Applied Mathematics for Engineering Problems in Biomechanics and Robotics 2021View this Special Issue
Nonsimilar Modeling and Numerical Simulations of Electromagnetic Radiative Flow of Nanofluid with Entropy Generation
Electromagnetic water/CNTs nanofluid flow across a convectively heated moving surface is reported in this communication. Aspect of thermal radiations is considered for heat transport analysis. The concept of nonsimilar boundary layer is executed to simplify the convoluted mathematical expressions. Also, an entropy generation model is considered since its reduction minimizes the loss of available energy, which improves thermal efficiency. The governing model is reduced to a dimensionless system by using an appropriate nonsimilarity transformation. The numerical solution for the velocity and temperature profiles has been obtained by implementing local nonsimilarity via finite difference based Matlab algorithm bvp4c for various quantities of the main emerging parameters. The outcomes are depicted in tabular and graphical formats to analyze impacts of different geometrical, thermophysical, and dynamical factors on temperature, velocity, frictional drag, entropy generation (EG), Nusselt number, and the Bejan number. The temperature profile is seen to rise with Biot number and thermal radiation. Higher radiation parameters and nanoparticle concentrations cause an increase in entropy generation. Horizontal plate with the wedge angle is the optimal geometry for minimizing entropy generation. The increase in the values electric field parameter leads to the rise in the skin friction coefficient. Also, Nusselt number declines when magnetic parameter and Eckert number are increase. The authors discussed the local nonsimilarity approach for simulating the dimensionless nonsimilar structure. To the best of authors’ knowledge, no such study has yet been published in the literature. To show the originality of results, the current numerical findings are compared with the published research for some limiting cases and are found to be in excellent alignment. This study could be useful for examining the impacts of nanofluids in a thermal transport analysis.
Carbon nanotubes are an allotrope of carbon with tubal nanostructure. Carbon nanotubes have a broad variety of uses, including oil and gas industry, conductive fabrics, atomic force microscope tips, flat-panel displays, radar-absorbing coating, longer-lasting batteries, structural composite materials, antifouling paint, ultracapacitors, and medical instruments and biosensors due to their higher chemical compatibility with biomolecules such as proteins and DNA, as well as purification of tainted drinking water. Compared with other nanostructured compounds, nanotubes have the best thermal conductivity enhancement property. Iijima  discovered carbon nanotubes in 1991 by observing the electrical discharge between two carbon electrodes. CNTs are characterized as single wall carbon nanotubes and the multiwall carbon nanotubes, all of which depend on concentric layers of rolled graphene lamina. Carbon nanotubes’ thermal conductivity is found to be higher than that of pure fluids, and as a result, there is rising preference in SWCNTs and MWCNTs applications in industry, electronics, and medicine [2–4]. Meanwhile, significant work on CNTs boundary layer flow has been documented by [5–7]. Shahsavar et al.  examined thermal conductivity of a magnetic hybrid (CNTs + Fe3O4) nanofluid flow experimentally. Lu et al.  discussed the CNTs-suspended nanofluids flow with binary chemical reactions and Cattaneo-Christov heat flux condition. Haq et al.  analyzed the velocity slip effects on two-dimensional (2D), steady, incompressible, nanofluid flow containing CNTs over a flat stretching plate. Mosayebidorcheh and Hatami  investigated nanofluid (CNTs-water) flow between two parallel rotating disks. Iqbal et al.  numerically analyzed the bioconvection, CNTs nanofluid flow across a stretching surface. Lu et al.  evaluated numerically the nanofluid flow comprising CNTs with magnetohydrodynamics (MHD) and homogeneous-heterogeneous reactions. Ahmed et al.  examined incompressible, 2D, steady boundary layer flow with base fluid (engine oil) in the existence of CNTs with double stratification effects. Upreti et al.  assessed the 3D Darcy–Forchheimer water-CNTs (SWCNT and MWCNT) nanofluid flow across 2D stretchable surface with the consequences of Ohmic heating and nonuniform heat source/sink.
Entropy generation (EG) is accounted in technology, heat-related industries, and everyday life events. Thermal energy is converted to mechanical work through entropy, which is the indicator of thermal system disruption. In the form of a campfire and the combustion of solid wood into ash, smoke, and fumes, human everyday life is inextricably linked with entropy. Regarding fluid efficiency in thermal systems, liquids are treated in such a manner that energy waste is minimized. In support of this idea, the concept of EG has gained prominence in the thermal world. Bejan  initially investigated entropy production and developed a thermodynamic optimization process. Soomro et al.  numerically examined EG for mixed convection CNTs fluid flow over an inclined surface. Ishaq et al.  studied nanofluid flow with EG across an unsteady stretching surface. Farooq et al.  studied viscous dissipation and transpiration effects on EG in the nanofluid flow across a nonlinear stretched porous disk. Suleman et al.  discussed analytically, EG, and thermal radiation effects for nanofluid flow across an exponentially stretching surface. Sheikholeslami et al.  used a non-Darcy model to study EG and magnetic nanofluid flow in a porous enclosure. Khan et al.  evaluated the EG in nanofluid flow through porous medium comprising microorganisms and nanoparticles. Some important studies on EG have been conducted in [23–29].
Boundary layers theory plays a critical role in an extensive range of engineering and scientific fields. By observing high-temperature processes, such as nuclear power plants, energy storage, and gas turbines, the impact of thermal radiations on boundary-layer flows is crucial. Heat transfer through surface convection across different geometries has obtained a lot of consideration because of its possible uses in a variety of engineering technologies. Many researchers have conducted studies in this region. Ishak et al.  studied the convective flow across moving-plate with thermal radiation. Akbar et al.  analyzed the incompressible nanofluid flow over a cylinder along with convective boundary condition. Nadeem and Haq  explored the magnetohydrodynamic nanoliquids transport across stretching-sheet with the impact of convective-surface. Delouei et al.  investigated the boundary layer flow and heat transfer of non-Newtonian fluid over a heated cylinder by employing the direct-forcing IB-TLBM. Oyelakin et al.  observed Soret and Dufour effects in Casson nanofluid flow across an unsteady stretched surface with thermal radiation effects. Zaib et al.  studied the impacts of mixed convection and thermal radiation on copper-water nanofluid flow over a porous shrinking cylinder with heterogeneous-homogeneous reactions. Jyothi et al.  numerically assessed the boundary layer nanofluid flow between two disks employing thermal radiation and convective boundary surface. Reddy et al.  examined the influence of Arrhenius activation energy, nonlinear thermal radiations, and binary chemical reaction on MHD boundary layer flow of Eyring-Powell fluid over a stretching surface. Saif et al.  reviewed viscous fluid flow across a stretchable curved surface through convective boundary conditions. Raees et al.  evaluated mixed convective boundary-layer magnetized second grade nanofluid flow over a vertically stretching sheet. Farooq et al.  analyzed nonsimilar boundary-layer flow of Casson nanofluid within non-Darcy porous media.
According to the previously mentioned literature review, nonsimilar EG analysis for an electromagnetic, radiating nanofluids (water-SWCNT/MWCNT) flow across convective heated-wedge, vertical plate, and horizontal-plate has not been published. Thus, the current work aims to address this void in the literature by conducting a systematic review of boundary-layer flow and EG research. The governing equations are numerically solved by utilizing local nonsimilarity method via in-built MATLAB bvp4c. For SWCNT/MWCNT-water nanofluids, the impact of incorporated physical parameters on velocity, temperature, viscous drag, Nusselt number, EG, and Bejan number has been examined.
2. Mathematical Modeling
Consider the 2D, incompressible, viscous carbon nanotubes-based flow over a wedge. Water is considered as base fluid, while SWCNTs and MWCNTs are picked as nanoparticles. Free stream and wedge velocities are and , respectively, where , and are constant is Hartree pressure gradient parameter; represented a total wedge angle. The case relates to a horizontal plate and indicates a vertical plate. We presume that axis is parallel to the wedge and axis in the normal direction of flow. Furthermore, it is anticipated that the wedge's lower surface is heated by heat convection from hot fluid with transfer coefficient at a temperature . The base fluid (water) and the nanoparticles (SWCNTs and MWCNTs) are likewise considered to be in equilibrium. Figure 1 reveals the schematic sketch of the problem.
Governing equations for the problems under consideration  are as follows:where and are the and directions velocity components, respectively. is the nanofluid temperature and , , , and are the nanofluid's dynamic viscosity, density, thermal diffusivity, effective thermal conductivity, and heat capacitance.
In an inviscid flow, velocity at edge of the boundary layer and mainstream velocity () are identical, and pressure is treated as constant. From (2), we have
Employing equation (4) into equation (2), we get
Rosseland approximation is considered :where and represents Rosseland absorption coefficient and Stephan–Boltzmann constant. Consider that the temperature variations in the flow are minimal, so can be written as
Employing equations (6)–(7) into equation (3), we get
The linked boundary conditions (BCs) are
Dimensionless transformations  are as follows:
Equation (1) is verified automatically, while (2)–(9) become
With boundary conditions, the following is obtained:
Here, denotes the dimensionless wedge angle; represents the velocity ratio parameter; depicts static wedge; and show moving-wedge situation in the same and opposite direction to free stream direction, respectively. expresses the radiation parameter, indicates the Reynolds-number, is the Biot number (surface convection parameter), and represent Eckert and Prandtl numbers, respectively, is the electric field parameter, and is the Hartmann number defined aswhere is the nanofluid volume fraction, and , , and are the density, thermal conductivity, and electrical conductivity of the base fluid (see Table 1).
2.1. First Truncation System
At the first truncation level, the terms accompanied by are considered approximately very small. This is particularly true when . Therefore, (11)–(13) become
Boundary conditions are as follows:
2.2. Second Truncation System
For the second truncation system, we assumed the followings:
Therefore, equations (11)–(13) take the following form:
Boundary conditions are as follows:
Frictional drag on the surface (local skin friction coefficient ) and rate of heat transfer (local Nusselt number ) are defined aswhere and are the shearing stress and heat flux at the surface of the wall, respectively. Putting in (15), we get
3. Entropy Generation Analysis
Entropy generation (EG) per unit volume for the problem under consideration is 
The dimensionless entropy generation number is defined aswhere represents the dimensionless temperature difference and denotes the characteristic entropy generation rate.
Dimensionless form of Bejan number () for the considered problem can be stated as
4. Numerical Solution Procedure
In this section, we will discuss the numerical solution procedure for the dimensionless equations (15) to (21) using Matlab algorithm bvp4c. The ordinary differential equations (ODEs) solver technique employs the Lobatto IIIA formula, which necessitated the initial guesses to justify the boundary conditions. Hereafter, we employ FDM, which updates the initial predictions for the subsequent iterations. We accomplish the process with the following assumptions:
5. Results and Discussion
Electromagnetic, radiating water-CNTs nanofluid flow over convective heated-wedge and vertical/horizontal surface under impact of entropy optimization has been investigated by employing the method of local nonsimilarity  up to the second truncation level via Matlab command bvp4c numerically. According to the total wedge angle , three cases are examined for boundary layer flow. The flow around a flat plate is represented by or (), or (), which shows the flow across a vertical plate, and for the wedge flow, we choose or (). Values of approximately correlate to the wedge angles , and . The Prandtl number () is fixed as 6.2, and nanoparticles volume fraction () is examined in range. The physical importance of various flow fields has been demonstrated for several relevant parameters. We set the parameters , and , unless otherwise specified in specific figures/tables. Table 2 implies the thermophysical properties of water and both CNTs. To assess the authenticity of the present solutions, Table 3 reports a comparison examination of for several values of with earlier published results, which validates our method. Table 4 depicts the skin friction coefficient's behavior in response to various estimations of , , , , and for both MWCNT + H2O and SWCNT + H2O. Viscous drag is reduced as and are increased. Further, rise in , and upgrades the viscous drag for both MWCNT/SWCNT-water nanofluid. The numerical value of the Nusselt number is shown in Table 5 for both SWCNTs and MWCNTs. Nusselt number () decreases for greater values of magnetic parameter and Eckert number, while it rises for increasing the values of dimension less electric field, surface convection, velocity ratio parameter, solid volume fraction, and radiation parameters.
Velocity () for several values of the () are shown in Figure 2. It shows that when rises, falls for both conducting SWCNT/MWCNT-water nanofluids flow by wedge and a horizontal /vertical plate . By raising , the fluid's viscosity enhances and so resists the flow under the impact of shear stress. As a result, fluid velocity diminishes. The significance of magnetic parameter () on the velocity field for all cases are outlined in Figure 3. For greater estimations of , velocity decreases. Increase in indicates a rise in resistive power (Lorentz force), and so, the velocity of the liquid decreases. Figure 4 indicates that the rise in upgrades velocity field irrespective of . This is because an increase in adds to a stronger electric field, which accelerates the flow of CNTs nanofluids. Figure 5 demonstrates the variance in velocity for growing velocity ratio parameter . It shows that velocity of nanofluids within the boundary layer rises with .
Influences of nanoparticles fraction on () are seen in Figure 6. Clearly, graphs show that increasing the nanoparticles concentration raises the temperature inside the boundary layer for all considered cases. This is because as increases, the thermal conductivity of the nanofluid rises, and as a result, the temperature of the fluid grows. Figure 7 outlines the significance of the temperature by considering an estimation of the radiation parameter . Clearly, increasing the radiation estimator value improves the temperature distribution and corresponding boundary layer thickness. It is evident that thermal radiation causes an upsurge in surface heat flux. Consequently, the temperature rises within the boundary layer region. Figure 8 portrays the effects of Hartmann on temperature profile θ(η). It is clear that as the value of increases, so does the temperature. The Lorentz force obstructs fluid flow, whereas rise in the Hartmann number increases the Lorentz force, resulting in a portion of energy being transferred as heat. Temperature profile rises as a result. An analysis of Eckert number on is portrayed in Figure 9. It is worth noting that when rises, increases for both SWCNT and MWCNT. Increase in the results in higher drag forces between the fluid materials. Consequently, more heat is generated, causing the temperature distribution to increase. The behaviors of in response to different for electromagnetic flow of CNTs nanofluids transport by convective heated-horizontal/vertical plates and wedge surface are drawn in Figure 10. In this case, the temperature distribution for both nanofluids declines as increases. The variations of with the strength of local Biot number () is seen in Figure 11. Thermal boundary layer thickness was shown to rise slightly with an escalation in the for all the considered cases. Biot number is defined physically as the relationship between conduction and convection at the surface. As predicted, stronger convection results in higher surface temperatures. Physically, the inclination in generates significant heat transfer through convection, which rises the thermal field.
Figure 12 displays the variation in entropy generation () versus nanofluid volume fraction parameter (), which are stated in terms of . It is obvious that as the increases, so does the . This is due to the high thermal conductivity () of SWCNT/MWCNT nanoparticles. Also, the entropy generation number increases with . Figure 13 represents the thermal radiation impact on the entropy generation number . An increasing trend in the causes an upsurge in the . The effects of entropy generation number versus Hartmann number () for electromagnetic SWCNT/MWCNT-H2O nanofluids flow are depicted in Figure 14. The increase in causes an escalation in . Physically, this means that the increasing estimations of accelerates the drag force that enhances the dissipation energy, which is the primary cause of irreversibility. Figure 15 shows the Eckert number's () impact on the entropy generation number (). For all considered cases, the rises with the rise in . Figure 16 describes the impacts of (), on (). We discover that have a minimum value when is small, indicating poor convection in the surfaces, and then begin to improve until stability is obtained when (isotherm-surface). All three flow instances exhibit the same behavior. As a result, minimizing convection via boundaries can reduce entropy generation. Figures 17(a) and 17(b) illustrate the effect of on Bejan number () for electromagnetic SWCNT−H2O and MWCNT−H2O nanofluids flow, respectively. In both situations, an increase in elevates . Figure 18 shows the thermal radiation's effect on the Bejan number . increases as the radiation parameter is increased. Greater radiation-absorption inside boundary zone raises system's energy, which causes an increase in . Figure 19 reveals the influence of the convective surface parameter () on the Bejan number (). When is small, we discover that has a minimal value. The behavior is consistent across all three flow cases.
In this study, we explored numerically the nonsimilar electromagnetic, radiating water/CNTs nanofluid flow across convective heated-moving horizontal/vertical plates and wedge with entropy generation influence. The following are the study's key findings:(i)Velocity field for electromagnetic SWCNT/MWCNT− H2O nanofluids flow became antithetical to the enhanced values of and (ii)With increasing estimations of the electric field parameter, the velocity field rises for SWCNT/MWCNT− H2O nanofluids(iii)Thermal field improves with the augmentation of , and , while it peters out via incremented (iv)Increment in leads to diminution of viscous drag while rise in , and upgrades the viscous drag for both MWCNT/SWCNT-water nanofluid(v)Nusselt number () decreases for greater estimations of magnetic parameter, while rises for increasing values of , , , , and (vi)The increase in viscous dissipation, measured by the Eckart number (Ec), reduces the heat transfer rate(vii)Temperature profiles for both SWCNT and MWCNT nanofluids fall as the angle of the wedge increases(viii)For increasing , , , and , entropy generation in boundary-layer nanofluid flow increases(ix)Entropy generation can be reduced by limiting convection across boundaries(x)The optimum geometry to minimize entropy generation is a horizontal plate corresponding to the wedge angle (xi)Bejan number () rises with an increment in and parameters for both SWCNT and MWCNT(xii)High radiation absorption within the boundary layer raises the system's internal energy, causing an increase in and
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through research groups program under grant number RGP.2/20/43.
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