Impact of Nonlinear Chemical Reaction and Melting Heat Transfer on an MHD Nanoﬂuid Flow over a Thin Needle in Porous Media

: A novel mathematical model is envisioned discussing the magnetohydrodynamics (MHD) steady incompressible nanoﬂuid ﬂow with uniform free stream velocity over a thin needle in a permeable media. The ﬂow analysis is performed in attendance of melting heat transfer with nonlinear chemical reaction. The novel model is examined at the surface with the slip boundary condition. The compatible transformations are a ﬃ anced to attain the dimensionless equations system. Illustrations depicting the impact of distinct parameters versus all involved proﬁles are supported by requisite deliberations. It is perceived that the melting heat parameter has a declining e ﬀ ect on temperature proﬁle while radial velocity enhances due to melting.


Introduction
The study of magnetohydrodynamics (MHD) explains the magnetic features of electrically conducting fluids. Normally, it affects the heat transfer and appears as Joule heating and Lorentz force. Cooling of the refrigerator, saltwater, plasma, tumor therapy, radiation of X-ray, and electrolytes are examples of MHD. Many valuable works done have been published highlighting varied aspects of MHD. Hayat et al. [1] highlight the unsteady MHD nanoliquid flow over a starching sheet in the attendance of viscous dissipation and stratification. Ramzan et al. [2] discussed steady MHD Jeffery nanofluid flow over a vertical inclined cylinder with chemical reaction, thermal radiation, and double stratification. They engaged the homotopy analysis method (HAM) to obtain series solutions. Shehzad et al. [3] discussed the impact of viscous dissipation on tangent hyperbolic non-Newtonian fluid over a stretching sheet with Joule heating. They obtained results by implementing numerical procedure (Keller-box method). They also examined that the fluid velocity is lowered by enlarging the Weissenberg number. Some more recent remarkable researches featuring MHD may be found at [4][5][6][7] and many therein.
The increasing rate of heat transfer by employing nanofluid rather than the real base fluids has attracted the attention of researchers around the world, which makes a clear difference between nanofluids and the base fluid. Nanofluids holding nanoparticles submerged into the base fluid are As the above-referred survey discloses, there are abundant studies available discussing the flow of nanofluids in varied scenarios with different geometries; however, less material exists conversing the nanofluid flows over thin needles. This channel becomes even narrower when we talk about nanofluid flow over a thin needle with the simultaneous impacts of melting heat and nonlinear chemical reaction in permeable media. The layout of the subject mathematical model consists of, firstly, the erection of the mathematical model. Secondly, the detail of the numerical scheme concerns the presented model. Thirdly, the results and discussion section and, lastly, the conclusions.

Mathematical Modeling
Assume an MHD steady, a thin needle immersed is in a nanofluid with a uniform stream velocity u ∞ of incompressible flow with permeable media in the existence of viscous dissipation and temperature-dependent heat sink/source. The effects of melting heat transfer and non-linear chemical reactions are also considered with the slip boundary. The strength of magnetic field B 0 is applied in the radial direction. Here, induced magnetic and electric fields are neglected due to our presumption of a small value of Reynolds number. The needle is seen as thin and its thickness is smaller than the boundary layer formed over it. A physical interpretation of the flow can be seen in Figure 1. Here, x− coordinate is measured as the axial and r− as the radial direction of the cylinder. Assume the function r = R(x) = (vxc/U) 1/2 determines the state of a slight needle in which U = u w + u ∞ is the composite velocity.
Appl. Sci. 2020, 10, x FOR PEER REVIEW  3 of 15 nanofluid flow over a thin needle with the simultaneous impacts of melting heat and nonlinear chemical reaction in permeable media. The layout of the subject mathematical model consists of, firstly, the erection of the mathematical model. Secondly, the detail of the numerical scheme concerns the presented model. Thirdly, the results and discussion section and, lastly, the conclusions.

Mathematical Modeling
Assume an MHD steady, a thin needle immersed is in a nanofluid with a uniform stream velocity u  of incompressible flow with permeable media in the existence of viscous dissipation and temperature-dependent heat sink/source. The effects of melting heat transfer and non-linear chemical reactions are also considered with the slip boundary. The strength of magnetic field 0 B is applied in the radial direction. Here, induced magnetic and electric fields are neglected due to our presumption of a small value of Reynolds number. The needle is seen as thin and its thickness is smaller than the boundary layer formed over it. A physical interpretation of the flow can be seen in Figure 1. Here, x  coordinate is measured as the axial and r  as the radial direction of the cylinder.
n BT x r rr r r r r r r n m DD uC vC rC C r rT T r The suitable boundary conditions are described by: Appl. Sci. 2019, 9, 5492 4 of 14 The suitable boundary conditions are described by: Similarity transformations are defined as: Using the above transformation, the incompressibility Equation is satisfied, and Equations (2)-(4) are transformed into: 2 The pertinent boundary conditions are: Similarity transformations are defined as: Using the above transformation, the incompressibility Equation is satisfied, and Equations (2)-(4) are transformed into: The pertinent boundary conditions are: , Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 15 Similarity transformations are defined as: Using the above transformation, the incompressibility Equation is satisfied, and Equations (2)-(4) are transformed into: The pertinent boundary conditions are: , Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 15 Similarity transformations are defined as: Using the above transformation, the incompressibility Equation is satisfied, and Equations (2)-(4) are transformed into: The pertinent boundary conditions are: ( , ).
Similarity transformations are defined as: Using the above transformation, the incompressibility Equation is satisfied, and Equations (2)-(4) are transformed into: The pertinent boundary conditions are: Using the above transformation, the incompressibility Equation is satisfied, and Equations (2)-(4) are transformed into: The pertinent boundary conditions are: The heat transfer rate and skin friction are classified by: where Through the transformations defined in Equation (5), the drag force and heat transfer rate in dimensionless form are appended below: where

Numerical Method
In the current model, the MATLAB built-in-function bvp4c is used to solve coupled ordinary differential equations (ODEs) (Equations (7)-(9)) with mentioned boundary conditions (Equation (10)). For computational purposes, the infinite domain is restricted to η = 4, which is enough to indicate the asymptotic behavior of the solution. The theme numerical scheme needs initial approximation with tolerance 10 −6 . The initially taken estimation must meet the boundary conditions without interrupting the solution technique.

Results and Discussion
This segment is prepared to investigate the impacts of involved parameters M, λ, Q, Pr, N t , N b , γ, M 1 , k 0 and Sc on dimensionless velocity, temperature and concentration, Nusselt number, and skin friction coefficient. The influence of the velocity ratio parameter (λ) on velocity distribution is drawn in Figure 2. It is observed that when 0 < λ < 0.5, the axial velocity expands just close to the surface of the needle and diminishes far from it. However, a contrary pattern is noticed for the values of ratio parameter greater than zero.   Figure 3 portrays the impact of different estimations of the velocity ratio parameter (λ) on the temperature profile. It is witnessed that for escalating values of ratio λ, the temperature profile is enhanced. In Figure 4, the output of λ on the nanoparticle concentration profile is demonstrated. It is examined that for mounting values of λ, the concentration profile ϕ(η). decreases. Figure 5 reveals the needle velocity for various estimations of the melting parameter (M e ). It is examined that for the mounting valuation of M e , the velocity profile increases. For a stronger M e , more molecular motion is observed, thus increasing the fluid velocity. In Figure 6, a variation of the magnetic parameter (M) on the velocity field is portrayed. It is found that an increase in (M) causes a decreased velocity profile. Sturdier Lorentz force hinders the movement of the fluid motion and thus decreases the velocity of the fluid. Figure 7 is plotted to show the trend of the porosity parameter (k 0 ) on the velocity profile. It is reported that the fluid velocity dwindles for the mounting estimation of (k 0 ). Physically, the motion of the fluid is hindered due to the presence of porous media, which is why a reduced velocity profile is witnessed. In Figure 8, a variation of heat absorption/generation coefficient (Q 0 ) on the temperature distribution is plotted. For rising estimates of Q 0 , a higher temperature profile is observed. Figure 9 depicts the attributes of M e on the temperature. It is examined that with an increase in values of M e , the temperature profile reduces due to enhancement in thermal boundary layer thickness. Moreover, a sheet on normal temperature when dipped in the hot water. This causes the temperature to decrease while melting heat values may increase. The impact of Prandtl number (Pr) on the temperature profile is depicted in Figure 10. The increasing estimations of Pr results in a reduction of temperature field. Higher estimates of Prandtl number lower the thermal diffusivity, thus declining the temperature of the fluid. Figure 11 is illustrated to observe the behavior of non-linear chemical reaction (n) on the concentration profile. It is witnessed that the thickness of the species distribution increases as n increases. Figure 12 indicates the concentration profile for varying thermophoresis parameter (N t ). The temperature gradient is directly proportional to the larger values of N t that, accordingly, upsurges the concentration profile and its related concentration boundary layer thickness. Figure 13 is designed to highlight the Schmidt number (S c ) behavior on the nanoparticle concentration φ(η). S c is the ratio of momentum to the mass diffusion. For a growing estimation S c , a reduction in the mass diffusion coefficient causes a thinner concentration boundary layer. The response of chemical reaction parameter (γ) on the concentration profile ϕ(η) is observed in Figure 14. It is clearly examined that concentration ϕ(η) is decreased via γ. In Figure 15, a retarding effect of λ against magnetic parameter M can be seen for skin friction. It is reported that the thinner boundary layer is accompanying larger values of λ, which results in a higher velocity gradient close to the wall. That is why drag force reduces against λ. An analysis of the impact λ and porosity parameter (k 0 ) on skin friction is detected in Figure 16. A dwindling effect of λ against k 0 can be noticed for skin friction.                                 An excellent consensus is achieved in this regard.   Table 1 portrays the values of f "(m) with those of Ishak et al. [30] and Rida et al. [31] when λ = 0. An excellent consensus is achieved in this regard. Table 2 displays the behavior of different parameters such as M, λ, Q, Pr, m, N t . on the local Nusselt number. It is witnessed that the rate of heat transfer enhances for a larger estimation of λ. The increasing values of λ show that the rate of cooling of the needle can be improved by reducing the external flow velocity or by enhancing the needle velocity. It is analyzed that the heat transfer rate is proportional to the values of Prandtl (Pr). It enhances for the growing values of Prandtl number. It is further reported that viscosity of the fluid is escalated as Pr increases. It is depicted that the heat transfer rate declines for peak estimation of thermophoresis parameter (N t ). Pr is a ratio of the momentum to thermal diffusivity. For increasing values of N t , a heated needle with an opposite trend for negative values of N t is observed. The next four entries show that the Nusselt number is enhanced when the needle size (m) is reduced from 0.1 to 0.001. For increased values of heat absorption/generation coefficient (Q 0 ), the Nusselt number increases.

Conclusions
We consider an incompressible steady MHD nanofluid flow over a thin needle immersed in permeable media and viscous dissipation in the existence temperature-dependent heat source/sink. The effects of melting heat transfer slip boundary condition and non-linear chemical reaction are also discussed. The conclusive remarks of the current study are presented as follows: • Magnetic parameter (M) and porosity parameter (k 0 ) have a diminishing effect on radial velocity;