Radiation effect on stagnation point flow of Casson nanofluid past a stretching plate/cylinder

The exclusive behaviour of nanofluid has been actively emphasized due to the determination of improved thermal efficiency. Hence, the aim of this study is to highlight the laminar boundary layer axisymmetric stagnation point flow of Casson nanofluid past a stretching plate/cylinder under the influence of thermal radiation and suction/injection. Nanofluid comprises water and Fe3O4 as nanoparticles. In this article, a novel casson nanofluid model has been developed and studied on stretchable flat plate or circular cylinder. Adequate rational assumptions (velocity components) are employed for the transformation of the governing partial-differential equations into a group of non-dimensional ordinary-differential formulas, which are then solved analytically. The momentum and energy equations are solved through the complementary error function method and scaling quantities. Using various figures, the effects of essential factors on the nanofluid flow, heat transportation, and Nusselt number, are determined and explored. From obtained results, it is observed that the velocity field diminishes owing to magnification in stretching parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document}B and Casson fluid parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda$$\end{document}Λ. The temperature field increases by amplifying radiation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{r}$$\end{document}Nr, and solid volume fraction parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}ϕ. The research is applicable to developing procedures for electric-conductive nanomaterials, which have potential applications in aircraft, smart coating transport phenomena, industry, engineering, and other sectors.

The exclusive behaviour of nanofluid has been actively emphasized due to the determination of improved thermal efficiency.Hence, the aim of this study is to highlight the laminar boundary layer axisymmetric stagnation point flow of Casson nanofluid past a stretching plate/cylinder under the influence of thermal radiation and suction/injection.Nanofluid comprises water and Fe 3 O 4 as nanoparticles.In this article, a novel casson nanofluid model has been developed and studied on stretchable flat plate or circular cylinder.Adequate rational assumptions (velocity components) are employed for the transformation of the governing partial-differential equations into a group of non-dimensional ordinary-differential formulas, which are then solved analytically.The momentum and energy equations are solved through the complementary error function method and scaling quantities.Using various figures, the effects of essential factors on the nanofluid flow, heat transportation, and Nusselt number, are determined and explored.From obtained results, it is observed that the velocity field diminishes owing to magnification in stretching parameter B and Casson fluid parameter .The temperature field increases by amplifying radiation N r , and solid volume fraction parameter φ .The research is applicable to developing procedures for electric- conductive nanomaterials, which have potential applications in aircraft, smart coating transport phenomena, industry, engineering, and other sectors.The fluid movement near a solid surface's stagnation area is referred to as stagnation point flow.The stagnation flows occur in a variety of application fields, such as submarines, aircraft, and flows over the tips of rockets.The concern of stagnation point flow has been expanded in a variety of approaches, including boundary layer.Historically, Hiemenz 1 was a pioneering researcher who considered the issue of two-dimensional stagnation point flow towards a flat surface and found that the Navier-Stokes formulas may accurately examine this problem.Homann 2 expanded this issue to include the scenario of axisymmetric stagnation-point circulation.After that, Eckert 3 strengthened the suggested formulation by inserting both energy and momentum.Very recently, Norzawary 4 investigated the difficulties of stagnation point flow in carbon nanotubes through a stretching/ shrinking sheet and the effects of suction/ injection on it.Shah 5 tackles the problem of boundary layer flow and heat transmission analysis of coupled stress fluid stagnation point flows through a continuously stretched surface.Soid 6 employs a numerical model to deliberate an axisymmetric stagnation point flow past a stretching/shrinking sheet through second order slip as well as a thermal jump.Another popular topic of current research activity is stagnation-point flows across extending/contracting bodies.For instance, Merkin 7 addresses the movement of the boundary layer and the transfer of heat through a cylinder that is exponentially expanding and contracting.Turkyilmazoglu 8 has looked at how exactly the solution arises from stagnation-point flows through stretched flat plates or circular pipes.

List of symbols
The casson fluid constitutes one of the most prominent non-Newtonian rheological concepts, and it is a plastic fluid with significant yield stress and shear-subordinate characteristics.The Casson fluid flow happens whenever the shear tension exceeds the yield stress.The Casson model, which is developed for fluids with bar-like texture materials along it, is commonly used to simulate plasma flow and other real-world scenarios, including manipulating liquid chocolate and comparable meals in the present era.The presence of a stretched barrier expands under a variety of conditions, including the flow induced by the removal of polymers, the designing of copper cables, the persistent extension of synthetic films and reconstituted threads, the movement of hot glass fibers, the expulsion of metal, and material twisting.Researchers are more interested in conducting theoretical and practical studies on the flow properties of such complicated fluids, as non-Newtonian fluids are used in industrial applications more frequently every day.The boundary layer expressions among motion generated viz stretched plastic sheets in the polymer industry were analytically solved by Sakiadis 9,10 and Crane 11 .Very recently, Mahanta 12 demonstrated the effect of magnetohydrodynamic stagnation point flow on non-Newtonian liquid past a strengthening sheet.Mahabaleshwar 13 has analytically presented radiative magnetohydrodynamic axisymmetric flow of Casson fluid over a stretching/shrinking sheet in porous medium.Rehman 14 investigates concurrent heat and mass transfer properties of the flow of Casson fluid approaching aligned stretched flat and cylindrical surfaces.Devdas 15 has developed computational dual remedies for stagnation point flow and molten thermal expansion of non-Newtonian liquid due to a stretched surface.
On the other hand, in order to control conventional heat transfer fluids like ethylene glycol, oil, and water, nanoparticles with an average size of less than 100 nm are suspended in the fluids.They are distinguished by the enrichment of typically used fluids, such as H 2 O , toluol, ethylene alcohol, and fuel, along with nanomaterials of different kinds, like metals, oxides, carbides, and carbon.TiO 2 , CuO, Al 2 O 3 , and ZnO are examples of common nanofluids, along with water.Nanofluids are currently intended to have huge implications in pharmaceuticals, biomedicines, deterrence, nuclear power generation, and, more specifically, any heat dissipation associated with industrial uses.Choi 16 , the person who introduced the term "nanofluid, " has conducted substantial studies on non-Newtonian flow developments and applications as well as optimizing the nanofluids heat conductivity.Yang 17 scrutinized the heat transfer in casson nanofluid flowing at a stagnation point flow across a shrinking sheet with viscous dissipation.Mahabaleshwar 18,19 highlights the erratic stagnation point flow of a hybrid nanofluid through a stretching/shrinking sheet inserted in a permeable material with mass transpiration and chemical processes.Poornima 20 studied steady two dimensional, immiscible boundary layer stagnation point flow of regular and hybrid CuO + MgO nanoparticles across a stretching/shrinking cylinder.As a result of this, several more analytical and numerical research on the flow of nanoparticles has been widely published, such as in references [21][22][23][24][25][26] .
The effect of radiation on the flow of a nanofluid has become very important industrially.At very high operating temperature, the effects of radiation can be very significant.In engineering areas many processes occur at high temperature and the knowledge of radiation heat transfer becomes very important for designing of reliable equipment's, nuclear plants, turbines of gas and several devices of propulsion or aircraft, missiles, satellites and space vehicles.Nayak 27 presents the impacts of slip velocity, radiation, and magnetohydrodynamic consequences when they are taken into account at the stagnation point flow over the stretching sheet.Mahabaleshwar 28 also describes an analytical method for quantifying the impact of incompressible, casson nanofluid fluid through stretching/shrinking sheet in the existence of thermal radiant mass transmission characteristics.Kirankumar 29 considers the unstable magnetohydrodynamic boundary layer stagnation point flow of nanofluid across a nonlinear elongated sheet along a permeable media with a fluctuating sheet.Daniel 30 deliberates on the impact of radiation and MHD on a flow toward an extended surface in a two-dimensional stagnation point.Kaneez 31 explored the thermal efficiency of Non-Newtonian fluid by submerging trihybrid nanoparticles Fe 3 O 4 -Al 2 O 3 -TiO 2 uniformly.Aside from these, there have been a few recent studies [32][33][34][35][36][37][38] on the impact of thermal radiation on the various nanoparticles flow past a stretching/shrinking surface or different geometries.Recently, Maranna 39 presented an effect of radiation and mass suction/injection on two dimensional laminar flow of ternary nanoparticles due to porous stretching/shrinking sheet.
The above-mentioned literature survey reveals that no research work exists on axisymmetric stagnation point flow of Casson nanofluid over a stretching flat plates/circular cylinder.Also, there does not exist any solution to such a type of problem analytically in the complementary error function form.Hence, the novelty of this paper is to examine the axisymmetric stagnation point flow of Casson nanofluid over a stretching flat plates/circular cylinder.For both cartesian/circular cylindrical coordinate systems, the regulating PDEs are turned into a system of ODEs form via a velocity component and then solved analytically.For both momentum and energy fields, precise solutions are desired.The momentum and energy equations have precise solutions as a result of the established analytical solutions, which is important.The effects of the stretching parameter, Casson parameter, mass suction/injection, solid volume fraction, and radiation will also be analyzed through graphs.
The following are the innovative aspects of the planned investigation: • The importance of heat radiation effects on axisymmetric stagnation point flow of non-Newtonian liquid through a stretching flat sheet/circular cylinder.• The exact solution also determines whether to injection/suction into the walls and when stretching is not present, these solutions tend to the well-known particular limitations.• Consider Fe 3 O 4 nanoparticles suspended in base fluid as water.
• The collective consequences of the stretching, permeability factors, solid volume fraction, and thermal radia- tion are ameliorated in this simulation.• The use of nanoparticles in microprocessors, fuel cells, pharmacological procedures, hybrid-powered motors, ignition stimulating, the control of vehicle temperature, domestic refrigerators, coolers, heat exchangers, atomic power plant, grinding, machining, satellite communications, defensive performance, ships, and boiler flue gas thermal reduction have all been the subject of ongoing studies.

Mathematical modelling and solution
Assume two dimensional laminar axisymmetric stagnation point flow of non-Newtonian fluid across a flat stretching sheet /a circular cylinder that can be stretched.Allow an encroaching flow on a sheet to collide with an approaching flow with axial velocity u = U w (x) = αx .The surfaces should extend with an assumed veloc- ity u = U w (x) + u 0 , in which u 0 is fixed in both flow configurations as shown in Fig. 1.The two dimensional momentum field involves axial x factor as u and radial y factor as v .In addition, using both radial and axial dimensions, p indicates the pressure exerted on the system.T is considered to be fluid temperature, uniform wall temperature is signified as T w and T ∞ is the ambient temperature, respectively.

Inferences and model expressions
The followings are the aspects and problems that concern the mobility system: • Two dimensional laminar boundary layer axisymmetric stagnation point flow.

Rheological expression for Casson fluid
Assuming rheological equation of non-Newtonian Casson fluid as 40,41 : In the above equation, µ B is the dynamic viscosity of the biviscous Bingham fluid, fluid yield stress is p y , π is the result of multiplying the deformation rate components by itself, notably, π = e ij e ij , e ij is the i, j th compo- nents of the deformation rate, and π c is the critical values of the π.

Spherical coordinates
Under the foregoing assumptions, the governing boundary layer equations can be described 8,42 as

Cylindrical coordinates
Under certain assumptions, the fundamental equations in cylindrical coordinates (r, x) are as follows 7,8,53,54 : (1) ∂u ∂x It should be highlighted that the hydraulic and thermal characteristics are supposed to be constant and independent of any external factors.
Given the aforementioned reasoning assumptions, in both configurations, the axial velocities are being addressed in the following manner 8 : And the solution for radial velocity can be described as 8 : where velocity constituents in x, y are (u, v) correspondingly, (x, r) are axial and radial constituents of (u, v) .According to both axial and radial dimensions, p is symbolizes the pressure that is exerting itself on the system.ρ nf is the nanofluid density, ν nf kinematic viscosity of the nanofluid is denoted as is the thermal diffusivity of the nanofluid, whereas κ nf is the thermal conductivity of the nanofluid and ρC p nf is heat capacity of nanofluid.It is noted that the stretched flow over a smooth surface across v 0 in ( 11) may have a cumulative rate of suction/injection.At velocities less than − αR 2 , only suction can be imparted to the pipe surface with R is the pipe radius.Furthermore, the stagnation-point flow occurs beneath the fluid flows provided by Systems (6-9) and (11).

Boundary constraints
The related boundary constraints 8 are

Similarity transformation for energy equation
The desired similarity quantity 37 is whenever the flat-plate scenario and the Navier's-stokes equations in (2-9) are taken into account, the pressure solution is obtained 8 .
where p 0 and δ 1 are being constants.

Analysis of Rosseland approximations
While this is happening, the Rosseland approach for heat radiation is provided [43][44][45][46] as where σ * is the Stefan-Boltzmann constant and k * is known as mean absorption coefficient.Suppose that the temperature fluctuations inside the flow are negligibly infinitesimal, allowing T 4 to be represented as a temper- ature-dependent linear function  10) and (11), axial velocity profile U y ought to be deduced from the boundary layer problem.
where Ŵ = µ nf /µ f ρ nf /ρ f is the dummy variable, whereas µ nf is the dynamic viscosity of nanofluid.The condition 12) is due to the velocity-dependent asymptotic decay characteristic.The non-dimensional variables should be established and stated 8 as follows: where L is the reference length scale, incorporating these scalings and eliminating the bars, Eqns.( 6) and ( 7) becomes Corresponding to transformed boundary conditions are where B = αL 2 ν f is the stretching parameter, permeability is designated as s = v 0 L ν f , and δ = δ 1 Ŵ 2 L 3 is the pres- sure parameter.

Analytical solution for the momentum equations
In light of the complementary error function Erfc = 1 − Erf , the axial velocity from Eq. ( 20) is determined in relation to stretching flow across an obstacle.which is analytical solution for momentum expression in spherical form.Now Eq. ( 23) can be reduced to following for keeping u 0 = s = 0.
The radial velocity as well as pressure in dimensionless representations, are integrated into the solutions in (23) and (24).
The results in (23) and (24) are similar in that they disappear at the limit B → ∞.

Analytical solution for the energy equation
After employing Eq. ( 13) into Eqs.( 5) and ( 9), yields The following are some of the factors that appear in Eqs. ( 26) and ( 27) are: (ρCp) nf (ρCp) f are the dummy variables.
2. Pr = Equations ( 26) and ( 27) differentiating twice, we have obtained following results and For the flat stretching sheet, the heat transfer rate from the body surface can then be quantified using normalized Nusselt numbers provided by Fourier's heat equation.

and for the circular cylinder
The present section briefly discussed about precise solution for the energy equation, in the next section we will discuss about outcomes of the present problem.

Results and discussion
To investigate the impact of non-dimensional regulating factors on Casson nanofluid flow across an extending flat plate or circular cylinder, analytical calculations are specified.In this article, the shapes of stretching flat plates and pipes will be explored in light of the derived velocity and energy solutions.Fe 3 O 4 is thought to be a nanofluid in the current problem, with water acting as a base liquid.Prandtl number Pr is considered to have a value of 6.2.The impact of the stretching parameter B , wall suction/injection parameter s , Casson fluid parameter , radiation N r , and solid volume fraction parameter φ on the momentum, temperature, and Nusselt number profiles are depicted in the plots.To obtain the results in terms of figures the value of parameter ranges between 0.1 ≤ B ≤ 10, −0.25 ≤ s ≤ 1, 0.1 ≤ � ≤ 1, 0.01 ≤ φ ≤ 0.3,and 1 ≤ N r ≤ 3 .The thermophysical properties of nanofluid are mentioned in Tables 1 and 2.
Figure 2 portrays how the stretching parameter B affects the normalised velocity distribution (24).We noted that the stretching parameter enhances as velocity declines, keeping other parameter values at u 0 = 0, s = 0, φ = 0.01, and = 1 .It is clear that B has a dampening impact on the intensity of momentum, and the asymptotic behaviour at infinity is accurate.We also noticed that the momentum boundary layer declined with increased strength B.
� ∂θ ∂y  .the temperature is found to decrease as s increases, and it decreases as the distance from the surface increases, and finally vanishes at some large distance from the surface.It is suggested that for case of suction (s > 0) , the increase of s results to the boundary layer thinning while in the case of injection (s < 0) , the presence of injection effect has promoted to the drastically increases in the Table 1.Mathematical expression for empirical correlation of nanofluid [47][48][49] .

Thermo-physical properties
Nanofluid ( Fe 3 O 4 )  boundary layer thickness.As observed from Fig. 5, the thermal boundary layer drops when there is an increment in stretching parameter B strength because stretching improves the nanofluid movement, which further raises the thickness of the thermal boundary layer, leading to significant thermal gradients on the surface.Figure 6, respectively, demonstrated the distribution of thermal radiation N r on temperature distribution.It has been noted that as the value of thermal radiation increases with temperature, so does the temperature distribution.Actually, radiation is a thermal transfer phenomenon that transmits heat via liquid particles.So it produces some heat in the flow.Hence, we perceive an increase in heat transfer for higher values of N r .Figure 7 presents the consequences of solid volume fraction φ on temperature distribution.It is evident from figure, hence, it can be shown that by improvement in nanoparticle volume fraction with increasing the temperature boundary layer.It is obvious that whenever volume fraction of nanoparticle grows, the thermal conductivity and thermal diffusivity is also enhanced.Due to rising values of φ , the temperature distributions become dis- torted resulting in an increase in the overall heat transfer.This result can be attributed to the dominance of the thermal conductivity property.It is worth noting that as the values of φ increase, the thickness of the thermal boundary layer near the top surface rises which indicates a steep temperature gradients.Figure 8a exemplify the characteristics of mass suction/injection parameter s on reduced Nusselt number at some specific values of Pr = 6.2, φ = 0.1 , and N r = 1 .Figure 8a display the rate of heat transmission declines uniformly on escalating the strength of the mass suction/injection.Rate of the heat transfer improved with incrementing value of radiation N r has been observed in Fig. 8b.

Figure 3
Figure3depicts the influence of the casson parameter on the normal velocity profile.This figure revealed that as the strength of the casson parameter enhances, the nanoparticle flow diminishes gradually because the sheet stretchiness slows moving fluid components.Substantially, in this scenario, the momentum barrier layer thickness falls as grows.It is a known fact that increasing the Casson parameter decreases the yield stress of the Casson fluid, and increasing it indefinitely will make the fluid behave as a Newtonian fluid.It is evident that fluid motion is slowed down in velocity due to increase in the value of the Casson parameter , which means decrease in the velocity profiles and leads to a decrease in the momentum boundary layer thickness.Figure4a, b display the fluctuation in the temperature profile owing to the mass suction/injection parameter s , when Pr = 6.2, φ = 0.01, and N r = 1 with B = 1 and B = 4. the temperature is found to decrease as s increases, and it decreases as the distance from the surface increases, and finally vanishes at some large distance from the surface.It is suggested that for case of suction (s > 0) , the increase of s results to the boundary layer thinning while in the case of injection (s < 0) , the presence of injection effect has promoted to the drastically increases in the

Figure 2 .
Figure 2.An impact of stretching parameter on normalized velocity distribution.

Figure 3 .Figure 5 .
Figure 3.An impact of casson parameter on velocity distribution.

Figure 6 .
Figure 6.An impact of radiation on temperature distribution.

Figure 8 .
Figure 8.(a) An impact of suction/injection parameter on Nusselt number profile.(b) An impact of radiation on Nusselt number profile.