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The impingement of Hiemenz stagnation-point flow and a flexible surface which has linear stretching/shrinking is studied. The flow is governed by the wall's velocity, gyration velocity, transpiration and velocity slip parameters. The stretching sheet problem offers single solution, whereas for the shrinking sheet dual solutions are obtained in the region s ∗ ≤ s < ∞ , and λ ≤ λ ∗. The stretching/shrinking strength parameter ( c ) under different values produces numerous exact solutions. The existence of critical numbers explains the increasing or decreasing nature of solutions. Dual solutions are found for skin friction , couple stress friction and temperature gradient . In the process of boundary layer, increasing suction or injection decreases velocity. The effect of velocity slip indicates a decrease in linear wall velocity. It was found that the number and formation of solutions are dependent on a parameter measuring correlation of the external flow strength to the surface stretching/shrinking parameter which is a key player in determining exact solutions. Exact solutions for the uniform surface temperature and linearly growing wall temperature are derived for different special cases. Introduction Non-Newtonian micropolar fluids are utilized in a range of engineering and industrial applications, including lubricants, fluid suspensions, polymer manufacturing, food production, and cosmetics. The micropolar fluid flow model, initially set by Eringen [1,2], accounts for the microscale properties of fluids arising from the local structure and micromovements of fluid particles. Ahmadi [3] examined the micropolar fluid flow on a boundary layer flat plate, followed by Jena and Mathur [4], who analyzed the vertical wall for free convective thermomicropolar fluid. Gorla [5] provided a solution for micropolar stagnation point boundary layer flow. The stretching sheet problem in a micropolar fluid environment was considered by Sankara and Watson [6], with the introduction of wall permeability by Heruska et al. [7]. The addition of heat transpiration into the permeable wall was given by Hassanien and Gorla [8], while the approximate solution by perturbation technique under constant wall transpiration was analyzed by Kelson and Desseaux [9]. The interesting work of shrinking sheet for exact solutions with wall mass flux was reported by Turkyilmazoglu [10], see also [[11], [12], [13]]. Turkyilmazoglu [14] also studied the permeable heated stretching surface of micropolar fluid. Turkyilmazoglu [15] extended the stretching sheet micropolar problem to a magnetohydrodynamic mixed convection permeable wall under heat transport with heat generation and absorption effects. Recently, Raza et al. [16] provided dual exact solutions for magnetohydrodynamic micropolar fluid stream in a porous medium with thermal and mass flux effects. The phenomenon of a boundary layer over a stretchable permeable membrane with heat transfer is a fundamental flow that is present in various scientific and engineering fields. This type of flow is found in nano/micro systems such as micro-pumps, hard-disk drives, and micro-nozzles. In non-Newtonian fluids, such as melting polymers, the customary no-slip condition is shaped by a nonlinear and monotone relationship between the slip velocity and adhesion. In particulate fluids, the usual no-slip condition is superseded by the slip effect at the boundary. This phenomenon is characterized by velocity slip effect at the surface, and it plays a crucial role in the transition flow mechanism. The notion of slip-free boundary is a vital component of fluid flow theory, where the viscous fluid sticks to the boundary in contact. However, in the case of a fluid being particulate, the no-slip state is replaced by the slip effect at the boundary. This idea is well-documented in several studies such as [[17], [18], [19], [20], [21], [22]]. The classical problem of a steady stream as it approaches a stationary solid and comes to a rest at the surface has been an area of immense interest among researchers working in fluid dynamics. The pioneering work of Hiemenz [23] marked the commencement of the study of the flow at the stagnation point. This was followed by Howarth [24], who successfully studied the two-dimensional steady flow over an obstacle. Further investigations have been conducted by numerous researchers, including Mahapatra and Gupta, who considered the combined effect of a stretching sheet and stagnation point, and Wang, who reported on the stagnation point flow for a shrinking sheet with convective heat transfer. Notably, the flow of a stretching or shrinking sheet with an obstacle, creating a stagnation point in a micropolar fluid, has also been reported by several researchers, as documented in [25,26,30,[32], [33], [34], [35], [36], [37], [38], [39]] and the associated references. The classical problem in fluid dynamics when the steady stream approaches a stationary solid and the stream is caused to rest at the surface of the solid has been of immense interest among researchers. Hiemenz [23] was first to pioneer the flow at stagnation point, followed by Howarth [24], who successfully studied the two-dimensional steady flow over an obstacle, see also [25,26]. The flow instigated by a stretchable surface, which has many engineering applications, admits similarity solutions. The integrated effect of stretching sheet and stagnation point was considered by Mahapatra and Gupta [27,28]. Wang [29] reported convective heat transfer amalgamating stagnation point with shrinking plate fluid flow. The flow of a stretching or shrinking sheet with an obstacle, creating a stagnation point in micropolar fluid also has been reported by numerous researchers, see [30,[32], [33], [34], [35], [36], [37], [38], [39]] and the references therein. The flow of micropolar fluids is a thrust area of importance for researchers, and much of the literature on the subject dominates the numerical treatment of physical problems [30,31,[40], [41], [42], [43], [44], [45]]. This is because the coupled nonlinear nature of flow equations makes it challenging to accurately obtain exact solutions. We believe that no work dealing with the impingement of surface flow, transpiration, velocity slip, stagnation-point flow, and heat transfer with constant and linearly growing surface temperature conditions has ever been reported. We have attempted to find new analytical exact solutions for elastic surface flow, which are unique in their presentation. We believe our work is distinct from the existing literature and provides new insights into the flow of micropolar fluids. The organization is as follows. The problem formation is given in Section 2. This is followed by exact similarity reduction given in Section 3. Exact solutions dependent on considered parametric behaviors are presented in Section 4. Discussion of results is given in Section 5 which is followed by conclusions in Section 6. Section snippets Model formulation The formulation of this problem is given in Cartesian 2D coordinate system ( x , y ) with respective velocities ( u , v ) and heat transfer at surface τ . An elastic permeable surface located at y = 0, stretches and shrinks in x -direction with rate U . The flow is instigated by linearly moving surface, mass transpiration, stagnation-point, or a combination all three. The respective velocities are u w x = Ux , v w = v w x , U s x = dx , where U and d are constants. The temperature distribution τ w x = τ w + x τ 0 is supposed to be Flow model transformation Consider similarity solutions of the following format ψ x η = ν U xf η , G x η = U ν Ux g η , θ η = τ − τ ∞ τ w − τ ∞ , η = U ν y with η the boundary layer variable, ψ the stream function which produces similarity solutions as u = ∂ y ψ = Ux f ′ η , v = ∂ x ψ = − νU f η . Inserting the ansatz (9) into the micropolar fluid system (4)–(6) gives the reduced coupled equations as 1 + λ f ‴ + f f ″ − f ′ 2 + λ g ′ + c 2 = 0 , 1 + 2 − 1 λ g ″ + f g ′ − g f ′ − λ 2 g + f ″ = 0 , θ ″ + Pr f θ ′ = 0 , under τ w x = τ w θ ″ + Pr f θ ′ − f ′ θ = 0 , under τ w x = τ w + x τ 0 with boundary conditions remodeled as f 0 = s , f ′ 0 = a + b f ″ 0 , f ′ ∞ = c , g 0 = − n f ″ 0 , g ∞ = 0 , θ 0 = 1 , θ ∞ = 0 . Exact solutions The wall boundary conditions suggest the following format of solutions f η = s + cη + a − c γ 1 + bγ 1 − e − γ η , g η = n a − c γ 1 + b γ e − γη , where, γ is to be found. Physical flow solutions are dependent on γ , and thus γ > 0. The expressions of f ( η ) and g ( η ) on substituting in (11) in turn, produces the following relation abλ + 2 ab − 2 bc − bcλ γ 3 + 2 a − 2 abcη − 2 abs + aλ + 2 bc 2 η + 2 bcs − 2 c − cλ γ 2 + 2 cs − 4 abc − 2 acη − 2 as + 4 bc 2 + 2 c 2 η γ − 2 a 2 + 2 c 2 = 0 . The following analysis is carried out for the weak concentration case n = 1/2. Exact solutions under particular Results and discussion Exact solutions for nonlinear partial differential system running the stretching/shrinking sheet flow, microrotation and thermal transport of micropolar fluid are derived under the wall mass flux, velocity slip, fixed heat transfer and linearly increasing temperature conditions. The cases when solution are possible to certain parametric situations only have been listed in detail with flow and temperature exact solutions. In particular, it is to emphasize that the numerical values of parameters Concluding remarks This work demonstrates a unified analysis of stretching/shrinking flexible surface and stagnation-point flow, where the two situations can occur together or separately. The number and existence of exact solutions solely depend on domains of parameters. Exact solutions for fluid flow, heat flow and reduced Nusselt numbers considering constant wall and linearly growing temperatures models are derived. The keys findings of this analysis are listed as: 1. Unique exact solution occurs in the physically Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The first author thanks the financial sponsorship by Nanjing Institute of Technology for Young Scholars Research Support Program under Grant No. YKJ202126 . References (53) M. 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Wang Flow due to a stretching boundary with partial slip–an exact solution of the Navier–Stokes equations Chem. Eng. Sci. (2002) A. Raza et al. Existence of dual solution for micro–polar fluid flow with convective boundary layer in the presence of thermal radiation and suction/injection effects Int. Commun. Heat Mass Transf. (2022) M. Turkyilmazoglu Mixed convection flow of magnetohydrodynamic micropolar fluid due to a porous heated/cooled deformable plate: exact solutions Int. J. Heat Mass Transf. (2017) M. Turkyilmazoglu Flow of a micropolar fluid due to a porous stretching sheet and heat transfer Int. J. Non-Linear Mech. (2016) E.H. Aly Dual exact solutions of graphene–water nanofluid flow over stretching/shrinking sheet with suction/injection and heat source/sink: critical values and regions with stability Powder Technol. (2019) E.H. Aly Existence of the multiple exact solutions for nanofluid flow over a stretching/shrinking sheet embedded in a porous medium at the presence of magnetic field with electrical conductivity and thermal radiation effects Powder Technol. (2016) M. Turkyilmazoglu A note on micropolar fluid flow and heat transfer over a porous shrinking sheet Int. J. Heat Mass Transf. (2014) View more references Cited by (0) Recommended articles (6) Research article Effect of variable gravity field on the onset of convection in a Brinkman porous medium under convective boundary conditions International Communications in Heat and Mass Transfer, Volume 144, 2023, Article 106777 Show abstract The effect of variable gravity on the onset of convection in a horizontal porous layer has been analyzed due to external heating. The third kind of thermal boundary conditions is considered on both the boundaries which are connected by Biot numbers B 0 and B 1. 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