Front. Phys.Frontiers in PhysicsFront. Phys.2296-424XFrontiers Media S.A.94990710.3389/fphy.2022.949907PhysicsOriginal ResearchHeat and Mass Transfer Analysis for Unsteady Three-Dimensional Flow of Hybrid Nanofluid Over a Stretching Surface Using Supervised Neural NetworksShoaib et al.Intelligent Networks for Hybrid NanofluidicsShoaibMuhammad^{1}AbukhaledMarwan^{2}RajaMuhammad Asif Zahoor^{3}*KhanMuhammad Abdul Rehman^{4}SabirMuhammad Tauseef^{1}NisarKottakkaran Sooppy^{5}*IltafIqra^{6}^{1}Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, Pakistan^{2}Department of Mathematics and Statistics, American University of Sharjah, Sharjah, United Arab Emirates^{3}Future Technology Research Center, National Yunlin University of Science and Technology, Douliou, Taiwan^{4}Department of ECE, COMSATS University Islamabad, Islamabad, Pakistan^{5}Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia^{6}Department of Mathematics, Riphah International University Islamabad, Islamabad, Pakistan
Edited by:Luigi Fortuna, University of Catania, Italy
Reviewed by:Arturo Buscarino, Università degli Studi di Catania, Italy
Salvina Gagliano, Università degli Studi di Catania, Italy
Lucia Valentina Gambuzza, University of Catania, Italy
*Correspondence: Muhammad Asif Zahoor Raja, rajamaz@yuntech.edu.tw; Kottakkaran Sooppy Nisar, ksnisar1@gmail.com
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics.
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.
The application of hybrid nanomaterials for the improvement of thermal efficiency of base fluid has increasingly gained attention during the past few decades. The basic purpose of this study is to investigate the flow characteristics along with heat transfer in an unsteady three-dimensional flow of hybrid nanofluid over a stretchable and rotatory sheet (3D-UHSRS). The flow model in the form of PDEs was reduced to the set of ordinary differential equations utilizing the appropriate transformations of similarity. The influence of the rotation parameter, unsteadiness parameter, stretching parameter, radiation parameter, and Prandtl number on velocities and thermal profile was graphically examined. A reference solution in the form of dataset points for the 3D-UHSRS model are computed with the help of renowned Lobatto IIIA solver, and this solution is exported to MATLAB for the proper implementation of proposed solution methodology based on the Levenberg–Marquardt supervised neural networks. Graphical and numerical results based on the mean square error (MSEs), time series response, error distribution plots, and regression plots endorses the precision, validity, and consistency of the proposed solution methodology. The MSE up to the level of 10–12 confirms the accuracy of the achieved results.
In the recent past, improvement in heat transfer for various engineering and industrial applications remains the main topic of research for scientists. Improvement in the heat transfer ability can be done through different methods such as use of microchannel, extended surface and through vibrating the surface. Thermal conductivity plays the most important role among all other characteristics of fluid for judging its heat transfer ability. Thermal conductivity of any nanofluid depends on the type, as well as the shape of nanoparticles. There will be less energy loss with the improved thermal efficiency of the fluid, which can further reduce the cost and increase the production in industrial applications. Various conventional fluids, such as water, kerosene oil, and ethylene glycol, are commonly used in many industrial and engineering applications (e.g., chemical manufacturing, water distillation, HVAC systems, medicine, and power-generation systems), but due to inadequate thermal transfer ability, these ordinary fluids lack good efficiency in systems having large thermal transfer requirements. As it is well known, the ability to conduct heat for metals is hundred times greater than that in liquids, which is why it is appropriate to use them in thermal systems [1–5]. Thus, the use of nanoparticles <100 nm not only improved the thermal capabilities of the fluid, but also positively improved the other rheological properties. Initially [6] was the first work to discuss the nature and characteristics of nanofluids. [7] compared the performance of heat transfer between the nanofluid comprised of a single wall and multiwall carbon nanotubes in a magnetohydrodynamic (MHD) flow under the influence of radiative flux.
A new type of nanofluids known as “hybrid nanofluids” is made by dispersing two different types of nanoparticles into base fluid to obtain various synergetic effect of both types of nanomaterial. The development of hybrid nanofluidic system is currently undergoing and under the evaluation to enhance the strength of thermal effects for different flow dynamics. Performance in terms of thermal energy utilization is being tested and assessed by various researchers. Due to its marvelous heat transfer abilities, this new type of nanofluid attracted the attention of researchers and scientists to investigate it in many of the industrial and engineering problems. Hybrid nanofluid have extensive range of application in scientific, industrial, engineering, and medical fields like medicine manufacturing, heating systems, transfer cooling, electronic chips, and solar panels [8]. Of late, several studies have been conducted to analyze the improvement of heat transfer capabilities for hybrid nanofluids as compared to conventional single-particle nanofluids [9–13]. The flow over rotating surfaces has many practical applications such as electrical appliances, cutting discs, data-storing devices, and heavy machinery parts. Song et al. [14] implemented a numerical shooting methodology to analyze the effects of various physical parameters on the velocity and thermal fields in the stagnation point flow of a hybrid nanofluid over a rotatory disc under radiative and activation energy effects. [15] numerically evaluated the three-dimensional MHD flow of hybrid nanofluid over a rotating and stretching sheet. In addition, the behavior of velocity and temperature profile depending on different physical factors has been highlighted through graphs and numerical results.
Any flow in which all of its parameters are independent of time is known as steady flow, whereas the time-dependent flow is known as unsteady flow. Due to excessive applicability in engineering and industrial applications, steady flow has gained a lot of concentration of researchers and scholars. However, recently many studies have been conducted on the behavior and characteristics of unsteady fluid flow systems, automobiles, power-generation systems, and aviation industry [16], explored the influence of variable thermal conductivity, viscosity, and Joule heating on the velocity and thermal field in the unsteady flow of 3D Maxwell nanofluid over a stretching surface. An enhancement in the velocity of the fluid has been observed with increased viscosity. [17] numerically investigated the biconvection flow of cross nanofluid and studied the effects of thermal radiation along with melting phenomenon over a cylinder. [18] studied the considerable effect of external magnetic field acting at an inclined angle on Williamson’s nanofluid over a rotating stretchable surface. [19] conducted a comparative analysis of heat transfer over a porous stretching surface between nanofluid containing nanoparticles of Graphene oxide and the combination of Ag–graphene oxide in kerosene oil as base fluid.
[20] described experimental research of various factors responsible for achieving better thermophysical properties and more stable results in terms of heat transfer in nanofluids and ionanofluids. There are a variety of nanoparticles available for manufacturing of nanofluids and hybrid nanofluid. Each type of material has its own benefits and limitations depending on the material’s characteristics. [21] presented the importance and significance of Aluminum nanoparticles in the industrial and engineering fields and their various advantages over other renowned nanomaterials. [22] explicated an analysis to show the physical and chemical stability and thermophysical properties based on thermal conductivity for the nanofluid comprising TiO_{2} nanoparticles. Furthermore, different experimental results showing the improvement in thermal properties have also been tabulated in the research. [23] presented a numerical investigation to examine the flow behavior, as well as heat transmission capabilities of a hybrid fluid over a gyrating surface under the influence of a uniform external magnetic field. [24] numerically explained the behavior of mass and heat transfer for the MHD flow of a nanofluid over stretched surface under the heating effects through the radiation phenomenon. [25] used state-of-the-art supervised neural network to solve the hybrid nanofluid flow model with Joule heating and MHD effects. A comparison with an already available solution has also been made to show the accuracy and performance of solution methodology. Among several other key factors, the concentration of nanoparticles also plays an important part in deciding the thermal capability of nanofluids. [26] presented experimental research showing a rapid increase in thermal efficiency of an oil-based nanofluid. In a similar manner, [27] explained the fact that for a nanofluid with TiO_{2} nanoparticles in water, the negative effect of using a lower concentration of nanoparticles can be overcome by the other important properties such as critical heat flux. [28] utilized the Laplace transformation method to solve the flow model of an incompressible non-Newtonian hybrid nanofluid over permeable surface revolving with uniform acceleration with velocity and thermal slip effects. [29] conducted a numerical analysis of activation energy for the two-dimensional flow of a hybrid nanofluid under the effect of buoyancy force and thermal radiation. Furthermore, the influence of key parameters, such as the Nusselt number and Sherwood number, on the heat and velocity profiles has also been investigated.
Most of the researcher used conventional solution methodologies to explain various fluid models describing the effects of entropy generation, rotating flow problems, and Joule heating [30–36]. The application of modern solution methodologies based on artificial neural networks to solve such problems is an inventive work. Of late, researcher applied these modern solutions depending on artificial intelligence to resolve the problem related to various field such as financial trading [37], rainfall prediction models [38], bioinformatics [39, 40], fluid dynamics [41, 42], energy, HIV virus spread models [40, 43, 44], and coronavirus perdition model [45] (also see [46, 47, 49]). In this study, the authors intended to solve the 3D-UHSRS fluid problem for the first time, to the best of our knowledge, by utilizing the Levenberg–Marquardt supervised neural networks (LM-SNNs) bases solution technique. Theses modern solution approaches were based on state-of-the-art computational algorithms that can easily tackle the nonlinear behavior of flow model’s equations.
The basic purpose of this research work is to investigate and explain the flow feature along with the heat and mass transfer in 3D-UHSRS. The basic feature of purposed solution methodology for 3D-UHSRS are as follows:
• A state-of-the-art mathematical model for the 3D-UHSRS problem has been developed and expressed in terms of PDEs, which are further converted into a set of nonlinear ordinary differential equations (ODEs) using dimensionless similarity variables.
• A numerical solution of the 3D-UHSRS problem was obtained by implementing the Lobatto IIIA solution methodology based on the bvp4c solver in MATLAB. The solution datasets of the 3D-UHSRS problem is then subjected for LM back propagation to carry out the training, validation, and testing of the data set points.
• Different plots describing the effect of physical constants on velocity as well as thermal profile have been presented. The performance, convergence, and accuracy of the solution approach were validated thorough residual error, the number of grid points, ODEs, and BCs evaluations.
• Numerical and graphical results in the form of the MSE curve, error histograms, regression and error plots were authenticated from the performance, precision, and convergence of the proposed LM-SNNs methodology.
Problem Formulation
Assume a 3D unsteady flow of a hybrid nanofluid over a stretchable surface, as shown as Figures 1and 2. The surface is stretched in the xy-plane of the Cartesian coordinate system. The surface is rotating with a uniform velocity “w” about an axis for which z=0. Here, (u,v,w) are the components of velocity in the direction along (x,y,z). Stretching velocities of the surface and in x and y direction are represented by u and v. Temperature of fluid at stretching surface is Tw, whereas the temperature of the ambient fluid is T∞.
Geometrical interpretation for magnetohydrodynamic-HNRD.
Working flow chart.
According to abovementioned assumptions, described flow model can be expressed in term of following set of mathematical equations (50) and (51):∂u∂x+∂v∂y+∂w∂z=0,ρhnf(∂u∂t−2ωv+u∂u∂x+v∂u∂y+w∂u∂z)=μhnf∂2u∂z2,ρhnf(∂v∂t+2ωu+u∂v∂x+v∂v∂y+w∂v∂z)=μhnf∂2v∂z2,ρhnf(∂w∂t+u∂w∂x+v∂w∂y+w∂w∂z)=μhnf∂2w∂z2,∂T∂t+u∂T∂x+v∂T∂y+w∂T∂z=khnf(ρCp)hnf[∂2T∂z2]+1(ρCp)hnf[∂qr∂z].
In the above system of equations, qr exhibits radiative heat flux known as Rosseland approximation, which can be mathematically expressed asqr=−4σ∗3k∗∂T4∂y
Here k∗ is the Stephen–Boltzmann constant, and σ∗ represents the mean absorption coefficient. Expending T4 and ignoring the higher order terms, we get T4=4TT∞3−3T∞4.
Using the value of T4 in Eq. (5), we obtain∂T∂t+u∂T∂x+v∂T∂y+w∂T∂z=1(ρCp)hnf(khnf+16σ∗T∞33k∗)∂2T∂z2
Along with the following boundary conditions of the system:Whenz=0Thenu=uw=cx1−αt,v=0,w=0,T=TwWhenz→∞Thenu→0,v→0,T→T∞
Here, “c” is the stretching coefficient of the surface, qr is the amount of radiative flux, ω is angular velocity, and T_{∞} is temperature of the ambient fluid. Undermentioned linear transformation is designed to transmute the modeling ODEs (1–5) into a nonlinear equivalent system of PDEs.u=ax1−αtf′(η),v=ax1−αtg(η),w=−aνf1−αtf(η),η=aνf1−αtz,θ(η)=T−T∞Tw−T∞.}
In the above equations, ρhnf is density, μhnf is dynamic viscosity, and (ρcp)hnf is the heat capacity of the hybrid nanofluid. Whereas ϕ1 represents the concentration of Al nanoparticles, and ϕ2 represents the concentration of nanoparticles of TiO_{2}. As “hybrid nanofluid” is a combination of two different types of nanoparticles, ϕhnf is the total concentration of nanoparticle in the base fluid that can be calculated as the sum of concentrations of both types of nanoparticles, i.e., ϕ1+ϕ2. All mathematical relationships expressing thermophysical properties are expressed in Table 1 [51]:
Mathematical expression of various thermophysical quantities.
Values regarding density, electrical, and thermal conductivity and specific heat against used nanoparticles and base fluid has been mentioned in Table 2.
Numerical values for various physical and chemical properties [48, 49].
Material
Density
Specific Heat
Electrical Conductivity
Thermal Conductivity
SI unit
(Kg/m^{3})
(J Kg^{−1}K^{−1})
(s/m)
(W m^{−1}K^{−1})
Water (H_{2}O)
997
4,179
5.5 × 10^{–6}
0.613
Al Nanoparticles
3,970
765
5.96 × 10^{7}
40
TiO2 Nanoparticles
4,250
686.2
2.38 × 10^{6}
8.953
Following important physical terminologies representing the skin friction and Nusselt number can be written as:Cfx=μhnfρfuw2(∂u∂z)z=0,Cfy=μhnfρfuw2(∂v∂z)z=0,Nux=−rkhnfkf(Tf−T∞)(∂T∂z)z=0+x(qr)z,Cfx
The mathematical evaluation of Eqs. 1 and 2)–5 givesμhnfμfρhnfρff′′′−f′2+ff″+2Ωg−β(f′+η2f″)=0,μhnfμfρhnfρfg″−f′g+fg′−2Ωf′−β(g+η2g′)=0,1Pr(ρCp)f(ρCp)hnf(khnfkf+43Rd)θ″−βη2θ′+fθ′=0,
The physical parameters of 3D-UHSRS flow problem involved in Eqs. 8–12 areλ=ca,Ω=ω∗c,Rd=4σ∗T∞2k∗kf,β=αa,Pr=υf(ρcp)fkf=υfαf
Here, Rd represents the radiation parameter, Pr is the Prandtl number, β is the unsteadiness parameter, Ω is the rotation parameter, λ decides the stretching or shrinking ability of the sheet, and λ=0 corresponds to the static nature of the revolving surface. The nondimensionalized form of Skin friction coefficient and Nusselt number are as follows:CfxRex1/2=[μhnfμf]f″(0),CfyRex12=[μhnfμf]g′(0)NuxRex−12=−(khnfkf+43Rd)θ′(0),CfxRex1/2
in which Rer=uxυf represent the local rotational Reynold number.
Results and Discussion
Solution of the system of ODE resenting the flow model has been accomplished in two major phases, in the first phase, set of nonlinear ODEs along with their relevant boundary conditions (Eqs. (1)–(7) are transformed into the first-order ODEs for the solution by using the renowned solution technique “Lobatto IIIA” with the use of MATLAB software. Graphical variation of velocity and temperature profile against various important physical parameters are shown in Figures 3 and 4. Various scenarios and cases are generated based on the variation of involved physical parameters, as described in Table 3.
Variation of velocity and temperature profile against β, ω, and, Rd.
Variation of velocity and temperature profile against λ andPr.
Variation of parameters for the unsteady three-dimensional flow of hybrid nanofluid over a stretchable and rotatory sheet problem.
Scenario
C-I
C-II
C-III
C-IV
I
β = −1.5
β = −1.2
β = −0.8
β = −0.5
II
λ = 0.6
λ = 0.8
λ = 1.0
λ = 1.4
III
ω = 2.0
ω = 2.3
ω = 2.6
ω = 3.0
IV
Rd = 2.0
Rd = 2.3
Rd = 2.6
Rd = 3.0
V
Pr = 6.0
Pr = 6.0
Pr = 6.0
Pr = 7.0
Figure 3A ∼ c) presented the impact of β (unsteadiness parameter) on the velocities in x and y directions, as well as on the temperature of the fluid. It was observed that, initially, the velocity of the fluid showed a decreasing trend with the increase in the values of β; after that, a further increase in β will produce a direct change in the velocity of the fluid because an increase in β will result in thickening of the velocity boundary layer. In addition, higher values of β will also results in the fluid temperature increase. Figure 3D ∼ e) revealed the rising trend of velocities against the increasing values of ω (the rotation parameter). In reality, centrifugal forces increase with the increase in ω, which tends the particles of fluid to move into the y direction. Figure 3F exposed the influence of ω (the rotation parameter) on the temperature profile of the fluid. As with the increase in the rotation parameter, the radial velocity and hence the kinematic energy of the fluid rises, which produces more heat; therefore, the overall temperature of the fluid rises.
Figures 4A,B presents the influence of λ (the stretching parameter) on the velocity and thermal profile of the fluid. It was observed that initially velocity of the fluid increases with the increase in values of stretching parameter and the temperature of the fluid shows a slight decline against the rising values of stretching parameter. Figure 4C portrayed the variation in temperature profile of the fluid against the variable values of Rd (the radiation parameter). It is understood that radiation parameter expresses the amount of heat contributed to fluid via thermal radiation. Due to this fact, increasing the radiation parameter will generate more heat in the fluid. Figure 4D was drawn to study the effect of the Prandtl number (Pr) on the temperature profile of the fluid, and it has been observed that with the rise in the Prandtl number, the overall temperature of the fluid shows a declining behavior. As the Prandtl number varies inversely to the thermal diffusivity of the fluid, therefore, with the rise in the Prandtl number, the viscosity of the fluid also rises, causing the temperature of the fluid to drop.
In the second phase, numerical results of the solution containing datasets for each involved variable against all scenarios and cases are generated between a fixed domain, and these dataset points are further subjected to the proper execution of innovative LM-SNNs-based solution. During the operation of proposed methodology out of total points, 80% points were subjected under the network training, whereas the leftover 10% dataset points were used for the operation of validation and testing processes. The number of neuron and delay steps in the calculations were adjusted as per complexity of the problem and required level of accuracy. A two-layer internal structure of the LM-SNNs is presented in Figure 5. Table 4–8 presents the numerical performance indicators comprising of MSE, gradient, Mu, and the number of epochs for scenario (I–V), respectively.
Internal structure of NAR.
Numerical performance indicators for scenario I.
Scenario
Cases
Number of Neurons
Mean Square Error
Grad
Mu
Total Epochs
Time (s)
Trng
Valid
Test
I
I
30
1.351e-10
1.713e-10
2.043e-10
9.675e-08
1e-12
71
14
II
30
1.873e-10
1.873e-10
2.180e-10
9.995e-08
1e-11
82
15
III
30
1.452e-10
1.611e-10
3.129e-10
9.045e-08
1e-11
87
15
IV
30
1.314e-10
9.439e-11
1.317e-10
9.287e-08
1e-12
64
14
Numerical performance indicators for scenario II.
Scenario
Cases
Number of Neurons
Mean Square Error
Grad
Mu
Total Epochs
Time (s)
Trng
Valid
Test
II
I
30
3.835e-11
5.491e-11
6.678e-11
9.675e-08
1e-12
71
14
II
30
2.832e-11
3.725e-11
3.888e-11
9.668e-08
1e-12
64
12
III
35
1.682e-10
1.886e-10
2.093e-10
9.963e-08
1e-11
89
15
IV
30
1.488e-10
1.5190e-10
1.706e-10
9.761e-08
1e-11
90
15
Numerical performance indicators for scenario III.
Scenario
Cases
Number of Neurons
Mean Square Error
Grad
Mu
Total Epochs
Time (s)
Trng
Trng
Trng
III
I
30
2.201e-11
2.539e-11
2.425e-11
9.935e-08
1e-12
76
14
II
30
2.180e-11
3.142e-11
3.352e-11
9.808e-08
1e-12
70
14
III
30
1.655e-10
1.667e-10
1.929e-10
9.865e-08
1e-11
90
16
IV
30
2.132e-11
2.220e-11
2.958e-11
9.460e-08
1e-12
83
15
Numerical performance indicators for scenario IV.
Scenario
Cases
Number of Neurons
Mean Square Error
Grad
Mu
Total Epochs
Time (s)
Trng
Valid
Test
IV
I
30
1.967e-10
2.059e-10
2.113e-10
9.887e-08
1e-11
88
15
II
30
1.933e-11
2.755e-11
2.328e-11
9.742e-08
1e-12
75
13
III
30
2.201e-11
2.492e-11
3.086e-11
9.716e-08
1e-12
70
13
IV
30
2.158e-11
2.158e-11
2.948e-11
9.783e-08
1e-12
72
13
Numerical performance indicators for scenario V.
Scenario
Cases
Number of Neurons
Mean Square Error
Grad
Mu
Total Epochs
Time (s)
Trng
Valid
Test
V
I
30
2.241e-11
1.968e-11
2.265e-11
9.827e-08
1e-12
73
14
II
30
1.607e-10
1.982e-10
3.659e-10
9.652e-08
1e-11
92
15
III
30
1.653e-10
1.881e-10
1.780e-10
9.821e-08
1e-11
91
15
IV
30
1.484e-10
3.305e-10
1.700e-10
9.929e-11
1e-11
101
16
Figure 6A ∼ d) presents the graphical details of the time series response, error distribution plot, MSE, and gradient plots against the second case of the first scenario. Furthermore, MSE for any computational methodology represents the mean of square of differences between the actual value and estimated value. Accuracy and stability of any method can be judged through the MSE value encountered during the computational process. Smaller MSE values correspond to a better solution and an accurate and reliable solution technique. Figures 7A–D were designed to illustrate the MSE plots of the first case of each scenario for training, validation, and testing to compare performances based on MSE for each case.
Performance and accuracy plots of C-2 S-1.
Mean square error bases performance of second cases for all scenarios.
The reference solution of the problem is available in the form of dataset points that are further categorized into training, validation, and testing processes at a specific ratio. In time series response, a close relationship between target and output values of training, validation, and testing depict the accuracy and precision of the solution methodology. Time series response of any variable is the statistical measure of any characteristics with respect to time. In addition, time series response help the researchers to estimate and understand the performance of a variable through any calculated data. Through these fitness plots one can easily observe the separate accuracy of all training, validation, and testing points as compared to available graphical solution. Figures 8A–D illustrate the fitness plots for first cases of all scenarios.
Fitness performance plots of second cases for all scenarios.
Error histograms are another way to measure the closeness of predicted values with the reference values. These histograms are actually the distribution of errors of all these computed values from a zero error point. Errors in term of all the achieved values are classified into 20 portions, which are aligned across a line representing the zero error line. More values that lie close to the zero error line indicate more accuracy and precision of the solution methodology. Figures 9A–D were designed to illustrate a comparison of error histograms for first cases of all scenarios.
Error distribution plots of second cases of all scenarios.
Regression is a graphical way to present the precision of the predicted values to the reference values separately for training, validation and testing points. In these plots, an available reference solution has been shown by a straight line, whereas the predicted values are shown by dots or small circles. Accuracy and precision of the computation can also be judged through the numerical value of regression. R = 1 means that the predicted values are very close to reference values, and R = 0 means there is a very poor relationship between reference and predicted values. Figures 10A–D exhibit a comparison of regression plots between first cases of all scenarios.
Regression plots of second cases of all scenarios.
Gradient is actually a vector responsible for guiding the network in the right direction with an accurate magnitude to reach the required solution as early as possible, whereas mu is a factor that controls a certain algorithm. The value of mu directly portraits the convergence of the solution. Figures 11A–D reflect a comparison between the gradient and mu plots of first cases of all scenarios.
Plot for gradient and mu of second cases of all scenarios.
Figure 12 portrays the graphic comparison of proposed LM-SNNs base solutions with the already available solutions for all scenarios, and additional comparison of errors for all cases of each scenario are also placed opposite showing the accuracy and precision of the proposed solution methodology.
Comparison of the Levenberg–Marquardt supervised neural network solution with reference solution along with relative errors for various scenarios. (a), (b).
Conclusion
Here, we employed a numerical study using a novel LM-SNNs–based methodology to investigate the 3D-UHSRS by modeling it in terms of PDEs, which were further reduced to ODEs. A well-renowned solution technique “Lobatto IIIA” was implemented to solve these sets of equation. The influence of various involved physical parameters on velocity and thermal performance are visualized and studied. The solution in respect of each variable in the form of dataset points are acquired and placed in MATLAB for proper operation of the LM-SNNs solution by training, validation, and testing of these dataset points at 80%, 10%, and 10%, respectively. Accuracy, precision, and cogency of the solution were validated through various graphical results consisting of time series, MSE, error distribution, and regression plots.
Followings are few important research outcomes:
• Higher values of ω(rotational velocity) will make the fluid velocity to decrease, whereas temperature of the fluid shows an increasing trend for a similar change.
• An increase in velocity and decline in the temperature field were observed with the increasing values of λ (the stretching parameter).
• A boost in the fluid temperature was observed with increasing values of Rd (the radiation parameter) whereas reverse behavior has been noted for the Pr.
Solution methodologies working on the principle of artificial intelligence and machine learning can be more beneficial and valuable in solving the problems related to nano [52–54] and micro fluids [55–61].
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
MS, MR, and MK contributed to conception and design of the study. SS organized the database. II performed the statistical analysis. MS and II wrote the first draft of the manuscript. KN, MR, and SS wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.
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.
The work in this study was supported, in part, by the Open Access Program from the American University of Sharjah”. This study represents the opinions of the author(s) and does not mean to represent the position or opinions of the American University of Sharjah.
ReferencesWenDDingY. Effective Thermal Conductivity of Aqueous Suspensions of Carbon Nanotubes (Carbon Nanotube Nanofluids). RajaBGodsonLLalDMWongwisesS. Experimental Investigation on the Thermal Conductivity and Viscosity of Silver-Deionized Water Nanofluid. XingMYuJWangR. Experimental Study on the Thermal Conductivity Enhancement of Water Based Nanofluids Using Different Types of Carbon Nanotubes. AgarwalRVermaKAgrawalNKDuchaniyaRKSinghR. Synthesis, Characterization, thermal Conductivity and Sensitivity of CuO Nanofluids. AgarwalRVermaKAgrawalNKSinghR. Sensitivity of thermal Conductivity for Al2O3 Nanofluids. ChiSUEastmanJA. HussainAHassanAAl MdallalQAhmadHRehmanAAltanjiMHeat Transport Investigation of Magneto-Hydrodynamics (SWCNT-MWCNT) Hybrid Nanofluid Under the Thermal Radiation Regime. BabarHAliHM. Towards Hybrid Nanofluids: Preparation, Thermophysical Properties, Applications, and Challenges. BabuJARKumarKKSrinivasa RaoS. State-of-Art Review on Hybrid Nanofluids. GulzarOQayoumAGuptaR. Experimental Study on Stability and Rheological Behaviour of Hybrid Al2O3-TiO2 Therminol-55 Nanofluids for Concentrating Solar Collectors. ShahTRAliHM. Applications of Hybrid Nanofluids in Solar Energy, Practical Limitations and Challenges: A Critical Review. YangLJiWMaoMHuangJN. An Updated Review on the Properties, Fabrication and Application of Hybrid-Nanofluids along with Their Environmental Effects. NagoorAHAlaidarousESSabirMTShoaibMRajaMAZ. Numerical Treatment for Three-Dimensional Rotating Flow of Carbon Nanotubes with Darcy-Forchheimer Medium by the Lobatto IIIA Technique. SongY-QAli KhanSImranMWaqasHUllah KhanSIjaz KhanMApplications of Modified Darcy Law and Nonlinear Thermal Radiation in Bioconvection Flow of Micropolar Nanofluid over an off Centered Rotating Disk. HussainAArshadMRehmanAHassanAElaganSKAhmadHThree-Dimensional Water-Based Magneto-Hydrodynamic Rotating Nanofluid Flow over a Linear Extending Sheet and Heat Transport Analysis: A Numerical Approach. AhmadSCobanHHKhanMNKhanUShiQ-HMuhammadTComputational Analysis of the Unsteady 3D Chemically Reacting MHD Flow with the Properties of Temperature Dependent Transpose Suspended Maxwell Nanofluid. ImranMFarooqUWaqasHAnqiAESafaeiMR. Numerical Performance of Thermal Conductivity in Bioconvection Flow of Cross Nanofluid Containing Swimming Microorganisms over a Cylinder with Melting Phenomenon. SrinivasuluTGoudBS. Effect of Inclined Magnetic Field on Flow, Heat and Mass Transfer of Williamson Nanofluid over a Stretching Sheet. AhmadFAbdalSAyedHHussainSSalimSAlmatroudAO. The Improved Thermal Efficiency of Maxwell Hybrid Nanofluid Comprising of Graphene Oxide Plus Silver/Kerosene Oil over Stretching Sheet. BakthavatchalamBHabibKSaidurRSahaBBIrshadK. Comprehensive Study on Nanofluid and Ionanofluid for Heat Transfer Enhancement: A Review on Current and Future Perspective. FarhanaKKadirgamaKRahmanMMNoorMMRamasamyDSamykanoMSignificance of Alumina in Nanofluid Technology. YasinskiyANavasJAguilarTAlcántaraRGallardoJJSánchez-CoronillaADramatically Enhanced thermal Properties for TiO2-Based Nanofluids for Being Used as Heat Transfer Fluids in Concentrating Solar Power Plants. OuyangCAkhtarRRajaMAZTouseef SabirMAwaisMShoaibM. Numerical Treatment with Lobatto IIIA Technique for Radiative Flow of MHD Hybrid Nanofluid (Al_{2}O_{3}—Cu/H_{2}O) over a Convectively Heated Stretchable Rotating Disk with Velocity Slip Effects. ShoaibMRajaMAZSabirMTIslamSShahZKumamPNumerical Investigation for Rotating Flow of MHD Hybrid Nanofluid with Thermal Radiation over a Stretching Sheet. ShoaibMRajaMAZSabirMTNisarKSJamshedWFelembanBFMHD Hybrid Nanofluid Flow Due to Rotating Disk with Heat Absorption and Thermal Slip Effects: An Application of Intelligent Computing. PatelHEDasSKSundararajanTNairASGeorgeBPradeepT. Thermal Conductivities of Naked and Monolayer Protected Metal Nanoparticle Based Nanofluids: Manifestation of Anomalous Enhancement and Chemical Effects. AbbassiYTalebiMShiraniASKhorsandiJ. Experimental Investigation of TiO2/Water Nanofluid Effects on Heat Transfer Characteristics of a Vertical Annulus with Non-Uniform Heat Flux in Non-Radiation Environment. KrishnaMVAhammadNAChamkhaAJ. Radiative MHD Flow of Casson Hybrid Nanofluid over an Infinite Exponentially Accelerated Vertical Porous Surface. SuganyaSMuthtamilselvanMAlhussainZA. Activation Energy and Coriolis Force on Cu-TiO_{2}/Water Hybrid Nanofluid Flow in an Existence of Nonlinear Radiation. ShoaibMRajaMAZSabirMTAwaisMIslamSShahZNumerical Analysis of 3-D MHD Hybrid Nanofluid over a Rotational Disk in Presence of thermal Radiation with Joule Heating and Viscous Dissipation Effects Using Lobatto IIIA Technique. AlarifiIAbokhalilAOsmanMLundLAyedMBelmabroukHMHD Flow and Heat Transfer over Vertical Stretching Sheet with Heat Sink or Source Effect. YousifMAIsmaelHFAbbasTEllahiR. Numerical Study of Momentum and Heat Transfer of MHD Carreau Nanofluid over an Exponentially Stretched Plate with Internal Heat Source/Sink and Radiation. PalDChatterjeeDVajraveluK. Influence of Magneto-Thermo Radiation on Heat Transfer of a Thin Nanofluid Film with Non-Uniform Heat Source/Sink. IqbalSASajidMMahmoodKNaveedMKhanMY. An Iterative Approach to Viscoelastic Boundary Layer Flows with Heat Source/Sink and Thermal Radiation. RamadeviBKumarKASugunammaVSandeepN. Influence of Non-Uniform Heat Source/Sink on the Three-Dimensional Magnetohydrodynamic Carreau Fluid Flow Past a Stretching Surface with Modified Fourier's Law. KhanIRajaMAZShoaibMKumamPAlrabaiahHShahZDesign of Neural Network with Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations. BukhariAHRajaMAZSulaimanMIslamSShoaibMKumamP. Fractional Neuro-Sequential ARFIMA-LSTM for Financial Market Forecasting. BukhariAHSulaimanMIslamSShoaibMKumamPZahoor RajaMA. Neuro-Fuzzy Modeling and Prediction of Summer Precipitation with Application to Different Meteorological Stations. RajaMAZShahFHSyamMI. Intelligent Computing Approach to Solve the Nonlinear Van der Pol System for Heartbeat Model. UmarM. Stochastic Numerical Technique for Solving HIV Infection Model of CD4+ T Cells. AhmadIIlyasHUroojAAslamMSShoaibMRajaMAZ. Novel Applications of Intelligent Computing Paradigms for the Analysis of Nonlinear Reactive Transport Model of the Fluid in Soft Tissues and Microvessels. RajaMAZManzarMAShahSMChenY. Integrated Intelligence of Fractional Neural Networks and Sequential Quadratic Programming for Bagley–Torvik Systems Arising in Fluid Mechanics. UmarMSabirZRajaMAZShoaibMGuptaMSánchezYG. A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics. CheemaTNRajaMAZAhmadINazSIlyasHShoaibM. Intelligent Computing with Levenberg-Marquardt Artificial Neural Networks for Nonlinear System of COVID-19 Epidemic Model for Future Generation Disease Control. ShoaibMRajaMAZSabirMTBukhariAHAlrabaiahHShahZA Stochastic Numerical Analysis Based on Hybrid NAR-RBFs Networks Nonlinear SITR Model for Novel COVID-19 Dynamics. MaqsoodNMustafaMKhanJA. Numerical Tackling for Viscoelastic Fluid Flow in Rotating Frame Considering Homogeneous-Heterogeneous Reactions. Mohd SohutNFHSoidSKAbu BakarSIshakA. Unsteady Three-Dimensional Flow in a Rotating Hybrid Nanofluid over a Stretching Sheet. WainiIIshakAPopI. Unsteady Flow and Heat Transfer Past a Stretching/Shrinking Sheet in a Hybrid Nanofluid. HuminicGHuminicA. Heat Transfer Capability of the Hybrid Nanofluids for Heat Transfer Applications. AwaisMRajaMAZAwanSEShoaibMAliHM. Heat and Mass Transfer Phenomenon for the Dynamics of Casson Fluid through Porous Medium over Shrinking wall Subject to Lorentz Force and Heat Source/Sink. UddinIAkhtarRZhiyuZIslamSShoaibMRajaMAZ. Numerical Treatment for Darcy-Forchheimer Flow of Sisko Nanomaterial with Nonlinear thermal Radiation by Lobatto IIIA Technique. UddinIAkhtarRKhanMARZhiyuZIslamSShoaibMNumerical Treatment for Fluidic System of Activation Energy with Non-Linear Mixed Convective and Radiative Flow of Magneto Nanomaterials with Navier's Velocity Slip. SapuppoFBucoloMIntagliettaMJohnsonPCFortunaLArenaP. An Improved Instrument for Real-Time Measurement of Blood Flow Velocity in Microvessels. SchembriFSapuppoFBucoloM. Experimental Classification of Nonlinear Dynamics in Microfluidic Bubbles' Flow. CaironeFAnandanPBucoloM. Nonlinear Systems Synchronization for Modeling Two-Phase Microfluidics Flows. CaironeFGaglianoSBucoloM. Experimental Study on the Slug Flow in a Serpentine Microchannel. AnandanPGaglianoSBucoloM. Computational Models in Microfluidic Bubble Logic. CaironeFGaglianoSCarboneDCReccaGBucoloM. Micro-Optofluidic Switch Realized by 3D Printing Technology. AbukhaledMKhuriSRabahF. Solution of a Nonlinear Fractional COVID-19 Model. SaravanakumarSEswariARajendranLAbukhaledM. A Mathematical Model of Risk Factors in HIV/AIDS Transmission Dynamics: Observational Study of Female Sexual Network in India. MaryMLCDeviMCMeenaARajendranLAbukhaledM. Mathematical Modeling of Immobilized Enzyme in Porous Planar, Cylindrical, and Spherical Particle: A Reliable Semi-Analytical Approach. NomenclatureSymbolsΩ