On solution existence of MHD Casson nanofluid transportation across an extending cylinder through porous media and evaluation of priori bounds

It is a theoretical exportation for mass transpiration and thermal transportation of Casson nanofluid over an extending cylindrical surface. The Stagnation point flow through porous matrix is influenced by magnetic field of uniform strength. Appropriate similarity functions are availed to yield the transmuted system of leading differential equations. Existence for the solution of momentum equation is proved for various values of Casson parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}β, magnetic parameter M, porosity parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_p$$\end{document}Kp and Reynolds number Re in two situations of mass transpiration (suction/injuction). The core interest for this study aroused to address some analytical aspects. Therefore, existence of solution is proved and uniqueness of this results is discussed with evaluation of bounds for existence of solution. Results for skin friction factor are established to attain accuracy for large injection values. Thermal and concentration profiles are delineated numerically by applying Runge-Kutta method and shooting technique. The flow speed retards against M, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}β and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_p$$\end{document}Kp for both situations of mass injection and suction. The thermal boundary layer improves with Brownian and thermopherotic diffusions.

www.nature.com/scientificreports/ problem. we also evaluated the priori bounds for skin friction factor, As for as we know these aspects for the flow of Casson fluids are never explained in the existing studies. The innovation of the work highlighted the existence of solution with uniqueness of results and bounds for skin friction. Moreover, numerical solution of this work is obtained by employing shooting base numerical method codded in matlab script. This exploration may find application in blood rheology, food processing and metallurgy.

Mathematical analysis
In this segment, we are concerned with the following incompressible Casson nanofluid model 42 Consider an incompressible and electrically conducting Casson fluid which flows steady state across an axially extending cylinder of radius = a. Velocity of the stretching wall of the cylinder is U w = caγ . The mass suction across the wall is w w = 2cz , Here c is strain rate of the radial flow and γ is permeability parameter. The fluid flows through a porous medium of Darcy resistance. There is a non varying magnetic field of intensity B o that acts normally to the axis of symmetry (see Fig. 1). The temperature T w and concentration C w are taken at the cylinder and T ∞ and C ∞ are the far field temperature and concentration. Casson fluid parameter is β and k ′ is the porosity of medium. The formulation in (r, θ, z) is constituted keeping in view with the assumptions as mentioned above.
with boundary conditions: (1)  www.nature.com/scientificreports/ In order to yield dimensionless form, similarity transformations are entitled as: The first expression in (2) becomes an identity and the remaining's take the form as follows: Where the expression (3) are transformed: represents Lewis number. The physical quantities of interest are Cf x (skin friction coefficient), Nu x (local Nusselt number) and Sh x (local Sherwood number): where τ w , q w and q m denotes shear stress, surface heat flux and surface mass flux, On solving these quantities with the help of given similarity transformation, we obtain: where, (Re x ) = xU w ν is the local Reynolds number.

Existence
Consider the BVP (boundary value problem) with In order to get the corresponding IVP (initial value problem), the missing initial condition is assumed to be Here ǫ , is a free parameter is relevant to skin friction parameter and f (ξ ; ǫ) denotes the solution. It is because an IVP can be uniquely solved (locally). Thus, a topological shooting argument for some choice of ǫ . For convince, the dependence of f on ǫ may be skipped for some time. The existence of f ′ (ξ ; ǫ) for all ξ > 1 to satisfy Eq. (8).
It may yield a solution to BVP. Two sets X and Y are taken as: and Both of these sets are shown to be open and non-empty in the two Lemmas below:

Lemma 2 The set Y is open and non-empty.
Proof Equation (5) after integration yields as: and a subsequent integration by parts yields It is to show that there is ǫ < 0 , such that f ′ is equated to zero in the interval (1,2], say, before f ′′ = 0 in strict. Suppose this assertion is not true and consider. (1,2]. Then Eq. (13), provides: (1,2] and thus f ′ (2) < 0 which contradicts f ′ > 0 on (1,2].
Since f ′ is bounded below and decreasing, f ′ (∞; ǫ * ) = Z exists where 0 ≤ Z < 1 . It is to see that Z = 0 . We let 0 ≤ Z < 1 . As f ′′ < 0 for ξ > 1 , f ′ is bounded below by Z > 0 , and so, f approaches to positive infinity. Finally the term ff ′′ is negative. Equation (5) provides as below: for ξ to be large enough, there exists a point ξ 2 > 1 and ξ > ξ 2 to imply that By integrating the above expression Let ξ → ∞ then f ′′ → ∞ , it contradicts to the fact that f ′′ < 0 . Hence we have f ′ (∞; ǫ * ) = 0 the following theorem is established.

Uniqueness
Now, we prove uniqueness of results: The differentiation of Eq. (5) with respect to ξ yields: associated with Thus for ξ > 1 , we have v ′ positive and increasing and v ′ > 0 and increasing for ξ > 1.
It is to show a positive maximum does not exists for v ′ (ξ , ǫ * ) . Let a maximum exists at first point for which It becomes contrary and hence v ′ cannot have a maxima. So v ′ = ∂f ′ ∂α > 0. IF we let two solutions f ′ (ξ ; ǫ * ) and f ′ (ξ ; ǫ * * ) with ǫ * * > ǫ * , and using Mean Value Theorem where ǫ * <ǫ < ǫ * * . Now v ′ is bounded below by L > 0 for ξ large as it cannot have a maximum. Suppose M = L(ǫ * * − ǫ * ) and ξ → ∞ , From Eq. (18) It becomes contrary. It is to mentioned that the bounds for skin friction factor are evaluated and presented in the next part. www.nature.com/scientificreports/ Bounds for skin friction factor. Bounds are derived for coefficient of skin friction f ′′ (1) = ǫ * . As f ′ (ξ ; ǫ * ) be a solution of the BVP to satisfy f ′′ (1; ǫ * ) = ǫ * < 0 and cannot have a maximum. It is claimed that for a solution to company the boundary condition (8), yields (20) to get: Then initially, f ′′′ < 0 for ξ > 1 , and f ′′′ cannot change sign.
Case-2: Solution for which f ′ (ξ ; ǫ * ) < 0 : let f ′′′ (1) < 0 and f ′ is down concave initially. Because there exist a first point ξ 4 such that f ′ (ξ 4 ) = 0 and f ′′ (ξ 4 ) < 0 , f ′ is not positive for all ξ . Also, f ′ should be concave up to satisfy Eq. (8) for some ξ > ξ 4 and it attained a minimum. As f ′ does not attain maximum, f ′ necessarily increase from its minimum monotonically, and then tends to 0 from below to become concave down.
It becomes clear that, f ′′′ must change sign from minus to plus and back to minus. Thus a point ξ 5 is such that f ′′′ has a positive max, i.e., f ′′′ (ξ 5 ) > 0 , f iv (ξ 5 ) = 0 , and f . It can be noticed that both bounds converge to zero, and so, f ′′ (1) converges to zero as γ (γ < 0) tends to infinity. Computations of skin friction coefficient f ′′ (1) = ǫ * are provided in Table 1. Here sharpening of the bounds on f ′′ (1) is elucidated for a fixed a Re = 1 , as the parameter γ enhances.
Proof Using Lemma 3 results and letting ξ → ∞ in Eq. (31) Although the existence of solutions where f ′ (ξ ; ǫ * ) < 0 is yet an open problem. Suppose such solution exist, then a bound on the skin friction coefficient is established in next Theorem 4. Two lemmas are required for the proof of this bounds.

Results and discussion
The current results are checked for validation as listed in Table 2 and 3. Their acceptable accord with those by Mastroberardino and Siddique 41 has established the accuracy of the present numeric scheme. The pictorial representation for Casson nano-fluid's velocity, temperature and concentration of nano-entities graphed for two cases of mass transpiration (γ > 0, γ < 0).
The outcomes for velocity f ′ (ξ ) , temperature θ ′ (ξ ) and concentration φ ′ (ξ ) are sketched in Figs. 2, 3, 4 and 5 for two cases of γ (γ = −0.5andγ = 0.5) with the variation of other influential parameters. The velocity f ′ (ξ ) is vividly decelerated against the increments in magnetic parameter M as well as that of porosity parameter K p as seen in Fig. 2. The strength of M means growth of electromagnetic resistive force (Lorentz force) which inhibits the flow. Similarly, parameter of porous matrix (K p ) offers enhanced resistance to the velocity. There is sound www.nature.com/scientificreports/ reason behind this fact that K p is related reciprocally with permeability and hence higher inputs of K p means lesser permeability. Thus the flow decelerates in this case. The incremented values of Re and Casson parameter β also slowed the flow velocity f ′ (ξ ) as delineated in Fig. 3. Here the viscous effects are enhanced (to oppose to) momentum. Furthermore, it is noticed that velocity of flow is faster in case of injection (γ > 0) than for suction (γ < 0) . Figure 4 exposed that the nanofluid diffusion parameters namely Nb (Brownian diffusion) and Nt (Thermophoresis diffusion) are responsible to raise the temperature function θ(ξ ) but the progressive values of Pr reduced θ(ξ ) . The faster random motion of nano-entities is associated with larger Nb. This rapidity in the movement of these small material particles causes greater thermal distribution. In the similar behavior due to enhanced enhanced thermophoresis. The particles move fastly towards cooler regimes and hence raises the temperature. It is also noticed that the fluid temperature for suction is higher than for injection. The greater values of Le and Re diminish the nanoparticle concentration φ(ξ ) in the boundary layer region as depicted in Fig. 5. Physically, the      www.nature.com/scientificreports/ Lewis number Le is inversely related to diffusion coefficient of concentration and hence its development impairs φ(ξ ) . Moreover as seen above, the larger Re slows the flow which results in decrement of φ(ξ ) The absolute values of skin friction are augmented in direct proportion with K p , M, Re and β for three cases of γ (γ < 0, γ = 0, γ > 0) as enumerated in Table 4. Physically, K p signifies the resistance of porous matrix, M for electromagnetic resistive force, Re (Reynolds number) and β the non-Newtonian viscous effects (for Casson fluid). Hence the drag force enhances. Table 5 indicates that Nusselt number −θ ′ (0) increases with Pr but it diminishes against Nb and Nt. Physical ground for augmentation of Nusselt number with Prandtl number lies in the fact that thermal diffusivity being reciprocal to prandtl number is responsible to decrease the temperature of the fluid and more heat transfer takes place at the surface and hence the magnitude of Nusselt number is boosted. Further the growing thermopherotic and Brownian diffusion raise the fluid temperature and heat transfer rate at the surface is decreased and the Nusselt number decreases against these parameters. Also, the Sherwood number −φ ′ (0) exceeds directly with Le and Re (see Table 5).

Conclusion
We discussed the existence of solution for Casson fluid flow towards a porous stretching cylinder. The fluid flows through porous medium and it is influenced by magnetic field. It is shown that the boundary value problem for any Re > 0 and −∞ < γ < ∞ , to satisfy f ′ (ξ ) > 0 and f ′′ (ξ ) < 0 for all ξ > 1 . The uniqueness of the result is established in the sense that we cannot have two solutions for the boundary value problem if −∞ < γ < ∞ and Re > 0 . Moreover, the bounds for skin friction factor are evaluated. Numerical solution of the flow and heat transfer for Casson nano-fluid is also obtain to reveal that: • It is observed that for the magnetic parameter M and porosity parameter kp reduces velocity when takes large values for three cases of γ. • Velocity recedes with the higher inputs of Re and β.