Heat transfer analysis of buoyancy opposing radiated flow of alumina nanoparticles scattered in water-based fluid past a vertical cylinder

Cooling and heating are two critical processes in the transportation and manufacturing industries. Fluid solutions containing metal nanoparticles have higher thermal conductivity than conventional fluids, allowing for more effective cooling. Thus, the current paper is a comparative exploration of the time-independent buoyancy opposing and heat transfer flow of alumina nanoparticles scattered in water as a regular fluid induced via a vertical cylinder with mutual effect of stagnation-point and radiation. Based on some reasonable assumptions, the model of nonlinear equations is developed and then tackled numerically employing the built-in bvp4c MATLAB solver. The impacts of assorted control parameters on gradients are investigated. The outcomes divulge that the aspect of friction factor and heat transport upsurge by incorporating alumina nanoparticles. The involvement of the radiation parameter shows an increasing tendency in the heat transfer rate, resulting in an enhancement in thermal flow efficacy. In addition, the temperature distribution uplifts due to radiation and curvature parameters. It is discerned that the branch of dual outcomes exists in the opposing flow case. Moreover, for higher values of the nanoparticle volume fraction, the reduced shear stress and the reduced heat transfer rate increased respectively by almost 1.30% and 0.0031% for the solution of the first branch, while nearly 1.24%, and 3.13% for the lower branch solution.

www.nature.com/scientificreports/ in a PMA were studied by Ghosh and Mukhopadhyay 12 . They discovered double outcomes when the free-stream and plate travel in reverse directions. Waini et al. 13 examined the SR and DU impressions on the nanofluid flow past a slim movable needle through the Tiwari and Das model and presented binary outcomes for a single value of a parameter. In the presence of nanoparticles, it was found that the UBS of the friction factor and the HT rises while the coefficient of mass transfer falls. The impact of Lorentz forces on a cross 3D flow of streamwise direction via incorporating nanofluid using Koo-Kleinstreuer-Lee (KKL) correlation was inspected by Khan et al. 14 .
It was found that the rate of mass transfer drops but the rate of heat transfer augments due to the Soret number. Uddin et al. 15 scrutinized the impact of the magnetic field on the stagnation-point flow of nanofluid with heat transfer from a stretchable/shrinkable sheet and found dual solutions by utilizing an innovative Metaheuristic approach. Khan et al. 16 explored the bio-convection stimulus through directions of streamwise and cross-flow cooperating nanofluid and reported dual solutions existence. Reddy and Goud 17 explored the rule of radiation on 2D flow toward a SP induced by nanofluid across a stretchable cylinder. They observed that the temperature and the profile of nanoparticle fractions improve in response to rises the influences of the radiation parameter. Asogwa et al. 18 examined the features of EMHD on the radiative flow of Casson nanofluid through a reactive stretchable sheet. They have perceived that the gradients increase as the modified Hartmann number rises. Goud et al. 19 inspected the impact of radiation and Joule heating on the magneto flow of nanofluid across an exponentially stretched sheet with a medium of thermal stratified. With increasing values of the Eckert number, the TTBL (thickness of the thermal boundary-layer) is increased as a result of frictional heating. More about the significance of nanofluids can be observed in recent articles [20][21][22] with different aspects. The feature of radiation is among the most significant procedures in the movement of fluid flow and heat during a thermal system of a lofty temperature. The effect of radiation is a crucial tool for managing excessive heat emissions that has a wide variety of industrial applications. Thermal radiation has a significant impact on the structure of high-quality equipment, missiles, nuclear power plants, turbines of gas, satellites, and a wide range of complex systems of conversion mentioned in their studies [23][24][25][26] . Madhu et al. 27 probed the time-dependent flow in non-Newtonian nanofluids by taking radiation and magnetic effects into account. They explored that the friction factor decreases in the existence of magnetic, unsteady, and Maxwell parameters. The impact of thermal radiation through a horizontal infinite surface induced by non-Newtonian fluid was calculated by Jamshed et al. 28 . In his study, the most striking finding is that the water-based copper nanofluid was found to be a better thermal conductor compared to titanium nanofluid. Yanala et al. 29 scrutinized the slip and ramped temperature mechanisms on the transient flow past a vertical infinite plate with radiation and chemical reaction effects. A strong flow is created near the plate due to buoyancy and radiation effects, which are amplified by slip. Jamaluddin et al. 30 analyzed the rule of radiative phenomenon on buoyancy flow near a SP induced by Cross fluid past a porous shrinkable sheet. They observed that the range of existence for dual solutions is greatly influenced by factors such as the mass transpiration, the Prandtl number, and the Weissenberg number. Goud et al. 31 discussed the Dufour effect on the dissipative unsteady flow of Casson fluid across a laminated porous vertical laminate with a chemical reaction. The features of radiation and chemical reaction on the dissipative flow past a vertical infinite plate in a porous media with magnetic and Soret effects were analyzed by Goud et al. 32 . They found that the heat transfer rate declines as Eckert rises, while the contradictory pattern is being distinguished for radiation.
The transport features inside the region of stagnation, for instance, the polymer productivity and extrusion process are influential in the modern industry and need ongoing enhancement to preserve a high standard of quality 33,34 . As a result, the subject has piqued the researcher's interest in the present decade. The classic twodimensional (2D) SP flow problem was first discussed by Hiemenz 35 and Homann 36 . Since that time, several researchers have conducted several investigations on stagnation point flow in a variety of flow systems. Kumari and Nath 37 utilized the theory of boundary layer and the finite difference technique to simulate mixed convective SPF induced by non-Newtonian fluids (NNFs). They noticed that the buoyancy and magnetic parameters increase the gradient of surface velocity and heat transfer. The steady 2D flow near an SP simulated with nanomaterials past a stretchable/shrinkable sheet moves in its plane was extended by Bachok et al. 38 . They reported that the Cu nanoparticles outperformed the other nanoparticles in terms of the friction factor and heat transfer. Awaludin et al. 39 performed the stability analysis for the stagnation point flow (SPF) through a linearly shrinkable or stretchable sheet and presented more than one solution. Halima et al. 40 inspected the flow of stagnation point simulated with non-Newtonian fluid through a slippery stretching sheet with nanofluid subject to passive and active nanoparticle controls. It was discovered that due to a stagnant point, the ability of heat transport phenomena improved in both active and passive controls. The entropy generation on the SPF of a NNF suspended by nanoparticles through a moving surface with activation energy was discovered by Zaib et al. 41 . Lately, Zainal et al. 42 investigated the SPF of a nanofluid past a stretchable sheet induced by non-Newtonian fluid and observed dual solutions. To verify the dependability of the solutions, a stability study was performed.
Many researchers are concerned with the investigation of fluid flow past a vertical cylinder. Several research papers examined the fluid flow through a moving/static cylinder and discussed the interesting aspects of flow models. Wang 43 modeled and investigated the motion of the fluid problem over a stretchable cylinder. Ishak et al. 44 reconfigured and adorned the problem by incorporating supplementary effects of heat transfer. These estimates were taken for cylinders that are homogeneously porous and stretchable. They observed that if there is no forceful infusion, water cools more effectively than air does. More specifically, the chronological work of Ishak and Nazar 45 is incorporated here, where they asserted the motion of fluid along a stretchable cylinder. Wang and Ng 46 investigated the slip influence on the fluid flow from a stretchable cylinder at the fluid-solid interface. They found that the slip parameter declines the shear stress and velocities. The HT features and dynamics of fluid flow were inspected by Gorla and Bhattacharyya 47 . They discussed the flow properties through a shrinkable permeable cylinder. They discovered that the HT is augmented owing to curvature and suction parameters. Majeed et al. 48 explored the flow with heat transport past a stretchable cylinder by assuming partial slip and heat-flux conditions and utilized the Chebyshev spectral Newton iterative method to find the iterative solution. www.nature.com/scientificreports/ The viscous flow past a stretchable (shrinkable) permeable cylinder having an irregular radius was inspected by Ali et al. 49 . Reddy et al. 50 examined the impact of entropy on the radiative transient flow with mass and heat transfer induced by a couple-stress fluid across a vertical cylinder with magnetic effect. The outcome shows that magnetic and radiation parameters decrease and ultimately increase the entropy generation. The supercritical free convective flow of couple stress and Newtonian fluids around an isothermal cylinder was addressed by Basha et al. 51 . The current computational work demonstrates that a supercritical Newtonian fluid has transient and steady-state velocity fields that are significantly higher than coupling stress fluid. Waini et al. 52 where U ∞ is the characteristic velocity, T 0 < 0 (opposing flow) is called the characteristic temperature of the base nanofluid, while the free stream temperature (base nanofluid) is symbolized by T ∞ . The term radiation heat flux q r is also examined in this model. The nanofluid is prepared of water-based (H 2 O) containing the type of nanomaterials, namely, alumina (Al 2 O 3 ). However, the carrier-based water fluid and the posited scattered alumina nanoparticles are in thermal equilibrium (TEM). Further, the given nanoparticles have a uniform shape, and size in the TEM state.
The governing equations of nanofluid can be expressed as follows under the boundary layer scaling and certain aforesaid presumptions (see Mukhopadhyay and Ishak 55 ; Devi and Devi 56 ): The components of velocity along the axial and radial directions are symbolized by u and w , respectively. Meanwhile, T signifies the nanofluid temperature and g denotes the acceleration owing to gravity.
Further, the absolute viscosity µ nf , coefficient of thermal expansion (ρβ T ) nf , density ρ nf , specific heat capacity ρc p nf , and thermal conductivity k nf of the nanofluid are defined as follows (see Zaib et al. 57 , Chu et al. 58 ): Here, ρc p f , ρc p s are the heat capacitances, µ f the viscosity, k f , k s the thermal conductivities, ρ f , ρ s densities, and (ρβ T ) f , (ρβ T ) s the coefficients of thermal expansion for the base water fluid and solid nanoparticle, while φ signifies the solid nanoparticle volume fraction and their corresponding zero value reduces the nanofluid model to a common fluid. In addition, c p reflects the heat capacity (HC) at a uniform pressure. Also, Table 1 shows the experimentation physical aspects of alumina (Al 2 O 3 ) nanomaterials and a regular fluid.
Moreover, the term q r is defined below, which a simplified form is obtained from the Rosseland approximations (see Bataller 60 ; Ishak 61 ; Magyari and Pantokratoras 62 ): where σ b and k b indicate the uniform SBN and MAC, respectively. Then, the higher power term T 4 is treated mathematically about the point T ∞ to get T 4 ≈ 4T 3 ∞ T − 3T 4 ∞ using the Taylor series and ignores the higherorder terms. With aid of this, Eq. (3) yields: or As in Mukhopadhyay and Ishak 55 , it is suitable to announce the following requisite posited similarity transformations: where ψ is the stream function demarcated as ru = ∂ψ/∂r and rw = −∂ψ/∂x which satisfies the continuity Eq. (1) identically. Substituting (9) into (2) and (8) gives u e xυ f 1 2 , The occupied non-dimensional Eqs. (10) and (11) confined the following factors such as the radiation param- , the Prandtl number Pr = υ f /α f , and the mixed convection parameter = gβ T T 0 R/U 2 ∞ . Moreover, the mixed convection is the ratio of the Grashof number to the square of the Reynolds number.
The gradients of the current problem are the shear stress C f (skin friction) and the local Nusselt number Nu x , which are defined as: Using (9) and (13), we obtain where Re x = xu e /υ f corresponds to the local Reynolds number.
Numerical procedure. The complete solution procedure of the scheme as well as the confirmation or validity of the code is clarified in this section. The system of higher-order ODE Eqs. (10) and (11) along with BCs (12) are numerically solved via the built-in bvp4c MATLAB solver which is based on the Lobatto IIIA formula to obtain the numerical results (see Kumar et al. 63 ). In this procedure, the system of higher order ODEs is reduced to the first-order ODE equations by introducing new variables. To start the procedure, let f = A a , f ′ = A b , f ′′ = A c , θ = A d , and θ ′ = A e . With the assistance of these variants, the developed third and second order ODEs are converted to the following set of first-order ODEs with high nonlinearity as follows: with boundary conditions are Moreover, to start the working process of computing the solution, the values of the unknown conditions are calculated, and other key prominent parameters in transformed equations are set in a manner that would provide the necessary numerical convergence. The process of initial iteration is carried out for an appropriate finite value of ξ = ξ ∞ = 10 , and the result is acknowledged only when the criteria in Eq. (16) are asymptotically met. The tolerance requirement is set 10 −6 to yield precise numerical results. The first solution is comparatively easy to find within a CPU time of 20 s, however, for the second solution, it is very difficult to get a little more time i.e., 40 s. The detailed procedure of the scheme was given in the book of Shampine et al. 64   www.nature.com/scientificreports/ in Table 2. Hence, this outstanding assessment can give us confidence that the attained unavailable results for both solution branches are accurate.

Results and discussion
The comprehensive analysis and physical interpretation of the results for the two distinct branch solutions with influence of various parameters are presented in this section via several graphs and tables. The characteristics of the base fluid (water) and the alumina (Al 2 O 3 ) nanomaterials are shown in Table 1, while the comparison for the restrictive cases is being exhibited in Table 2. Therefore, Table 3 compares the results of the numerical calculation for RSS (upper branch solution) made using bvp4c and NDSolve. The two numerical procedures used to generate the reported RSS data exhibit excellent/outstanding concordance up to three decimal places. Further, this segment is aimed to see the stimulus of the involved factors like the curvature constraint γ , the radiation constraint R d , the nanoparticles volume fraction φ , and the mixed convection constraint on velocity profile, temperature profile, heat transfer and friction factor for the stable and unstable results as graphically decorated in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13. In the present simulation, the following numerical values are fixed such as γ = 0.05 , R d = 2.0 , = −2.0 and φ = 0.035 for the comprised governing parameters. Meanwhile, the Pr is taken to be 6.2 for the considered regular fluid (water). In addition, the reduced shear stress (RSS) and reduced heat transfer (RHT) computational values for both solution branches due to distinct varying parameters are presented in Tables 4 and 5, respectively. From these results, it is exposed that the RSS upsurges for the UBS are due to the bigger value of φ and γ while the trend of the upshots is contrary for the LBS. Oppositely, the RHT Table 2. Values of the reduced skin friction and reduced heat transfer for some selected values of γ when = 0, R d = 0 and φ = 0.   www.nature.com/scientificreports/       www.nature.com/scientificreports/ augments in both branches of outcomes with varying all three parameters. Moreover, the RSS increases by 1.30% and 1.24% in the respective UB and LB results due to the superior values of φ while it decreases by 0.77% for the LB and enhances by 19.44% for the LB with superior values of γ . On the other hand, the RHT escalates for the UB solutions in percentage-wise at around 0.0031%, 5.23%, and 11.42% owing to the higher values of φ , γ and R d , respectively. Meanwhile, for the branch of lower solutions, the RHT enriches by 3.13%, 13.01% and 51.44% for the continuous uplifting in the values of φ , γ and R d , respectively. The present paper reports the existence of two solutions, which are termed first and second solutions, or upper and lower solutions, based on how they   6, and 7, respectively, for the increasing value of φ when γ = 0.05 and γ = 0.1 . Thus, the magnitude of the BVs | C | intensify with the magnification of φ and γ . This tendency displays that the gap of the BLR (boundary-layer thickness) declines due to the larger impression of φ and γ . Physically, the advanced alumina nanoparticles make the fluid flow more viscous, which causes the required minimum BLR to fall. However, as the volume percentage of solid nanoparticles rises, the thermal conductivity increases, and the temperature profile rises as a result. Also, the figures suggest that higher amplification of φ and γ in the existing solution range for both branches to the similarity Eqs. (10) and (11). Between these two results, we expect that the UBS is physically acceptable and stable in the long run, while the LBS is not acceptable (unstable) as time evolves. Thus, Weidman et al. 69 and Khan et al. 70 for example have described the process for determining this temporal stability, therefore we will not repeat it here.
The impact of R d on the RHT of the Al 2 O 3 -water nanofluid for binary outcomes versus the buoyancy opposing flow is graphically illustrated in Fig. 8. The results justify that the RHT enriches for the upper solution with large values of R d , while the behavior for the LBS is reversed, owing to greater augmentation for the selected choices of the parameter R d . In addition, the magnitude or absolute of the BVs reduces with larger R d . This pattern suggests that the boundary layer separation improves here with the superior impact of R d . Physically, the effective fluid receives additional heat through the radiative heat flux. Consequently, a wider thermal boundary layer is observed. Figures 9 and 10 describe the impacts of φ on f ′ (ξ ) and θ (ξ ) fields of the Al 2 O 3 -water nanofluid for the solid and dashed line curves, respectively. From these figures, it is noticed that the velocity declines but the curves of temperature boosted up for the UB and LB solution with superior impact on the nanoparticle volume fraction. Moreover, the separation gap for the LBS is slightly better than for the UBS. Physically, the higher concentration of the nanoparticle volume fractions creates an extra improvement in the thermal conductivity due to which the thermal boundary layer thickness and the temperature distribution escalate.
The curvature γ and radiation R d parameters influence on f ′ (ξ ) and θ(ξ ) fields of the Al 2 O 3 -water nanofluid for the UB and LB outcomes are exhibited graphically in Figs. 11, 12, and 13, respectively. Noted from the output curves that the velocity f ′ (ξ ) ascents for the UBS for higher impacts of γ while the velocity behavior of the curves slow down for the lower branch solutions. In contrast, the temperature acts differently in both branches of the solution as compared to velocity when the rule of γ increases. Meanwhile, the temperature distribution curves intensify for both explanations owed to bigger implementation of R d . Generally, the higher values of R d boosted the thermal conductivity (TCN), and as a consequence, the behavior of the temperature and the BLR thickness rises.

Conclusions
The idea of this work is to study the buoyancy effects of a stagnation point water-based alumina nanoparticles flow and HT aspects over a vertical cylinder with prescribed surface temperature, radiation effect, and external flow was theoretically examined. The key findings for the considered model are summarized as follows: • Multiple (lower and upper) branch solutions are observed to survive when the buoyancy or mixed convective parameter is negative [case of opposing flow or cylinder is cooled ( T w (x) < T ∞ )]. www.nature.com/scientificreports/ • The magnitude of the heat transfer and drag forces for the upper branch solutions is augmented with higher influences of the nanoparticle volume fractions, however, the trend is changed for the LBS. • The RSS and RHT escalate owing to the solid volume fraction of nanoparticles by almost 1.30% and 0.0031% for the UBS, while for LBS, it is heightened by almost 1.24% and 3.13%, respectively. • For the superior influence of the radiation parameter, the reduced heat transfer improves by almost 11.42% for the UBS and 51.44% for the LBS. • The existing problem reduces to that of the special geometry (flat plate) when the effects of the curvature parameter are taken to be zero. • Both gradients in magnitude-wise at the surface are lower for a flat plate compared to a cylinder.
• The BVs upsurge with φ but reduce due to the larger value of the radiation factor.
• The temperature augmented in the UB and LB outcomes owing to larger radiation parameter.
• For higher volume fractions of nanoparticles, the velocity curves moderate for the branch of binary outcomes but the temperature increases.
We believe that the current outcomes will offer significant information for complex issues within computer routines involving nanofluid with buoyancy force because of their several applications in processes of heat transfer, heat exchanger, solar collector, etc., and also utilize these results in experimental studies.

Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.