass fiber, researchers take more interest in the flow of nanofluids over the extending surface and attended by Lorentz’s effect. Makinde and Aziz (2011) used a stretching surface to study the boundary-layer flow of a nanofluid. Rana and Bhargava (2012) used a non-linear stretching surface to study the flow and heat transfer of a nanofluid. Khan and Pop (2010) used a stretching surface to study boundary-layer flow of a nanofluid. Makinde et al. (2016) used a stretching surface to study the MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat. Besthapu et al. (2017) used a stretching surface to study the MHD flow of the nanofluid. Acharya et al. (2016) used a stretching surface to study ramification of variable thickness on MHD TiO2 and Ag nanofluid. Acharya NDas and Prabir (2016) used two parallel plates to study squeezing flow of Cu–water and Cu–kerosene nanofluids. Das et al. (2016) used a shrinking sheet to study the onset of the nanofluid in the presence of a heat source/sink. Ishfaq et al. (2016) used a stretching surface to study the estimation of the boundary-layer flow of a nanofluid. Rana P Bhargava and Beg (2012) used a pour surface to study the mixed convective boundary-layer flow of a nanofluid numerically. The researchers take more interest in magneto-Marangoni convection which is produced due to the surface tension. The purpose behind this is some important applications such as, scattering of the thin liquid layer, atomic reactor, the processing of semiconductors, dynamic use in the welding process, crystal growth, material science, varnish, and silicon melt. The other important application of Marangoni convection is fine art mechanism, for instance, pigment on the ground. Pop Postelnicu (2001) studied the effect of Marangoni convection. Al-Mudhaf and Chamkha (2005) with the help of a porous medium investigated the Marangoni convection effect. Wang (2006) used a series solution method to study the influence of Marangoni convection. Chen (2007) discussed the inspiration of Marangoni convection. Magyari and Chamkha (2007) using the high magnitude of Re discussed the inspiration of Marangoni convection. Zheng et al. (Lin et al., 2013; Lin et al., 2014) studied the MHD Marangoni convection along with the thermal gradients. Aly and Ebaid (2016) used the Laplace transform to study the Marangoni flow over a permeable surface. Ellahi et al. (2016) used the ethylene glycol-based nanofluid to study the different shapes of nano-scale materials. Xu and Chen (Jiao et al., 2016) used permeable media to study Marangoni convection. Sheikholeslami and Chamkha (2017) studied the MHD Marangoni convection along with the two-phase nanoliquid hydrothermal model. The researchers take more interest in the novel kind of nanofluid known as the hybrid nanofluid due to a high heat transmission relation, which plays a significant role in the field of science and technology (Hussanan et al., 2019). They studied novel kind known as the hybrid nanofluid flow for some useful applications in manufacturing. (Eastman et al., 1996; Eastman et al., 1999; Moldoveanu et al., 2019; Kumar et al., 2020; Wakif et al., 2020). The thermal conductivity of EG (ethylene glycol) is little as compared to the other nanofluid. The required demand of the heat transfer ratio for current technologies and the wide requirement for thermal energy cannot be satisfied by the usually used fluids. The heat transfer ratio of the base fluid increases when the base liquids are mixed by the addition of minor shaped particles (Buongiorno et al., 2009). Thus, the increase in thermal possessions of usual fluids advanced strong interest of researchers to conduct more research. The overflowing literature on nanofluid and CNTs both (SWCNTs and MWCNTs) is studied by numerous researchers. Normally used in the energy sector and nanoscience (Kandasamy et al., 2016), copper oxide water was studied by Animasaun et al. (2019), carbon nanotube/water by Aman et al. (2017), heat transfer enrichment using the carbon nanotube by Raza et al. (2019), and the investigation of
Mathematical formulation
Consider a two-dimensional time-independent laminar incompressible flow of
where
The boundary conditions for the given flow problem are as follows:
From Eq. 4, we see that the velocity of the fluid and temperature changes along the x axis and above the x axis, the temperature of the surface remains constant, and the velocity of the fluid particles is zero;
To convert the non-dimensionless form of Eqs 2, 3, we introduced a dimensionless function of
In Eq. 5,
The boundary conditions for the given flow problem are as follows:
where
The non-dimensionless forms of skin friction and Nusselt number are as follows:
Using Eq. 5 in Eq 9, we have the dimensionless forms of skin friction and Nusselt number.
Solution by the HAM
To solve Eqs 6, 7 along with the boundary conditions (8), we apply the homotopy analysis method (HAM) with the following technique. The solutions having the auxiliary parameters
The initial guesses are selected as follows:
The linear operators are taken as
which have the following properties:
where
The resultant non-linear operatives
The basic idea of the HAM is described in Liao (2003), Liao (2004b), Liao (2010), and Acharya et al. (2017); the zeroth -order problems from Eqs 6, 7 are as follows:
The equivalent boundary conditions are as follows:
where
Expanding
where
The secondary constraints
The
The corresponding boundary conditions are
Here,
Stability analysis
The time-based stability of the multiple solutions as time changes is studied. This analysis was first presented by Merkin (1986) and then followed by Weidman et al. (2006). In order to achieve the solution time-based stability, the unsteady form of Eqs 2, 3 must be considered by suggesting the new dimensionless time variable equation one which will be unchanged. The unsteady forms of Eqs 2, 3 are
First, consider the new variables as follows:
Now, using Eq. 29 in Eqs 30, 31, the unsteady forms of Eqs 29, 30 become
The unsteady boundary conditions are
Now, consider the following perturbation function (Harris et al., 2009):
Equation 34 is used to apply a small disturbance
Subject to the boundary conditions
Without the loss of generality, we set
The solution will be stable if and only if the eigenvalue is positive, which shows the initial decay as time passes, and if the eigenvalue is negative at that point, the flow solution shows the initial growth of development, and the solution is said to be unstable as time passes.
Results and discussion
The important aim of this study is to deliver the idea of boost of the heat transfer, which is used in the new technology. In our study, we used different types of hybrid nanofluids for the improvement of the heat transfer ratio, which play an important role, for example, not only is the achievement of energy enough but is also expected to adjust the consumption of energy, and this is possible only to improve the development of heat transmission liquids to mechanism of the expenditures of energy and improvement. Most of the heat transmission is the demand of the industry and other related scientific fields. The second and important feature of this research is the enhancement of heat to reduce energy consumptions. The flow problem is conducted using a shrinking and stretching surface. The outputs of this study will be used to reduce energy consumption in industry and other engineering fields. In our research study, we used a new type of nanofluid known as the hybrid nanofluid for the enhancement of the heat transfer ratio because the heat transfer ratio of hybrid nanofluid is more than that of the base fluids. The hard elements dissolve in the base liquid, and after its constant diffusion, the hybrid nanofluid is produced. To transform the non-dimensionless form of the differential equation to the dimensionless form of the differential equation, the authors used the defined similarity transformation. The transformed dimensionless form of the differential equations is solved by the approximate analytical method. The total results obtained from the given flow problem are presented through figures. (Buongiorno, 2006; Khan and Pop, 2010; Makinde and Aziz, 2011; Rana and Bhargava, 2012; Acharya et al., 2016; Acharya N Das and Prabir, 2016; Makinde et al., 2016; Besthapu et al., 2017), the influence of different parameters on the velocity profile is presented in figures (Buongiorno, 2006; Khan and Pop, 2010; Makinde and Aziz, 2011; Rana and Bhargava, 2012; Makinde et al., 2016), and the influence of different parameters on the temperature profile is presented in figures (Acharya et al., 2016; Acharya NDas and Prabir, 2016; Besthapu et al., 2017). Table 1 shows the thermo-physical properties of water,
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TABLE 2. Convergence of the defined problem for the velocity equation; from table 2, we noted that as we increase the number of iteration, the residual error decreases and strong convergence is obtained.
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TABLE 3. Convergence of the defined problem for the temperature equation; from table 3, we see that as we increase the number of iteration, the residual error decreases and strong convergence is obtained.
![www.frontiersin.org](https://www.frontiersin.org/files/Articles/964543/fmats-09-964543-HTML/image_m/fmats-09-964543-t004.jpg)
TABLE 4. Consequence of different parameters such as
![www.frontiersin.org](https://www.frontiersin.org/files/Articles/964543/fmats-09-964543-HTML/image_m/fmats-09-964543-t005.jpg)
TABLE 5. Consequence of
Table 6 shows the comparison of the present work with already published work for the velocity equation.
Table 7 shows the comparison of the present work with already published work for the temperature equation.
There are two solutions in a certain range of parameters; an investigation of stability is conducted to find the most stable solution between them. To solve Eqs 24, 25 and the boundary conditions ((Liao, 1997)), the bvp4c solver in MATLAB software is used. This tool helps solve the equations numerically. Table 8 shows the certain values of
Figure 2 shows variation in
Figure 3 shows the variation in nanoparticle volume fraction
Figure 4 shows the variation in couple stress
Figure 5 shows the impact of the solar radiation parameter on velocity distribution for both
Figure 6 shows the variation in Marangoni convection
Figure 7 shows the relation between the solar radiation parameter and temperature profile for both
Figure 8 shows the relation between Eckert number and temperature profile for both
Figure 9 shows schemed
Figure 10 shows that the fluid velocity increases when the effect of the shrinking and shrinking parameters
Conclusion
This study investigates the influence of Marangoni convection, solar radiation, and viscous dissipation on the bioconvection couple stress flow of the hybrid nanofluid over a shrinking surface is investigated analytically. The novelty of current research is that for the first time, the influence of Marangoni convection, solar radiation, and viscous dissipation on the bioconvection couple stress flow of the hybrid nanofluid over a stretching surface is investigated analytically. To transform the non-dimensionless form of the DE (differential equation) to the dimensionless form of DE (differential equation), we used similarity transformation. The transformed dimensionless form of the DEs (differential equations) are solved by the approximate analytical method. In the future, we will face the problem to adjust the consumptions of energy; in our research article, we used a hybrid nanofluid which helped in the enhancement of the heat transfer ratio, which have some important applications, for example, not only the achievement of energy is enough but is also expected to adjust the consumptions of energy, and this is possible only to approve the development of heat transmission liquids to the mechanism of the expenditures of energy and improvement. Most of the heat transmission is the demand of the industry and other related scientific fields. Former to the application of nanotechnology, analysts and engineers have challenged such huge numbers of questions recognizing with heat transmission fluids. Still, with the development of nanometer-sized particles, and its uses in the heat transfer fluids have overall improved thermal conductivity. The development of heat transport via nanofluid has involved a number of scientists due to a lot of uses in various sectors such as distillation and separation of bio-molecules, biosensors, atomic system cooling, manufacture of glass fiber, thermal storing, in solar water boiler, in the field of defense, MRI, thermal absorption process, drug delivery, and transportation (thermal management od vehicle and cooling of engine). The impression of different parameters is taken by graphs for the velocity equation and the temperature equation. The key findings of the present research article are as follows:
1.Increasing the value of the solar radiation parameter increases the velocity field.
2.Increasing the value of the magnetic field parameter decreases the velocity field.
3.Increasing the value of nanoparticle volume fraction decreases the velocity field.
4.Increasing the value of the couple stress parameter decreases the velocity field.
5.Increasing the value of the Marangoni convection parameter doubles the effect in velocity field.
6.Increasing the value of the Prandtl number decreases the temperature field.
7.Increasing the value of the solar radiation parameter increases the temperature field.
8.Increasing the value of the Eckert number increases the temperature field.
Data availability statement
The original contributions presented in the study are included in the article/supplementary materials; further inquiries can be directed to the corresponding author.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
The authors will pay all the outstanding dues after the acceptance of this manuscript.
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
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Nomenclature
Abbreviations
x, y Cartesian coordinates
u, v velocity components
Uw velocities of the stretching sheet
Keywords: hybrid nanofluid, shrinking surface, OHAM BVP2.0 package, Marangoni convection, solar radiation
Citation: Rehman A, Khan W, Abdelrahman A, Jan R, Khan MS and Galal AM (2022) Influence of Marangoni convection, solar radiation, and viscous dissipation on the bioconvection couple stress flow of the hybrid nanofluid over a shrinking surface. Front. Mater. 9:964543. doi: 10.3389/fmats.2022.964543
Received: 08 June 2022; Accepted: 29 July 2022;
Published: 21 October 2022.
Edited by:
M. K. Samal, Bhabha Atomic Research Centre (BARC), IndiaReviewed by:
Ashish Mishra, Tula’s Institute, IndiaSawan Rawat, KIET Group of Institutions, India
Yahaya Shagaiya Daniel, Kaduna State University, Nigeria
Copyright © 2022 Rehman, Khan, Abdelrahman, Jan, Khan and Galal. 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: Waris Khan, wariskhan758@yahoo.com