During the last decades, the topic of heat/mass enhancement through multi-phase flows (e.g., monotype/hybrid/ternary nanofluid flows) has attracted amazing consideration from the scientific communities around the globe owing to their widespread necessity in several practical domains (e.g., the ameliorating the combustion characteristics of diesel fuel in compression ignition engines). In this context, it was demonstrated through a benchmark experimental investigation [1] and other authenticated sources [2, 3] that regular liquids (e.g., oil, ethylene glycol, and water) are not preferred thermally as working fluids due to their weak thermal conductance. As technical propositions, it was advised that the insertion of tiny/nano-sized particles (e.g., alumina, graphene, copper oxide, gold, titania, silver, copper, as well as single- and multi-walled carbon nanotubes) is among the best feasible way for improving the thermal performances of pure fluids on the condition that the adding solid nanoparticles should have a higher thermal conductivity as compared with the host fluid. Such a biphasic mixture (i.e., nanoparticles and base fluid) was named a nanofluid for the first time in 1995 by Choi and Eastman [4]. Thanks to the latest progress in the mixture theory and the science therein, the existing experimental and theoretical literature surveys involve an exhaustive overview regarding the principal specific appearances describing persuasively the physical, thermal, and rheological aspects of nanofluids [5–10]. In this respect, it was reported that the thermophysical proprieties of nanofluids depend on several influential factors (e.g., shape/size of nanoparticles as well as their concentration and temperature) that can affect significantly the heat transfer within a nanofluidic medium and its flow pattern. By linking theoretically the bioconvection occurrence phenomenon to other thermal and mass transport processes, Waqas et al. [11] proposed advanced non-homogeneous flow models constrained by realistic physical impacts to explore the consequence of swimming motile microorganisms on the hydrothermal and mass features of slippery nanofluid flows near a variable thick surface of a rotating disk. More recently, Shah et al. [12] exploited the possibility of hybridizing the nanoparticles M W C N T and F e 3 O 4 in a suitable base fluid to examine thermodynamically the behavior of a magnetized hybrid nanofluid inside a porous cavity structure in the presence of a tilted magnetic field. The same solid mixture was employed also by Maneengam et al. [13] in pure water to study the irreversibility features and MHD flow patterns of the hybrid nanofluid ( F e 3 O 4 + M W C N T ) − H 2 O inside a lid-driven corrugated porous cavity. Further hydrothermal and entropic appearances of the nanofluid A l 2 O 3 − H 2 O were evidenced comprehensively by Alshare et al. [14] during its MHD natural convective motion inside a lid-driven wavy cavity with an elliptical obstacle by emplying a robust numerical code based on the Galerkin weighted residual finite element technique.

Convective flows over rotating heated disk-shaped bodies are considered among the main popular dynamical problems treated fundamentally in fluid mechanics and heat transfer since a long time ago. Further, the dealing of flows near rotating disks is of great interest to the majority of researchers not only for comprehending the occurring flow regimes but also for their diverse uses (e.g., stability control of swirling flows, domestic devices, rotating heat exchangers, visco-rheometers, chemical stirring operations, spinning disk reactors, vehicle engines, and productive aero-hydrodynamic turbines). In this respect, several examinations were accomplished roughly for such flow problems [15–17]. In 1921, Von Kármán [18] was the first pioneering scientist who discussed theoretically the dynamical axisymmetric aspect of steady swirling flows driven over a rotating infinite disk for incompressible viscous fluids. Based on the boundary layer theory, Navier-Stokes’s mathematical formulation of this famous fluid flow problem can be extended reasonably to the case of non-Newtonian fluids [19, 20] (e.g., Bingham and tangent hyperbolic fluid models). Mathematically, the simplified boundary layer equations can be reduced to a nonlinear system of coupled ordinary differential equations, whose similar solutions can be derived semi-analytically or numerically through the modified so-called Von Kármán’s integral momentum equation. Keeping in mind the rheological and thermal importance of nanofluids and their applicabilities in Von Kármán’s flow configuration, several scrutinizations were performed recently on nanofluid flows over a rotating disk. In this context, Waini et al. [21] performed a numerical stability analysis of unsteady axisymmetric swirling flows of ( A l 2 O 3 + C u ) − H 2 O hybrid nanofluids over an isothermally heated disk. By utilizing the same hybrid mixture, Kumar et al. [22] invoked the entropy minimization approach along with Von Kármán’s strategy to quantify the different thermodynamical irreversibilities that can be happened during the radiative magnetohydrodynamic slipping flows of ( A l 2 O 3 + C u ) − H 2 O hybrid nanofluids over a rotating heated disk when the strengths of Ohmic heating and viscous dissipation are significant. Similarly, Mandal and Shit [23] assumed the effective contribution of non-Newtonian viscous dissipation and Joule heating to carry out a numerical entropic scrutinization on unsteady axisymmetric MHD flows of radiative Casson nanofluids over a turning permeable disk, which was supposed to be heated convectively, stretched radially, and embedded horizontally in a quiescent nanofluidic medium containing ethylene glycol or water as a holding fluid and alumina or copper as nanoparticles. From another strategical point of view, Magodora et al. [24] preferred employing the non-homogeneous nanofluid model of Buongiorno [25] to show the significance of Brownian and thermophoresis diffusions of nanoparticles on the onset of heat and mass transport during the axisymmetric swirling motion of a chemically reacting nanofluid containing water and gold nanomaterials nanofluid near a rotating heated disk. In another multi-diffusive problem, Latiff et al. [26] considered Stefan’s blowing influence and bioconvection occurrence to examine the hydrothermal appearances along with the resulting mass transport phenomena during the unsteady nanofluid motion over a spinning stretchable disk.

Nonetheless, inclusive studies on the significant impacts of Arrhenius’s chemical reactive kinetics, the thermo-migration diffusive process, and the Brownian motion of nanoparticles on MHD Von Kármán non-homogeneous flows of chemically reacting nanofluids over a horizontal turning disk are still rare in the available literature, in the case of zero mass flux and thermal convective conditions, and especially when the electrically conducting nanofluidic medium is exposed to an adjustable magnetic source acting radially, which is enhanced thermally via an exponentially decaying space-dependent heat source. Motivated by this pending scientific concern, the present examination aimed surely to provide definitive answers on this open topic. Accordingly, an appropriate MHD nanofluid flow model has been developed properly in this regard by adopting the renovated version of Buongiorno’s approach. Based on the generalized boundary layer simplifications and other admissible physical assumptions, the leading conservation equations are derived mathematically in the form of coupled partial differential equations (PDEs) together with their corresponding realistic boundary conditions (BCs), which are rewritten thereafter as a set of strongly nonlinear ordinary differential equations (ODEs) having no closed-form solutions. For this reason, an advanced hybrid algorithm has been implemented numerically in Matlab software. After performing extensive validating tests, the generated datasets are presented skillfully to provide a comprehensive physical discussion through the following research questions:

1- How can formulate a proper MHD non-homogeneous Von Kármán flow model for a chemically reactive nanofluid by considering the effective contribution of Brownian and thermophoresis diffusions and utilizing the molar concentration notion, in the case where the external magnetic field is exerted radially in the presence of an exponentially decaying heat generation?

Let’s consider a steady laminar nanofluid flow over an impermeable disk of infinite extension, which is rotated uniformly around its symmetrical z − axis with an angular velocity Ω and heated convectively thanks to a surrounding hot working fluid of temperature T f . As an additional presumption, a destructive reactive process is taken place chemically throughout the Newtonian nanofluidic medium according to Arrhenius’s kinetics. Initially, the geometrical configuration is positioned horizontally in a quiescent nanofluidic medium, which is featured physically by the pressure p ∞ , the temperature T ∞ , and the nanoparticles’ molar concentration χ ∞ as schematized tri-dimensionally in Figure 1. For a better description of the proposed nanofluid model, an appropriate cylindrical frame is chosen herein to provide a proper mathematical formulation in the cylindrical coordinate system ( r , ϕ , z ) . Moreover, the convective swirling nanofluid motion has happened spatially nearby to a magnetic source applying radially a uniform magnetic field of intensity B in the presence of an exponentially decaying space-dependent thermal source. Furthermore, the wall characteristics ( T w , χ w ) corresponding to the nanofluid temperature and nanoparticles’ molar concentration at the contact surface are unknown as boundary conditions. However, their approximate values can be estimated computationally through the gradient expressions of the thermal and concentration boundary conditions (i.e., the convective heating and zero vertical mass flux conditions). For reducing the complexity of the nanofluid flow problem under consideration, the following physical presumptions are taken into account:

− The viscous nanofluid has a Newtinanan rheological trend and behaves as a weakly electrically conducting medium, in which the induced magnetic field can be ignored in front of the externally applied magnetic field.

Based on the above statements, the leading conservation PDEs are written as: ∂ u ∂ r + u r + ∂ w ∂ z = 0 , (1) u ∂ u ∂ r − v 2 r + w ∂ u ∂ z = − 1 ρ ∂ p ∂ r + υ (   ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r − u r 2 + ∂ 2 u ∂ z 2   ) , (2) u ∂ v ∂ r + u v r + w ∂ v ∂ z = υ ( ∂ 2 v ∂ r 2 + 1 r ∂ v ∂ r − v r 2 + ∂ 2 v ∂ z 2 ) − σ B 2 ρ v , (3) u ∂ w ∂ r + w ∂ w ∂ z = − 1 ρ ∂ p ∂ z + υ (   ∂ 2 w ∂ r 2 + 1 r ∂ w ∂ r + ∂ 2 w ∂ z 2   ) − σ B 2 ρ w , (4) u ∂ T ∂ r   + w   ∂ T ∂ z   = { κ ( ρ C P ) ( ∂ 2 T ∂ r 2 + 1 r ∂ T ∂ r   + ∂ 2 T ∂ z 2 ) +   ( ρ C P ) n p D B ( ρ C P ) δ χ   ( ∂ T ∂ r   ∂ χ ∂ r + ∂ T ∂ z   ∂ χ ∂ z   ) + ( ρ C P ) n p D T ( ρ C P ) T ∞   [ ( ∂ T ∂ r ) 2 + ( ∂ T ∂ z ) 2 ] + Q ( T f − T ∞ ) ( ρ C P ) e − δ Ω υ z } , (5) u ∂ χ ∂ r + w ∂ χ ∂ z = ( ∂ 2 ∂ r 2 + 1 r   ∂ ∂ r + ∂ 2 ∂ z 2 ) ( D B χ + δ χ D T T ∞ T   ) − K χ ( T T ∞ ) n ( χ − χ ∞ ) e − E A K B T . (6)

These conservation equations are governed by the following BCs: B C s ( u , v , p ) : { u ( r , z = 0 ) = 0 ,   v ( r , z = 0 ) = Ω r ,   w ( r , z = 0 ) = 0 , u ( r ,   z → ∞ ) → 0 ,   v ( r ,   z → ∞ ) → 0 ,   p ( r ,   z → ∞ ) → p ∞ } , (7) B C s T : { ∂ T ∂ z ( r , z = 0 ) = h f κ [ T ( r , z = 0 ) − T f ] ,   T ( r ,   z → ∞ ) → T ∞ } , (8) B C s χ : { ∂ χ ∂ z ( r , z = 0 ) = − δ χ D T D B T ∞ ∂ T ∂ z ( r , z = 0 ) ,   χ ( r ,   z → ∞ ) → χ ∞ } . (9)

For the sake of briefness, the meanings of the physical symbols and abbreviations used above are well regrouped in the nomenclature table. Further, it is preferable to introduce feasible similarity alterations into Eqs. 1–9 as suggested below: { ξ = Ω υ z , F ( ξ ) = u Ω r ,   G ( ξ ) = v Ω r ,   H ( ξ ) = w Ω υ , P ( ξ ) = p − p ∞ ρ υ Ω   ,   Θ ( ξ ) = T − T ∞ T f   − T ∞ ,   C ( ξ ) = χ − χ ∞ χ ∞ } . (10)

Accordingly, the following ODEs and BCs are yielded: H ′ ( ξ ) + 2 F ( ξ ) = 0 , (11) F ″ ( ξ ) + G 2 ( ξ ) − F 2 ( ξ ) − F ′ ( ξ ) H ( ξ ) = 0 , (12) G ″ ( ξ ) − M G ( ξ ) − 2 F ( ξ ) G ( ξ ) − G ′ ( ξ ) H ( ξ ) = 0 , (13) P ′ ( ξ ) − H ″ ( ξ ) + M H ( ξ ) + H ( ξ ) H ′ ( ξ ) = 0 , (14) Θ ″ ( ξ ) + Pr N T Θ ′ 2 ( ξ ) + Pr N B   Θ ′ ( ξ )   C ′ ( ξ ) − Pr ⁡ H ( ξ ) Θ ′ ( ξ ) + Q E   e − δ ξ = 0 , (15) C ″ ( ξ ) + N T N B Θ ″ ( ξ ) − S c H ( ξ ) C ′ ( ξ ) − Γ S c [ 1 + ϖ Θ ( ξ ) ] n C ( ξ ) e − E 1 + ϖ Θ ( ξ ) = 0 , (16) F ( ξ = 0 ) = 0 ,   G ( ξ = 0 ) = 1 ,   H ( ξ = 0 ) = 0 ,   Θ ′ ( ξ = 0 ) − B i Θ ( ξ = 0 ) = − B i ,   C ′   ( ξ = 0 ) = − N T N B Θ ′ ( ξ = 0 ) , (17) F ( ξ → ξ ∞ ) → 0 ,   G ( ξ → ξ ∞ ) → 0 ,   P ( ξ → ξ ∞ ) → 0   ,   Θ ( ξ → ξ ∞ ) → 0 ,   C ( ξ → ξ ∞ ) → 0 . (18)

For more clarification on the influencing parameters involved in Eqs. 13–17, a technical list of the pertinent flow parameters is provided properly as seen in Table 1.

Fundamentally, the total viscous frictional coefficient C f r and the thermal transfer rate N u r featuring hydrothermally the present nanofluid flow are defined locally by: C f r   = τ w r 2 + τ w ϕ 2 ρ ( Ω r ) 2 , (19) N u r   = r q T k ( T f − T ∞ ) . (20)

Besides, the wall shear stress components ( τ w r   , τ w ϕ   ) and the wall heat flux q T are given by: τ w r = μ ( ∂ u ∂ z + ∂ w ∂ r ) z = 0 , (21) τ w ϕ   = μ ( ∂ v ∂ z + 1 r ∂ w ∂ ϕ ) z = 0 , (22) q T = − k (   ∂ T ∂ z ) z = 0 . (23)

By injecting the transformations of Eq. 10 into Eqs. 19–23, we get the reduced forms : C f = F ′ 2 ( 0 ) + G ′ 2 ( 0 ) , (24) N u = − Θ ′ ( 0 ) . (25)

As underlined above, the reduced quantities ( C f , N u ) are given by: C f = R e r 1 2 C f r   , (26) N u = R e r − 1 2 N u r   . (27)

Before starting the numerical modeling of the present nanofluid flow problem, the spatial physical domain [ 0 , ξ ∞ ] should be altered to the computational domain [ 0,1 ] by introducing another spatial variable ς into the dimensionless functions { F ( ξ ) , G ( ξ ) , H ( ξ ) , P ( ξ ) , Θ ( ξ ) , C ( ξ ) } and their derivatives { F ( m ) ( ξ ) , G ( m ) ( ξ ) , H ( m ) ( ξ ) , P ( m ) ( ξ ) , Θ ( m ) ( ξ ) , C ( m ) ( ξ ) } as follows: { { F ( ξ ) G ( ξ ) H ( ξ ) P ( ξ ) Θ ( ξ ) C ( ξ ) } = { F ( ξ ∞ ς ) G ( ξ ∞ ς ) H ( ξ ∞ ς ) P ( ξ ∞ ς ) Θ ( ξ ∞ ς ) C ( ξ ∞ ς ) } = { F ¯ ( ς ) G ¯ ( ς ) H ¯ ( ς ) P ¯ ( ς ) Θ ¯ ( ς ) C ¯ ( ς ) } ,   w h e r e   { F ( m ) ( ξ ) G ( m ) ( ξ ) H ( m ) ( ξ ) P ( m ) ( ξ ) Θ ( m ) ( ξ ) C ( m ) ( ξ ) } = 1 ξ ∞ m { F ¯ ( ς ) G ¯ ( ς ) H ¯ ( ς ) P ¯ ( ς ) Θ ¯ ( ς ) C ¯ ( ς ) }   a n d   ς = ξ ξ ∞ } . (28)

By focussing only on the physical function { F ( ξ ) , G ( ξ ) , H ( ξ ) , Θ ( ξ ) , C ( ξ ) } and adopting the transformation of Eq. 28, the following differential system is obtained: { F ¯ ( ς ) = 0 , w h e n   ς = 0 ,   1 ξ ∞ 2 F ″ ¯ ( ς ) + G ¯ 2 ( ς ) − F ¯ 2 ( ς ) − 1 ξ ∞ H ¯ ( ς ) F ′ ¯ ( ς ) = 0 , w h e n   ς ≠ 0,1 , F ¯ ( ς ) → 0 , w h e n   ς → 1 ,   G ¯ ( ς ) − 1 = 0 , w h e n   ς = 0 ,   1 ξ ∞ 2 G ″ ¯ ( ς ) − M G ¯ ( ς ) − 2 F ¯ ( ς ) G ¯ ( ς ) − 1 ξ ∞ H ¯ ( ς ) G ′ ¯ ( ς ) = 0 , w h e n   ς ≠ 0,1 ,   G ¯ ( ς ) → 0 , w h e n   ς → 1 , H ¯ ( ς ) = 0 , w h e n   ς = 0 ,   1 ξ ∞ H ′ ¯ ( ς ) + 2 F ¯ ( ς ) = 0 , w h e n   ς ≠ 0 ,   1 ξ ∞ Θ ′ ¯ ( ς ) − B i Θ ¯ ( ς ) + B i = 0 , w h e n   ς = 0 ,   1 ξ ∞ 2 Θ ″ ¯ ( ς ) + Pr N T ξ ∞ 2 Θ ′ ¯ 2 ( ς ) + Pr N B ξ ∞ 2   Θ ′ ¯ ( ς )   C ′ ¯ ( ς ) − Pr ξ ∞ H ¯ ( ς ) Θ ′ ¯ ( ς ) + Q E e − δ ξ ∞ ς = 0 , w h e n   ς ≠ 0,1 , Θ ¯ ( ς ) → 0 , w h e n   ς → 1 , 1 ξ ∞ C ′ ¯   ( ς ) = − N T ξ ∞ N B Θ ′ ¯ ( ς ) , w h e n   ς = 0 ,   1 ξ ∞ 2 C ″ ¯ ( ς ) + N T ξ ∞ 2 N B Θ ″ ¯ ( ς ) − S c ξ ∞ H ¯ ( ς ) C ′ ¯ ( ς ) − Γ S c [ 1 + ϖ Θ ¯ ( ς ) ] n C ¯ ( ς ) e − E 1 + ϖ Θ ¯ ( ς ) = 0 , w h e n   ς ≠ 0,1 , C ¯ ( ς ) → 0 , w h e n   ς → 1 } . (29)

Also, Eq. 24 and 25 are altered to: C f = 1 ξ ∞ F ′ ¯ 2 ( 0 ) + G ′ ¯ 2 ( 0 ) , (30) N u = − 1 ξ ∞ Θ ′ ¯ ( 0 ) . (31)

Numerically, the precise approximate solutions satisfying the nonlinear coupled differential system of Eq. 29 can be developed easily via an advanced hybrid algorithm GDQM-NRIT, which is principally based on the generalized differential quadrature method (GDQM) and Newton-Raphson’s iterative technique (NRIT). For this purpose, a non-uniform distribution of spatial collocation nodes ς i along the reduced computational domain [ ς 1 , ς N ] is considered based on Gauss-Lobatto’s grid points, which are defined as: ς i = 1   2   − 1   2   cos ( π i − π N − 1 ) , (32) [ ς 1 , ς N ] = [ 0,1 ] = ∪ p = 1 p = N − 1 [ ς p , ς p + 1 ] , (33) where 1 ≤ i ≤ N .

In the framework of the proposed numerical procedure, the following generalized differential quadrature approximations are invoked during the ς − discretization of the resulting equations in the one-dimension Gauss-Lobatto space: F ¯ ( m ) ( ς i ) = ∑ j = 1 N d i j ( m ) F ¯ j , (34) G ¯ ( m ) ( ς i ) = ∑ j = 1 N d i j ( m ) G ¯ j , (35) H ¯ ( m ) ( ς i ) = ∑ j = 1 N d i j ( m ) H ¯ j , (36) Θ ¯ ( m ) ( ς i ) = ∑ j = 1 N d i j ( m ) Θ ¯ j , (37) C ¯ ( m ) ( ς i ) = ∑ j = 1 N d i j ( m ) C ¯ j , (38) in which {   F ¯ ( ς j ) = F ¯ j , G ¯ ( ς j ) = G ¯ j , H ¯ ( ς j ) = H ¯ j , Θ ¯ ( ς j ) = Θ ¯ j , C ¯ ( ς j ) = C ¯ j } . (39)

Computationally, the mathematical algorithm of the weighing GDQ coefficients d i j ( m ) of Eqs. 34–38 is structured as follows: { d i j ( m ) = ∏ p = 1 ,   p ≠ i N ( ς i − ς p ) ( ς i − ς j ) ∏ p = 1 ,   p ≠ j N ( ς j − ς p ) ,   f o r   m = 1 ,   i ≠ j ,   a n d   1 ≤ i , j ≤ N , w h e r e   d i j ( 1 ) = Q i j , d i j ( m ) = m [ d i i ( n − 1 ) d i j ( 1 ) − d i j ( n − 1 ) ( ς i − ς j ) ] ,   f o r   m > 1 ,   i ≠ j ,   a n d   1 ≤ i , j ≤ N , w h e r e   d i j ( 2 ) = R i j , d i j ( m ) = − ∑ p = 1 , p ≠ i N d i p ( m ) ,   f o r   m ≥ 1 ,   i = j ,   a n d   1 ≤ i , j ≤ N ,   w h e r e   ς i = 1   2   − 1   2   cos ( π i − π N − 1 ) , ς j = 1   2   − 1   2   cos ( π j − π N − 1 ) ,   a n d   ς p = 1   2   − 1   2   cos ( π p − π N − 1 ) } . (40)

Accordingly, we obtain the following nonlinear algebraic system: { F ¯ i = 0 , w h e n   i = 1 ,   1 ξ ∞ 2 ∑ j = 1 N R i j F ¯ j + G ¯ j 2 − F ¯ j 2 − 1 ξ ∞ H ¯ i ∑ j = 1 N Q i j F ¯ j = 0 , w h e n   i ≠ 1 , N , F ¯ i → 0 , w h e n   i = N , G ¯ i − 1 = 0 , w h e n   i = 1 ,   1 ξ ∞ 2 ∑ j = 1 N R i j G ¯ j − M G ¯ i − 2 F ¯ i G ¯ i − 1 ξ ∞ H ¯ i ∑ j = 1 N Q i j G ¯ j = 0 , w h e n   i ≠ 1 , N ,   G ¯ i → 0 , w h e n   i = N , H ¯ i = 0 , w h e n   i = 1 ,   1 ξ ∞ ∑ j = 1 N Q i j H ¯ j + 2 F ¯ i = 0 , w h e n   i ≠ 1 , 1 ξ ∞ ∑ j = 1 N Q i j Θ ¯ j − B i Θ ¯ i + B i = 0 , w h e n   i = 1 ,   1 ξ ∞ 2 ∑ j = 1 N R i j Θ ¯ j + Pr N T ξ ∞ 2 ( ∑ j = 1 N Q i j Θ ¯ j ) 2 + Pr N B ξ ∞ 2   ∑ j = 1 N Q i j Θ ¯ j ∑ j = 1 N Q i j C ¯ j − Pr ξ ∞ H ¯ i ∑ j = 1 N Q i j Θ ¯ j + Q E e − δ ξ ∞ ς i = 0 , w h e n   i ≠ 1 , N , Θ ¯ i → 0 , w h e n   i = N , 1 ξ ∞ ∑ j = 1 N Q i j C ¯ j = − N T ξ ∞ N B ∑ j = 1 N Q i j Θ ¯ j , w h e n   i = 1 ,   1 ξ ∞ 2 ∑ j = 1 N R i j C ¯ j + N T ξ ∞ 2 N B ∑ j = 1 N R i j Θ ¯ j − S c ξ ∞ H ¯ i ∑ j = 1 N Q i j C ¯ j − Γ S c ( 1 + ϖ Θ ¯ i ) n C ¯ i   e − E 1 + ϖ Θ ¯ i = 0 , w h e n   i ≠ 1 , N ,   C ¯ i → 0 , w h e n   i = N } . (41)

An efficient NRIT algorithm has been developed properly to generate accurate discrete solutions { ( F ¯ i , G ¯ i , H ¯ i , Θ ¯ i , C ¯ i ) ,   w h e r e   1 ≤ i ≤ N   } for the above gigantic algebraic system with a higher order of exactitude. Sequel to this numerical treatment, the following square residual errors (SREs) are assessed numerically: Δ F ¯ = ∫ 0 1 [ 1 ξ ∞ 2 F ″ ¯ ( ς ) + G ¯ 2 ( ς ) − F ¯ 2 ( ς ) − 1 ξ ∞ H ¯ ( ς ) F ′ ¯ ( ς ) ] 2 d ς , (42) Δ G ¯ = ∫ 0 1 [ 1 ξ ∞ 2 G ″ ¯ ( ς ) − M G ¯ ( ς ) − 2 F ¯ ( ς ) G ¯ ( ς ) − 1 ξ ∞ H ¯ ( ς ) G ′ ¯ ( ς ) ] 2 d ς , (43) Δ H ¯ = ∫ 0 1 [ 1 ξ ∞ H ′ ¯ ( ς ) + 2 F ¯ ( ς ) ] 2 d ς , (44) Δ Θ ¯ = ∫ 0 1 [ 1 ξ ∞ 2 Θ ″ ¯ ( ς ) + Pr N T ξ ∞ 2 Θ ′ ¯ 2 ( ς ) + Pr N B ξ ∞ 2 Θ ′ ¯ ( ς ) C ′ ¯ ( ς ) − Pr ξ ∞ H ¯ ( ς ) Θ ′ ¯ ( ς ) + Q E e − δ ξ ∞ ς ] 2 d ς , (45) Δ C ¯ = ∫ 0 1 [ 1 ξ ∞ 2 C ″ ¯ ( ς ) + N T N B ξ ∞ 2 Θ ″ ¯ ( ς ) − S c ξ ∞ H ¯ ( ς ) C ′ ¯ ( ς ) − Γ S c [ 1 + ϖ Θ ¯ ( ς ) ] n C ¯ ( ς ) e − E 1 + ϖ Θ ¯ ( ς ) ] 2 d ς , (46)

Briefly, the most important strategic steps involved in the suggested methodological solution are outlined in Figure 2.

Once the computed SREs reach very low values, the following dimensionless physical quantities can be deduced accurately: { F ( ξ i ) = F ¯ ( ς i ) ,   w h e r e   1 ≤ i ≤ N   a n d   ξ = ξ ∞ ς , G ( ξ i ) = G ¯ ( ς i ) ,   w h e r e   1 ≤ i ≤ N   a n d   ξ = ξ ∞ ς , H ( ξ i ) = H ¯ ( ς i ) ,   w h e r e   1 ≤ i ≤ N   a n d   ξ = ξ ∞ ς , Θ ( ξ i ) = Θ ¯ ( ς i ) ,   w h e r e   1 ≤ i ≤ N   a n d   ξ = ξ ∞ ς , C ( ξ i ) = C ¯ ( ς i ) ,   w h e r e   1 ≤ i ≤ N   a n d   ξ = ξ ∞ ς } , (47) F ′ ( 0 ) = 1 ξ ∞ ∑ j = 1 N Q 1 j F ¯ j , (48) G ′ ( 0 ) = 1 ξ ∞ ∑ j = 1 N Q 1 j G ¯ j , (49) H ( ∞ ) = H ¯ ( ς N ) (50) C f = 1 ξ ∞ ( ∑ j = 1 N Q 1 j F ¯ j ) 2 + ( ∑ j = 1 N Q 1 j G ¯ j ) 2 , (51) Θ ( 0 ) = Θ ¯ ( ς 1 ) , (52) Θ ′ ( 0 ) = 1 ξ ∞ ∑ j = 1 N Q 1 j Θ ¯ j , (53) N u = − 1 ξ ∞ ∑ j = 1 N Q 1 j Θ ¯ j , (54) C ∗ ( 0 ) = χ w χ ∞ = C ¯ ( ς 1 ) + 1 , (55) C ′ ( 0 ) = 1 ξ ∞ ∑ j = 1 N Q 1 j C ¯ j . (56)

A perfect authentication of the executed GDQM-NRIT code has been demonstrated evidently in Table 2 for the quantities { F ′ ( 0 ) , G ′ ( 0 ) , H ( ∞ ) , Θ ′ ( 0 ) } by comparing the computed GDQM-NRIT values with those estimated by previously Turkyilmazoglu [27] in a certain special case study using the semi-analytical ESAT procedure. As anticipated, the extensive comparative tests performed quantitatively between the results of GDQM-NRIT and ESAT reflect forcefully the correctness of our outputted GDQM-NRIT outcomes, which are evaluated accordingly with very small square residual errors { Δ F , Δ G , Δ H , Δ Θ } . To check again the correctness of the present GDQM-NRIT numerical simulation, a general corroboration has been carried out for the studied nanofluid flow problem as shown in Table 3 by examining the accurateness order of the results given by the proposed GDQM-NRIT algorithm to those outputted additionally via another most commonly used method based on an efficient RKFM-ST numerical subroutine. Quantitatively, it is found an outstanding agreement between the results of GDQM-NRIT and RKFM-ST. Thus, the prime preliminary emphasized objectives of the present investigation can be accessed accurately via the developed GDQM-NRIT code. Furthermore, it is worth noting here that all the tabular results are provided with an absolute accuracy level of the order of 10 − 10 .

This illustrative section is dedicated especially to the physical deliberations of MHD Von Kármán’s flow that can be happened suddenly within a chemically reactive nanofluidic medium in the presence of an internal heat generation. In this respect, several physical aspects have been explored for this kind of axisymmetric flow under the significant impact of a radial magnetic source. To strengthen the onset of heat and mass transport phenomena within the nanofluidic medium, a chemical reaction mechanism is taken place destructively according to Arrhenius’s kinetics and the thermal aid of an exponentially decaying space-dependent heat generation source, in the case where the disk surface is heated convectively and unaffected by the vertical nanoparticles’ mass flux. Based on the generated GDQM-NRIT outputs, numerous dimensionless demonstrations are portrayed properly for the radial, azimuthal, and transverse velocity fields { F ( ξ ) , G ( ξ ) , H ( ξ ) } , the nanofluid temperature and nanoparticles’ molar concentration profiles { Θ ( ξ ) ,   C ( ξ ) } , the local frictional and heat transfer rate factors { C f ,   N u } , as well as the wall characteristics { Θ ( 0 ) , C * ( 0 )   } as revealed graphically and tabularly in Figures 3–15, Table 4 and Table 5. To provide noticeably comparable graphical upshots, the curves of Figures 3–15 are plotted skilfully via professional graphical tools (e.g., Grapher and Surfer) to evidence the responses of dimensionless quantities { F ( ξ ) , G ( ξ ) , H ( ξ ) , Θ ( ξ ) ,   C ( ξ ) , C * ( 0 ) } against the mounting parametric values of M − Magnetic parameter, Q E − Heat generation parameter, B i − Thermal Biot number, N T − Thermophoresis parameter, E − Activation energy parameter, N B − Brownian motion parameter, and S c − Schmidt number by varying one of these parameters and keeping the others fixed at their default values underlined in Table 1.