Effects of chemically reactive and thermally radiative MHD Prandtl nanouid by a vertically heated stretchable surface with convective conditions

: The existing investigations purpose to disclose the interaction effects of transverse magnetic and hydrodynamic flow of Prandtl nanofluid subjected to convective boundary conditions over a vertical heated stretching surface. Developing a fundamental flow model, a boundary layer approximation is done, which yields momentum, concentration, and energy expressions. Moreover, Brownian effect and thermophoresis influence are also taken into the account. The constitutive flow laws of nonlinear (PDEs) is altered into ordinary one via similarity transformation variables. The dimensionless nonlinear systems of (ODEs) are then solved through bvp4c numerical algorithm. Consequences of innumerable flow factors on steam wise velocity, thermal field, and concentration of nanoparticle are explicitly debated and plotted graphically. The drag force coefficient and heat transference rate are assumed and deliberated accordingly. It has been perceived that f higher estimation of thermophoresis parameter upsurges the internal thermal energy of the nanofluid and nanoparticles concentration field.


Introduction
Almost all fluids that occur in the process of radiation and chemical reactions unable to comply with classical Newton's law which categorized as non-Newtonian fluids. Due to the majority of applications of these fluids in different fields like technology, petroleum production, biochemical, pharmaceutical, food, and power engineering, considerable attention has been given by researchers and scientists to study the physical aspects of such fluids. Besides this, because of the budding utility of non-Newtonian fluids in large numbers of industrial and manufacturing practices, more attention paid to the analysis of heat transfer features of these fluids. However, a practical approach was taken into consideration, the various systems have been suggested to analyze the structural behavior of these fluids namely the Power-law model, Carreau model, the Rivlin-Ericksen model, Williamson model, Powell-Eyring model, Prandtl model, Ellis model. These models are basically characterized into viscoelastic and inelastic fluids, micro-structure fluids, and polar fluids. Among these, the visco inelastic fluids are important in the view of the mixed effects of elastic and viscous characteristics. Akbar et al. [1] analyzed the result of magnetized Prandtl flow driven by stretching sheet through the porous plate. Bilal et al. [2] studied the impact of double diffusion on Prandtl nanofluid considering stretched sheet under magnetic effects. Nadeem et al. [3] analyzed the peristaltic movement of a Prandtl fluid model analytically. Khan et al. [4] investigated heterogeneous and homogeneous reactions over Prandtl fluid at the stretched sheet. Further, Amanullah et al. [5] studied the Prandtl-Eyring fluid flow over an isothermal sphere under magnetic strength and permeable medium. Mukhopadhyay [6] showed the slip effects of magnetized flow past an exponential stretched sheet under radiative flux. Ali et al. [7] elaborated the unsteady Powell-Eyring nanofluid over a stretching sheet. The massive information on viscoinelastic model examined by many authors has been listed in [8][9][10][11][12][13][14][15][16]. Several examinations have been carried out by the researchers to enhance the thermo-physical characteristics in conventional heat transfer fluids. Unfortunately, the expectation of modern technology could not have satisfied by these fluids. Because most of the fluids like glycol, water, oil, and many more are low conductors of heat due to the poor thermal conductivity. Hence, to cope up with this situation, nanoparticles are introduced in the fluids. Nanofluids are developed by immersing tiny particles in the base fluid. Because of nanofluids, a reduction in thermal resistances is observed as they can easily flow through the microchannels benefited by their extra small size (10-50 nm). Choi [17] coined the phenomena of the "Nanofluid". He proved that the inclusion of the nanoparticles in the liquid causes to rise in thermal conductivity tremendously. Chiller, microelectronics, fuel cells, hybrid power industries are witnesses of such processes. Reddy and Sreedevi [18] presented the influence of chemical reaction on heat and mass transfer features of nanofluid past stretched sheet through porous media. Abbas et al. [19] analyzed the Darcy Forchheimer nanofluid flow under magnetism subject to stretching surface. Gireesha et al. [20] discussed the MHD stagnation flow consisting of magnetized nanofluid against a stretched surface. Sreedevi and Reddy [21] discussed the characteristics of thermophoresis and Brownian motion over the three-dimensional Maxwell nanofluid flow due to stretching sheet considering radiation and chemical reaction. Recently, various excellent investigations on vital features of nanofluid were reported by [22][23][24][25][26][27][28]. MHD flows in association with a stretching sheet have immense implications in technical and modern industrial practices such as hot rolling, glass fiber, cooling in nuclear reactors, plastic sheets extrusion, metal casting, glass blowing, and metallurgical casting. Magnetized nanofluid flow passed a perpendicular permeable surface was shown by Das et al. [29]. Ibrahim et al. [30] elaborated the power of chemical reaction on the mixed convective flow of Casson nanofluid over a stretched surface under magnetic strength effect. Kumar et al. [31] studied micro-polar fluid flow with magnetism and radiation at a stretching surface. Aziz and Afify [32] reported the Casson MHD flow driven by stretched sheet through slip velocity and claimed that for magnetic parameter less than one, heat transfer rate enhanced and vice versa. The study of heat transfer associated with radiation has gained considerable attention by the researchers because of vast utility in the technological applications which includes missiles, space technology, nuclear power plant, gas turbines. Mabood and Das [33] discussed the MHD viscoelastic fluid of Casson liquid in a porous regime under the radiation effects. Dogonchi and Ganji [34] analyzed MHD nanofluid flow past parallel plates caused by the radiation. Gupta et al. [35] presented Brownian movement and Thermophoresis effects on the nanofluid over inclined stretched sheet under chemical and radiative flux. Sheikhoslami and Shamlooie [36] explored natural convection flow of nanofluid under magnetism and radiation. Alkanhal [37] showed the thermal impact on MHD nanofluid flow over permeable surface under radiative heat sources and magnetic strengths.

Framework of the flow problem
Here, we have assumed two-dimensional unsteady viscoelastic incompressible boundary layer flow of non-Newtonian Prandtl nanofluid in view of vertically heated stretchable surface along with thermal radiation is considered in modeling for the flow problem the formulation formulated.
The fluid motion with stretching sheet is taken onwards the x axis − and y axis − is horizontal to it.
Physical configuration and Coordinate system is shown in Fig. 1. The flow under consideration is sketched below.
Then the governing model becomes, The particular extreme values are:

h t C C for y y and u U x t T T C C for y
The radiation heat flux ( ) In view of similarity transformation, the governing equations (3)- (7), reduced to dimensionless form by with the transformed boundary conditions, The pertinent flow parameters appearing in (9)- (12) are defined in (13) by , Pr Describes Brownian effect, thermophoresis influence, Prandtl number, Schmidt number, convective parameter/Biot number, fluid parameter, Prandtl fluid factor, unsteady parameter, stretching parameter, radiation parameter and Chemical reaction respectively.
The expressions of physical quantities ( ) , , Here, w  denote shear stress at the sheet, w q heat flux and m q mass flux of nanoparticle is written The simplified dimensionless form of (14) for ( ) , , x x x Cf Nu Sh using (9) we have

Numerical solution
The dimensionless nonlinear system of ( ) ODEs in (9) Let introduce new transformation variables , , w x y and z then the system of nonlinear equations (9)-(12) are transformed to first order linear differential equations as, with altered conditions, The process is continued unstill the wanted level of accuracy of    Fig. 8 and Fig. 9.
Clearly, both thermal field and concentration upsurges for higher estimations of unsteady parameter. Physically, higher unsteady parameter, the internal system becomes more heated due to which the fluid particles have higher kinetic energy releases. In consequences, thermal field increases. Moreover, larger ( )

Conclusion
In this study, numerical solution of Prandtl nanofluid flow with a vertically heated stretching sheet with thermal radiation, heat source, and MHD is analyzed. The boundary layer flow yields constitutive (PDEs) which were altered to the corresponding dimensionless nonlinear (ODEs) via similarity transformations. The system was solved numerically using the bvp4c technique. The outcomes are presented graphically with numerous system parameters. Following are the leading findings of this investigation have been observed: • It is perceived that fluid velocity augmented subject higher fluid parameter, whereas this parameter depicts an opposing appearance versus thermal field and concentration profile. • Similar features are witnessed qualitatively for larger thermophoresis versus temperature and solutal fields. • Higher Brownian motion parameter escalates fluid particles movement that developed the fluid temperature and fluid particles starts rapidly down from higher to lower regions.
• Magnetism influence diminishes the velocity stream significantly, whereas reverse trends for temperature and nanoparticles concentration observed.
• Larger Hartman number/magnetic parameter yields diminishes the surface drag force.

Data availability statement
The data used to support the findings of this study are included within the article.