Front. Mater.Frontiers in MaterialsFront. Mater.22968016Frontiers Media S.A.139106610.3389/fmats.2024.1391066MaterialsOriginal ResearchIrreversible mechanism and thermal crossradiative flow in nanofluids driven along a stretching/shrinking sheet with the existence of possible turning/critical pointsElattar et al.10.3389/fmats.2024.1391066ElattarSamia^{1}KhanUmair^{2}^{3}^{4}^{5}*ZaibAurang^{6}IshakAnuar^{2}AlwadaiNorah^{7}AlbalawiHind^{7}^{1}Department of Industrial and Systems Engineering, College of Engineering, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia^{2}Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia^{3}Department of Mathematics, Faculty of Science, Sakarya University, Serdivan, Türkiye^{4}Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon^{5}Department of Mechanics and Mathematics, Western Caspian University, Baku, Azerbaijan^{6}Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, GulshaneIqbal Karachi, Pakistan^{7}Department of Physics, College of Science, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia
Edited by:Adebowale Martins Obalalu, Augustine University Ilara Epe, Nigeria
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The significant increase in thermal efficiency and the rate of energy exchange used in fuel dynamics and automobile coolants are leading to a better understanding of nanofluids. This computational analysis explores the thermal conductivity performance for radiative crossflow of a nanofluid across an expanding/constricting sheet with a suction effect as a result of its application. To compute or calculate the magnificent point of nanofluid flow, the entropy, and asymmetrical heat source/sink effects are also elicited. The boundary layers traverse a streamwise procedure for expanding and contracting sheets. Additionally, the study examines the features of heat transfer and crossflow of nanofluids using numerical simulations. By employing similarity variables, the basic PDE equations of the current model are transformed into ODEs, and they are subsequently evaluated using the bvp4c method. Therefore, the effects of embedded flow variables on drag force, heat transfer rate, and entropy generation profiles have been framed using parametric research. Multiple solutions are offered for a specific range of the contracting parameter as well as the mass suction parameter. In addition, the heat transfer rate accelerates due to the heat source and decelerates due to the heat sink. The literature that is already published has been compared favorably, and it reveals many commonalities.
The nanofluid has a large number of nanoparticles and nanometersized molecules. Tiny particles are disseminated (added) in fluids to increase their heat transfer capabilities, even though the structure and content of nanomolecules depend on carbides, metals, and carbon nanotubes. The combinations that make up the nanofluid are typically combined with particles of average nanoscale size. Nanotubes, nanofibers, nanowires, nanoparticles, nanosheets, nanorods, and other materials are routinely made with nanofluids. The most effective and practical methods have been developed and put into practice for the modeling of nanofluid flow models, while the addition of specific materials of a particular kind has increased the thermal conductivity of various fluids. Several engineering procedures use nanoparticles in fluids to improve heat transfer. Equipment like heat exchangers in heavy machinery, automobiles, and industries is highly dependent on effective energy transfer. With these applications in mind, Choi and Eastman (Choi and Eastman, 1995) developed nanofluids to enhance the heat transfer capabilities of ordinary fluids. According to Eastman et al. (Eastman et al., 2001), adding copper nanoparticles to ethylene glycol at a concentration of 0.3 volumes results in a 40% increase in thermal conductivity. Khan and Pop (Khan and Pop, 2010) have numerically addressed the issue of laminar fluid flow that develops when a flat surface is stretched in a nanofluid. It was discovered that the reduced heat transfer is a diminishing function of every single dimensionless number. The numerical solution of the boundary layer flow caused by a linearly extending sheet in a nanofluid was inspected by Makinde and Aziz (Makinde and Aziz, 2011). They observed that the convective heating, thermophoresis, and Brownian motion all get more intense as the local temperature rises, which causes the thermal boundary layer to thicken. The thermal conductivity performance of carbon nanotubes in fluid flow over a stretching sheet was discovered by Haq et al. (Haq et al., 2015). Sheikholeslami et al. (Sheikholeslami et al., 2016) discovered the impacts of Lorentz forces on free convective flow in the presence of nanofluid with thermal radiation. A twodimensional timeindependent flow conveying nanofluid toward a thin needle was deliberated by Soid et al. (Soid et al., 2017) where the existence of multiple solutions was reported. Bakar et al. (Bakar et al., 2018) looked into the stability analysis of mass suction impacts through a shrinking/stretching cylinder considering the nanoparticles. Kamal et al. (Kamal et al., 2019) investigated the flow of nanomaterial through a stretchable/shrinkable sheet with a chemical reaction effect. A twodimensional magneto mixed convection flow induced by a shrinking/stretching plate in a nanofluid was investigated by Jumana et al. (Jumana et al., 2020). Johan and Mansur (Johan and Mansur, 2021) examined the features of dusty nanomaterial flow and thermal transport analysis past a stretchable sheet with a slip boundary. They used three kinds of particles, namely, alumina, copper, and titania. Shahzad et al. (Shahzad et al., 2022) scrutinized the influence of the slip effect on the flow and heat transport incorporated by copper nanofluid with different shape factors through a heated stretched sheet and found that the platelet shape factor particles have a greater rate of heat transfer compared to other shape factors. Shamshuddin et al. (Shamshuddin et al., 2024a) examined the ohmic and ferromagnetic effects on the flow of nanofluid through a porous rotating disk. They discovered that in normal fluid and nanofluid, the increased suction velocity suppresses the fluid velocity produced by the surface’s porosity. Recently, Shamshuddin et al. (Shamshuddin et al., 2024b) explored the impact of electromagnetic on the radiative flow of trihybrid nanofluid through a bidirectional stretchy surface in penetrable media. The factors of the surface frictional force of the ternary hybrid nanofluids are found to decrease with increasing the Lorentz force and number of surface pores.
The problem of sheets stretching or shrinking in a viscous fluid has caught a lot of concentration as it has numerous applications in physics, engineering, and other scientific disciplines. It frequently occurs in practical issues that have attracted a lot of research attention because of their wide range of significance in fields like the production of glass fibre, glass blowing, metal extrusion, transportation, microfluidics, paper production, hot rolling, space, and acoustics (see Fisher (Fisher, 1976)). The BLF over an ongoing solid kind surface flowing at uniform motion was initially studied by Sakiadis (Sakiadis, 1961) in light of these applications. Numerous authors (Chen and Char, 1988; Ishak et al., 2009; Mi, 2015; Guo and Fu, 2019; Wang, 2019; Zhao et al., 2019; Zi and Wang, 2019) have thought about different elements of this problem and found similarity solutions since the groundbreaking research conducted by Crane (Crane, 1970), who provided an exact solution for the 2D steady flow caused by a stretchable surface in a quiescent fluid. Instead of focusing on the scenario of a stretched sheet, researchers instead looked at the scenario of a shrinking sheet. According to Goldstein (Goldstein, 1965), this new kind of flow of shrinking sheet is fundamentally a backward flow. The steady flow across a shrinkable sheet was studied by Miklavčič and Wang (Miklavčič and Wang, 2006). They discovered that mass suction is necessary to continue the flow across a shrinkable sheet. Waini et al. (Waini et al., 2019) examined the dependency of timevarying flow along with thermal transport across a shrinking/stretching sheet incorporated with hybrid nanofluids and provided multiple solutions.
The examination of crossflow began following the early studies by Prandtl (Prandtl, 1946a) and Blasius (Blasius, 1908) that included the flow over a smooth surface caused by thin viscosity. Prandtl (Prandtl, 1946b) is believed to be the initial researcher to publish the findings for uniform pressure gradients flowing through a finite yawed cylinder. Weidman et al. (Weidman, 2017) examined the boundary layer via crossflow generated by transverse plate motions. The work of Weidman was recently expanded by (Roşca et al., 2021) by taking into account rotational stagnationpoint flow that transports hybrid nanofluids along a permeable shrinking or stretching surface. It was discovered that both stretching and shrinking surfaces can have multiple solutions to the fundamental similarity equations.
The phenomena of an irregular heat sink or source have applications in both engineering and medicine, involving the recovery of crude oil, the construction of thrust bearings, and the cooling of metallic sheets, etc. In the presence of an irregular heat sink/source, Tawade et al. (Tawade et al., 2016) addressed the motion of the MHD unsteady thin film and heat transfer past a stretchable sheet. It was determined that irregular heat parameters are crucial to the effectiveness of heat transfer. Thumma et al. (Thumma et al., 2017) revealed that the stretching of a sheet caused the MHD convective motion of nanofluid to have a changeable heat sink or source. To obtain the solution, a wellknown KellerBox numerical approach is utilized. Kumar et al. (Kumar et al., 2020) looked at the movement of hybrid ferrofluids film and heat transfer in the inclusion of radiation and erratic heat source/sink (EHSE/EHSK). The rate of heat transfer is believed to be greater in hybrid ferrofluids than in ferrofluids. In addition, the velocity of the fluid and temperature tend to decline as film thickness parameters increase. Areekara et al. (Areekara et al., 2021) investigated the impact of an irregular heat source/sink on the fluid flow of nanofluid past a nonlinear stretching sheet. They observed that positive correlations exist between the radiative heat flux and the rate of heat transfer. A negative sensitivity to the rate of heat transfer is shown by the exponential heat source. Akram et al. (Akram et al., 2022) discussed the concepts of nonlinear stretching and EHS/SHS to describe the heat transfer through the stretchable cylinder. It is discovered that the temperature distribution in the fluid region is being disrupted by the nonlinear stretching rate and the source of heat.
The scrutiny of second law analysis or, EG (entropy generation) in fluid flow and heat transport is a prominent area of study. Energy losses resulting from chemical processes, diffusion, solidsurface friction, and the viscosity of fluids all contribute to the production of entropy in thermodynamic systems. As a result, the formation of entropy generation (EG) has drawn a lot of attention to applications including heat exchangers, turbo machinery, and electronics cooling. Aiboud and Sauoli (Aiboud and Saouli, 2010) scrutinized the entropy optimized in viscoelastic flow through a flexible surface subjected to the magnetic field. The effect of slip across a heated vertical surface in entropyoptimized flow was calculated by But et al. (Butt et al., 2012). Slips have been found to allow for the control and adjustment of entropy formation in thermal systems. Abolbashari et al. (Abolbashari et al., 2014) employed HAM to examine, EG in magneto nanofluid flow near an unsteady stretched surface with H_{2}Obase liquid and several nanoparticle types. Tlau and Ontela (Tlau and Ontela, 2019) examined the role of magnetohydrodynamics on nanomaterial entropyoptimized flow from an inclined channel with a heat source/sink embedded in a porous media. Entropy optimization of nanomaterials flow across two stretchable rotating disks with effects of bioconvection was examined by Khan et al. (Khan et al., 2020). Mondal et al. (Mondal et al., 2021) used trapezoidal liddriven enclosures with, EG to study the Lorentz forces on the constant buoyant flow of Al_{2}O_{3} nanoparticles. According to the calculations, the average Nusselt and Sherwood numbers, and aspect ratio all decrease with increasing the percentage of nanoparticle volume.
The literature that is currently accessible indicates that no exploration has been performed on the entropy generation of the radiative crossflow stimulated with nanofluid from a porous expanding/contracting sheet with an asymmetrical heat source/sink. The present investigation fills a research gap by demonstrating crossflow and heat transfer towards a stagnation point of nanofluid via an expanding/contracting sheet with asymmetrical heat sink/source and thermal radiation. The proposed model is originally described via a highly nonlinear system of PDEs. The PDEs are rehabilitated into ODEs by using the proper similarity variables and then solved by employing a bvp4c solver. Multiple solutions are provided for certain values of the parameters such as mass suction and shrinking sheet. The study described in this paper was driven by the following research questions.
• What impact does have on the skin friction and Nusselt number by raising the suction parameter against the stretching/shrinking parameter?
• How can the separation of the boundary layer be controlled in the presence of nanoparticles and stretching/shrinking parameters along with suction?
• What influence do Brinkman, temperature difference parameters, and TiO_{2} nanoparticles have on the entropy profiles?
2 Description of the flow problem
The nanofluids’ stagnation points radiative cross flow and thermal system characteristics past a contracting/expanding surface with the mutual influence of mass suction and EHSE/EHSK are taken in this study. As portrayed in Figure 1, xa,ya are Cartesian coordinates measured along the horizontal and vertical surface of the sheet, respectively, with the flow occurring in the domain ya≥0. Nanofluid is a mixture of regular fluid (water) and single titanium dioxide (TiO_{2}) nanoparticles. The investigational features of the (water/TiO_{2}) nanomaterials are taken to be uniform. In addition, the horizontal surface of the sheet is assumed to have a variable velocity of ua=εb2/3νf1/3xa1/3γb=uwaxaγb, where γb refers to the expanding/contracting factor with γb<0, γb>0 and γb=0 signify the particular cases of the shrinking, stretching, and stationary/static sheet, respectively. In the meantime, εb indicates the positive constant and υf is the kinematic viscosity of the regular (water) fluid. The ambient or farfield (nanofluid) is also supposed to have a linear velocity of uexya=εbya, see Weidman et al. (Weidman, 2017). Moreover, the mass suction/injection or transpiration velocity at the surface of the sheet is vwaxa with vwaxa<0 refers to the case of injection and vwaxa>0 refers to the case of suction while vwaxa=0 indicating the impermeable surface of the sheet. It is also supposed that the constant temperature of the sheet is Twa, while T∞ represents the free stream temperature (inviscid fluid). With the help of these aforesaid assumptions, the governing equations in the Cartesian form are written as (Weidman, 2017; Roşca et al., 2021):∂ua∂xa+∂va∂ya=0,ρnfua∂ua∂xa+va∂ua∂ya=μnf∂2ua∂ya2,ρcpnfua∂Ta∂xa+va∂Ta∂ya=knf∂2Ta∂ya2−∂Qrad∂ya+knfuwaxaxavnfAb*Twa−T∞e−η+Bb*Ta−T∞,with boundary conditions (BCs) are:ua=γbuwaxa,va=vwaxa,Ta=Twa at ya=0,∂ua∂ya→∂uex∂ya=εb,Ta→T∞ as ya→∞.
Physical model of the problem.
In Eqs 1–4, ua and va are the nanofluid velocities in the corresponding xa, and ya directions, Ta the temperature of the nanofluid, Ab* the exponentially decaying space coefficients, and Bb* the temperaturedependent heat source/sink. Therefore, the heat source or absorption phenomenon is produced due to the positive value of Ab* and Bb* while the phenomenon of heat generation or sink is found by the negative value of both Ab* and Bb*.
The radiation heat flux Qrad is expressed by the Rosseland approach as:Qrad=−4σa3ka∂Ta4∂ya.
Here, the Stefan Boltzmann constant and the mean absorption coefficient are denoted by ka and σa, respectively. Moreover, the term Ta4 is simplified further by using the Taylor series at T∞ and overlooking the power of higherorder yields:Ta4≅4TaT∞3−3T∞4.
In addition, executing Eqs 5, 6 into Eq 3 yields the final form:ρcpnfua∂Ta∂xa+va∂Ta∂ya=kfknfkf+43Nr∂2Ta∂ya2+knfuwaxaxavnfAb*Twa−T∞e−η+Bb*Ta−T∞,where Nr=4σaT∞3kakf signifies the thermal radiation parameter.
Furthermore, knf indicates the thermal conductivity of the essential posited nanofluid (NFD), ρcpnf indicates the heat capacitance of the NFD, ρnf indicates the density of the NFD, and μnf indicates the absolute viscosity of the NFD. The correlation of these NFDs is written as follows:knf=kTiO2+2kf−2φTiO2kf−kTiO2kTiO2+2kf+φTiO2kf−kTiO2,ρcpnfρcpf=φTiO2ρcpTiO2ρcpf+1−φTiO2,ρnfρf=φTiO2ρTiO2ρf+1−φTiO2,μnf′μf=1−φTiO2−2.5.
Here, kf,ρf, and μf refer to the thermal conductivity, the density, and the absolute viscosity of the base (water) fluid while the heat capacity at constant pressure is represented via cp. Therefore, φTiO2 symbolizes the titania nanoparticles volume fraction and the special case φTiO2=0 reduces the Eqs 8 to a normal or a regular fluid (water). In addition, Table 1 displays the physical data of the titania (TiO_{2}) nanoparticles and the regular fluid (water).
The physical aspects of (TiO_{2}/water) nanofluid.
Physical properties
Water
TiO_{2}
ρkg/m3
997.1
4,250
cpJ/kgK
4,179
686.2
kW/mK
0.613
8.9528
Pr
6.2

For the considered model, the similarity transformations that can be expressed to further simplify the procedure for mathematical analysis are as follows:η=εb/υf1/3yaxa1/3,ua=εb2/3υf1/3xa1/3G′η,Hη=Ta−T∞Twa−T∞,va=−εb1/3υf2/33xa1/32Gη−ηG′η,which provides the opportunity to describe flows extremely broadly, regardless of the system size. Also, the prime corresponds to the derivative with respect to η, H is the nondimensional temperature distribution, G describes the nondimensional quantities, and G′ is the dimensionless velocity profile. However, the mass suction/injection velocity at the surface of the sheet is written as:vwaxa=−23εbυf2xa13fwa.
In Eq. 10, fwa is the constant mass suction/blowing constraint with fwa=0, fwa<0 and fwa>0 describe the phenomena of impermeable, blowing, and suction, respectively.
With the help of similarity transformations (9), the continuity Eq 1 of the governing model is satisfied while the rest of Eqs 2, 7 equations change to the resulting known ordinary (similarity) differential equations (ODEs) as:μnf/μfρnf/ρfG‴+23GG″−13G′2=0,knfkf+43NrH″+23PrρcpnfρcpfGH′+knfkfρnfρfμnfμfAb*e−η+Bb*H=0,along with border conditions:G0=fwa,G′0=γb,H0=1 at η=0,G″η→1,Hη→0 as η→∞.
In addition, Eq 11 for the case of φTiO2=0 is the same as equation (6.1, see Weidman et al. (Weidman, 2017)) when α=1, but Eq 12 with some special effects has been not taken in the same reference paper. Moreover, the dimensionless model comprised the following distinct parameters, the suction/injection fwa, the expanding/contracting γb, and the Prandtl number Pr=υf/αf.
2.1 Gradients
The two vital physical aspects, namely, heat transfer rate and shear stress of the assumed model are of practical significance to apply by scientists or engineers. They are defined as follows:Cf=μnfρfuwa2∂ua∂yaya=0,Nuxa=−xaknfkfTwa−T∞−knf∂Ta∂ya+Qradya=0.
By incorporating Eq 9 into Eq 14, the following dimensionless form yields:CfRexa1/2=μnfμfG″0,Rexa−1/2Nuxa=−knfkf+43NrH′0.
Hence, Rexa=xauwaυf refers to the local Reynolds number.
3 Second law analysis
Entropy generation (E.G.,), also known as the second law analysis, is a necessary instrument for measuring energy loss and depreciation in the effectiveness of engineering and industrial systems, such as rate and transport operations. As a result, the systems expend less energy, making, E.G., analysis and comprehension crucial. Taking into account the scenario of, E.G., for the viscous Newtonian liquid with the inclusion of nanoparticles (Abolbashari et al., 2014; Tlau and Ontela, 2019).EG=1T∞2knf+163σaT∞3ka∂Ta∂ya2+μnfT∞∂ua∂ya2.
Two fundamental elements are principally responsible for the, E.G., in the contemplated crossflow of viscous Newtonian nanofluid. The first term in the statement, which is on the righthand side, denotes the ensuing local heat transfer, and the final term, the consequent fluid friction or viscous dissipation. The second law analysis is defined as follows in the dimensionless form:NG*=xa2T∞2kfTwa−T∞2EG.
The following formulas are obtained by incorporating the similarity transformations from Eq 9 into Eq 16. Hence,NG*=knfkf+43NrRexaH′2+μnfμfRexaBrbΩbG′′2,where Ωb the dimensionless temperature difference and Brb the Brinkman number. Consequently, they are represented mathematically as:Ωb=Twa−T∞T∞,Brb=μfuwa2kfTwa−T∞.
4 Methodology
This section demonstrates the analysis of the assumed crossflow and the suspension of nanofluid for heat transfer. The requisite model equations are expressed as highly nonlinear ODEs (11) and (12) along with BCs (13) using similarity variables (9). A builtin function named bvp4c included in the MATLAB software is used to work out these equations numerically. It ought to be noted that the scheme of the finite difference is the foundation for the bvp4c package, which is further highlighted by the 3stage Lobatto IIIA procedure. To instigate the bvp4c method, the transmuted ODEs are modified into a firstorder system by launching newfangled variables. By establishing this process, letG=A1,G′=A2,G″=A3,H=A4,H′=A5.
Substituting Eq. 20 into Eqs 11, 12 along with BCs Eq. 13 we obtain the set of firstorder ODEs as follows:ddηA1A2A3A4A5=A2A3ρnf/ρfμnf/μf13A22−23A1A3A51knfkf+43Nr−23PrρcpnfρcpfA4A2−knf/kfρnf/ρfμnf/μfAb*e−η+Bb*A4,with BCsA10=fwa,A20=γb,A40=1,A3∞=1,A4∞=0.
The code desired initial estimations at the posited mesh point to solve Eq 21 and the corresponding conditions Eq. 22. The polynomial used in the collective type yields a continuous result. A fourth order accuracy set that is equally distributed over the spatial intervals where the function is integrated provides the result. The limitation at a distance η→∞ is replaced by the value η=η∞=8 in many successful boundary layer theory applications, and the relative tolerance error is predefined as 10^{−6}. Additionally, the residual of the smooth output serves as the basis for both the mesh selection and error control. The initial mesh comprises four identical discredited points on the range 0,η∞=8, and the mesh selection is then automatically changed using the bvp4c package. The problem now has two possible solutions, which means that the bvp4c package needed two alternative guesses for the unstable and stable solutions. The early initial presumes for the upper solution is moderately straightforward, whilst selecting a guess for the second solution is fairly challenging. Merkin et al. (Merkin, 1986) and Weidman et al. (Weidman et al., 2006) claim that the upper solution is physically stable and reliable but the lower solution is unstable and not physically dependable since the outcome only exists for a specific range of shrinking sheets.
4.1 Validation of the MATLAB bvp4c solver
This subsection of the work specifies the rationality or validity, accuracy, and correctness of the considered MATLAB bvp4c solver for the special limiting case. To confirm this rationality, the friction factor outcomes for both branches (stable and unstable) results owing to several values of the shrinking constraint with prior research work when fwa, and φTiO2 are equal to zero. Therefore, Table 2 displays the results as well as a comparison to those of previous research work/literature. Thus, we can conclude that our results are trustworthy because our data closely aligns with those that have already been published.
Numerical comparison of outcomes for friction factor owing to several values of the shrinking parameter when fwa=0, and φTiO2=0.
γb<0
Waini et al. (Waini et al., 2020)
Present
Stable
Unstable
Stable
Unstable
0.1
0.993440
−0.017703
0.993440
−0.017703
0.2
0.971925
−0.018388
0.971925
−0.018388
0.3
0.931424
−0.000045
0.931424
−0.000045
0.4
0.864452
0.044824
0.864452
0.044824
0.5
0.752585
0.134657
0.752585
0.134657
5 Analysis of the results
The current portion of the work describes the binary (stable and unstable) solutions for a certain region or area of the dimensionless mass suction parameter as well as the contracting parameter γb due to the variations in one physical parameter at the time of computations while the other factors are taken to be fixed. To ease analysis, the values of the basic physical influential parameters are φTiO2=0.025, γb=−2.0, fwa=1.5, Ab*=0.1, Nr=0.05, and Bb*=0.1. Table 1 reveals the experimental data of titania (TiO_{2}) nanoparticles. Meanwhile, the comparison of the upshots for the unusual case is demonstrated in Table 2. Besides, the outcomes of the friction factor, heat transfer, and entropy generation are captured in various graphs (see Figures 2–8) of the nanofluid for the unstable and stable branches owed to the influence of the several comprised factors while their quantitative outputs are shown in Tables 3, 4. However, the branches of stable solutions (SBES), as well as solutions of unstable (USBES) are categorized by the black solid and black dash lines, respectively. The position in the graph or picture where both (SBES and USBES) curves meet at a single point is called the bifurcation or critical point. In this study, the SBES and USBES are invented only for the case of shrinking parameters.
Impact of fwa on CfRexa1/2versusγb.
Impact of fwa on NuxaRexa−1/2versusγb.
Impact of γb on CfRexa1/2versusfwa.
Impact of γb on NuxaRexa−1/2versusfwa.
Impact of Brb on NG*versusη.
Impact of φTiO2 on NG*versusη.
Impact of Ωb on NG*versusη.
Numerical outcomes are made for the shear stress with φTiO2 and fwa when γb=−2.0, Nr=0.05, Ab*=0.1, and Bb*=0.1.
φTiO2
fwa
Shear stress
Stable
Unstable
0.025
1.5
5.6460767
1.0653406
0.030

5.8683445
1.0791223
0.035

6.0989946
1.0931549
0.025
1.5
5.6460767
1.0653406
0.030
2.0
8.1019827
0.7430102
0.035
2.5
10.531567
0.5628759
Numerical outcomes are made for the heat transfer with φTiO2 and Ab*,Bb* when γb=−2.0, Nr=0.05, and fwa=1.5.
φTiO2
Ab*,Bb*
Heat transfer
Stable
Unstable
0.025
0.5
4.6336867
15.001956
0.030

4.6360107
15.273743
0.035

4.6387796
15.562330
0.025
0.5
4.6336867
15.001956

0.7
4.3834284
10.021935

0.9
4.1226401
8.2234605
0.025
−0.5
5.7491768
4.6342404

−0.7
5.9488839
5.2367610

−0.9
6.1418652
5.6895740
The numerical data of the gradients (skin friction and heat transfer rate) with the impression of the several distinguished factors corresponding to waterbased Titania nanofluid are illustrated in Tables 3, 4 for the SBSE and USBES, respectively. Upshots divulge that the friction factor upsurges for the SBSE owed to the superior values of φTiO2 and fwa while the branch of USBES behaves distinctly with variations in the mass suction parameter fwa but similarly with higher impacts of φTiO2. Notably, the shear stress of the nanofluid is highest and lowest for the SBES and USBES with mass suction fwa. In contrast, the heat transfer escalates for both (SBES and USBES) results with superior consequences of φTiO2. Therefore, owing to the rise in the heat source parameter Ab*,Bb*>0, the heat transfer shrinkages in the SBES as well as the USBES are endlessly enriching due to the higher role of the heat sink parameter Ab*,Bb*<0. In addition, the lowest and highest heat transfer approximations are perceived for the branch of SBES and USBES with the following selected values Ab*,Bb*=0.9, and φTiO2=0.035.
Figures 2, 3 show the impact of fwa on the shear stress and heat transfer corresponding to (TiO_{2}/water) nanofluid for the SBES as well as the USBES, respectively. In this study, the dual (SBES and USBES) outcomes are possible to occur for a certain domain of the specific set of physical parameters. Therefore, the nonunique outcomes in either pictures or graphs exist for a posited shrinkable sheet. Moreover, it is clear from the above graphs that the position where both solution curves meet is at a point called the critical point. Mathematically, this point is expressed by the symbol γbC, where the solutions are unique γb=γbC. The nonunique solutions (dual) and no solutions are possible to exist for the range γbC<γb<∞ and −∞<γb<γbC, respectively. Besides, the outcomes refer to the shear stress and heat transfer escalating for the SBES due to the larger impact of fwa while they are declined for the USBES. Physically, the motion of the nanofluid stops due to the inspiration of fwa shifting the particles of the liquid moving toward the surface of the sheet and sticking with it. Hence, the friction and motion/velocity of the nanofluid hold the inverse relations, as a result, the shear stress is enhanced. Furthermore, the next eight distinct critical values −2.1903, −2.2941, −2.4016, −2.5125, −2.6274, −2.7463, −2.8686, and −2.9951 are obtained for the respective change value of fwa. With the rise of fwa causes an increment in the absolute value of γbC. This behavior corresponds to the superior inclusion of fwa decelerating the boundary layer separation.
The impressions of γb on CfRexa1/2 and NuxaRexa−1/2versusfwa of (TiO_{2}/water) nanofluid for both (SBES and USBES) results are typified in Figures 4, 5, respectively. The dual (SBES and USBES) results are shown in both graphs for the case of fwa due to the variations in the shrinking parameter. In both graphs, it is seen that the SBES and USBES curves congregate at a point called the critical point which is denoted by fwaC. Meanwhile, the outcomes are unique for the case when fwa=fwaC, but the phenomena fwaC<fwa<∞, and −∞<fwa<fwaC indicate the nonunique and no solutions, respectively. Besides, the shear stress decays and rises for the SBES due to the higher values of γb while it shrinks for the USBES. Alternatively, with the increase of γb, the thermal transport phenomenon uplifts for the SBES and declines for the USBES. More significantly, it is understood from the diagrams that bifurcation values like 2.1615, 2.0760, 1.9888, 1.8986, 1.8057, 1.7096, 1.6098, and 1.5066 are found due to the several values of γb. Also, it is noted here that the magnitude of the critical values fwaC is weakened once the shrinking parameter is boosted. This further specifies that the growth in the impacts of γb hastens the boundary layer (BL) separations.
With the assistance or support of entropy generation or second law analysis (SLA), the thermodynamic system performance of waterbased Titania nanoparticles can be improved. Figures 6–8 describe the consequence of parameters Brb, φTiO2, and Ωb on SLA (second law analysis) corresponding to the posited nanofluid for the SBES and USBES, respectively. As premeditated in Figures 6, 7, an improvement in both parameters of Brb and φTiO2 results in an improvement of SLA for the branch of stable as well as unstable solutions. The second law analysis is also more susceptible to changes in this parameter Brb value at the place close to the wall surface of the sheet. As increases η, susceptibility rapidly decreases. Therefore, the physical data and prior research findings are compatible with this occurrence. Conversely, Figure 8 is designed to inspect the parameter Ωb on second law analysis of the (TiO_{2}/water) nanofluid for both (SBES and USBES). In general, the lower temperature difference between the wall and its surroundings is generally caused by larger values of Ωb, which reduces second law analysis or, EG. The aforementioned discussion leads to the conclusion that by altering the related parameters, the system’s second law analysis value can be decreased to increase the solar radiation utilization system’s effectiveness.
6 Conclusion
The theoretical inspection on radiative cross flow and heat transfer incorporated waterbased TiO_{2} nanofluid through a permeable stretching/shrinking sheet with irregular heat source/sink have been explored. The entropy generation was used to analyze the heat transfer process after the development of a computational model. Combined impacts of pertaining governing parameters like suction, expanding/contracting parameter, irregular heat source/sink parameter, radiation parameter, and volume fraction of the nanoparticles on shear stress and the heat transfer have been analyzed. The important conclusions of our research can be summed up as follows.
• The results suggest that the shear stress enhances due to fwa and φTiO2, whilst the heat transfer accelerates due to φTiO2.
• The heat transfer rate accelerates due to the heat source Ab*,Bb*>0 and decelerates due to heat sink Ab*,Bb*<0.
• The entropy generation increases in the presence of Brinkman number and nanoparticle volume fraction in both solutions, while decreasing due to the higher impacts of the difference of temperature parameter in both solutions.
7 Future work
Further, this problem can also be studied by incorporating unsteady flow or mixed convection flow in addition to incorporating various physical elements such as the slip effect, chemical reaction viscous dissipation, etc. Also, this model can be extended to any nonNewtonian models of the problem along with several other impacts like shape factors of the nanoparticles, waste discharge concentration, and thermophoresis particle deposition effects.
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University, through the Research Groups Program Grant no. (RGP14440060).
This work was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University, through the Research Groups Program Grant no. (RGP14440060).
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.
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Ab*
Exponentially decaying space coefficients
Cf
Skin friction
Nuxa
Local Nusselt number
Bb*
Temperaturedependent heat source/sink
Brb
Brinkman number
Nr
Thermal radiation parameter
Qrad
Radiation heat flux
cp
Specific heat at constant pressure (J kg^{−1} K^{−1})
Ab*,Bb*>0
Heat source parameter
Ab*,Bb*<0
Heat sink parameter
fwa
Suction parameter
G′
Dimensionless velocity
Ta
Temperature of the nanofluid (K)
Twa
Constant temperature (K)
H
Dimensionless temperature
Pr
Prandtl number
Rexa
Local Reynolds number
T∞
Free stream temperature (K)
vwaxa
Mass suction/injection or transpiration velocity at the surface of the sheet ua, vaVelocities in the xa−, and ya−directions, respectively (m/s)
uwaxa
Variable velocity at the surface of the sheet (m/s)