EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5700 ISSN 1307-5543 – ejpam.com Published by New York Business Global Numerical Analysis of Dissipative and Magnetized Reiner-Philippoff Nanofluid with Activation Energy and Cattaneo-Christov Double Diffusion Model M. Adel1,∗, M. M. Khader2,3, I. Alradaddi1, A. Alaidrous4, N. A. Mohammed4, G. M. Ismail1 1 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, Saudi Arabia 2 Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt 4 Mathemtaics Department, Faculty of Sciences, Umm Al-Qura University, Makkah, Saudi Arabia Abstract. We perform an investigation using numerical simulations to examine the influence of magnetohydrodynamics and thermal radiation on the mass transport and thermal energy properties of non-Newtonian Reiner-Philippoff nanofluids. We thoroughly examine the species response con- cerning activation energy, thermal radiation at the surface, viscous dissipation, Cattaneo-Christov double diffusions, and mass and energy transfer. This analysis also examines the impacts of an ap- plied transverse magnetic field and Ohmic heating. Using appropriate similarity variables converts the specified governing system of PDEs into a non-linear system of ODEs. We numerically solve the governing equations using the Mohand transform (MT) in conjunction with the Adomian decom- position method (ADM). The sophisticated Modified Decomposition Method (MDM) streamlines complex equations for computational solutions. It uses ADM and the MT to make sure that the series converges, giving a solution very close to the exact solution. We illustrate the temperature, species distributions, and flow velocity for the relevant parameters governing the Reiner-Philippoff model on two-dimensional charts to understand the influence of dimensionless parameters on these values. The tabulation, depiction, and interpretation of the local Nusselt number, local Sherwood number, and skin friction coefficient exemplify further engineering inquiry. We have utilized a table to illustrate the concordance between the current numerical data and previously published findings. 2020 Mathematics Subject Classifications: 41A30, 76F12, 65M60, 65N12. Key Words and Phrases: Activation energy, Nanofluid, Cattaneo-Christov double diffusions, Reiner-Philippoff model, Mohand transforms, ADM. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5700 Email addresses: m.adel@iu.edu.sa and adel@sci.cu.edu.eg (M. Adel), mmkhader@imamu.edu.sa (M. M. Khader), ialraddadi@iu.edu.sa (I. Alradaddi), aaaidrous@uqu.edu.sa (A. Alaidrous), namohammed@uqu.edu.sa (N. A. Mohammed), gismail@iu.edu.sa (G. M. Ismail) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 2 of 19 1. Introduction The study of non-Newtonian fluids has attracted a lot of interest because of its nu- merous uses in various industrial settings, such as drilling rigs, food processing, cooling systems, nuclear reactors, and organic material handling. These fluids are especially well suited for these intricate and demanding situations because of their distinctive flow behav- iors. Because of this, it is now essential to comprehend their characteristics and behaviors to maximize productivity and efficiency in these sectors. Non-Newtonian liquids can be represented by several models that are frequently used in technology and engineering do- mains. Examples of these models include the Gingham plastic model, Sisal model, Carrell- Cauda model, Jeffrey model, Caisson model, Power-law model, Ellipse model, and Jeffrey model. We explore a particular kind of non-Newtonian model called the Reiner-Philippoff fluid [16] in this work. This model stands out in particular because of its distinct rhe- ological characteristics, which make it useful for a wide range of real-world applications. The goal of this study is to gain a better understanding of the Reiner-Philippoff fluid and its possible use in industrial processes where conventional fluid assumptions are not applicable. Several researchers have thoroughly examined the Reiner-Philippoff model. For in-depth analyses and research on this subject, see references like ([2], [12], [13], [20]). These studies offer insightful information about the properties and uses of the Reiner- Philippoff fluid, facilitating a better comprehension of its behavior in diverse settings. In the same area of research, there are many researchers studied the same and similar problems for example ([11], [22], [21]): The authors conducted a comparative analysis of flow and Cattaneo-Christov heat flux in the presence of a magnetic field, taking into account the influence of nonmaterial and carbon annotations. The researchers also looked at magnetic swirling flow and the Cattaneo-Christov heat and mass flux over a stretchable cylinder. They also looked at how heat moves and melts in a Reiner-Philippoff fluid on a Darby-Herxheimer medium. One important natural occurrence is the movement of heat between two objects or within one object. Numerous academic fields rely heavily on this process: thermodynam- ics, where it is critical to comprehending energy systems; meteorology, where it impacts weather patterns and climate; engineering, where it is essential to the design and operation of machinery and structures; environmental science, where it impacts ecological dynamics; and material science, where it affects the characteristics and capabilities of various mate- rials. The Cattaneo-Christov heat flux model is the name given to this intricate mathe- matical structure. It offers a more realistic representation of non-Fourier heat conduction phenomena by taking into account thermal relaxation effects, which is an advanced method of explaining heat transfer [9]. Accurately describing thermal conduction-especially in dy- namic situations is one of the advantages of the Cattaneo-Christov heat flux model for heat transfer studies. For accurate forecasts in sectors like biotechnology and high-speed thermal processes, it is crucial to consider time-dependent effects including thermal relax- ation, which is not possible with Fourier’s equation. Numerous investigations ([7], [8], [19]) have been motivated by the significance and broad uses of this concept. Also, different nu- merical techniques studied many important problems like ([6], [10]), the authors presented M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 3 of 19 numerical methods for solving the Laplace equation’s IBM; finite difference methods un- derpin the method. Also, the authors numerically solved the period of a simple pendulum in a magnetic field. These days, nanofluid investigations are receiving a lot of interest due to their amaz- ing applications in engineering, science, and technology. Chop [5] came up with the idea of nanofluids first. Because of their improved thermal and lubricating qualities, panoti- tides are used in machining processes such as milling and turning. By examining the consequences of suspending different nonmaterials in fluids, scientists ([1], [4], [15]) have significantly advanced our understanding of nanofluid flow. With continuous study aimed at exploring and optimizing the efficiency of these sophisticated fluids in industrial appli- cations, this focus has grown, especially in the current decade. The concerned model simplifies to a system of extremely nonlinear ordinary differen- tial equations. These equations are inherently nonlinear, hence it is impossible to discover an accurate analytical solution. As such, to derive an approximation of the solution, a strong numerical technique must be used. We can manage the system’s complexity and obtain practical numerical results with this method. We addressed the given problem analytically by applying a recently developed methodology. The modified decomposition method replaces the classic Adomian decomposition technique with the Mohand trans- form, employed in this strategy. This new approach offers greater accuracy and efficiency, which makes it a significant advance in solving difficult analytical problems. Through the application of the Mohand transform ([17], [18]), the enhanced decomposition method op- timizes the solution process and yields more precise outcomes across a broader spectrum of applications. The MDM provides multiple-form solutions, culminating in the exact form solution. The solution to the resultant nonlinear system of ordinary differential equations confirms the efficacy and applicability of this method. Tables and charts are employed to compare the collected data. Building upon the insights from prior research, this study addresses a significant gap by exploring the flow dynamics of a non-Newtonian Reiner-Philippoff nanofluid over a nonlinear stretching sheet, a scenario not extensively studied in the literature. Unlike previous works, this analysis incorporates the complex effects of activation energy and the Cattaneo-Christov double diffusion model, alongside the influences of Ohmic heating and viscous dissipation, which are often overlooked in similar studies. By employing an enhanced decomposition method refined with Mohand transforms, this work provides a robust framework for tackling the challenges posed by these nonlinearities. Numerical solutions are derived and examined for a wide range of parameter values, offering new insights into the interplay of these effects. Furthermore, the method’s accuracy is vali- dated by demonstrating strong agreement with existing solutions in specific limiting cases, thereby filling a critical gap in understanding the behavior of Reiner-Philippoff nanofluids under these unique conditions. M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 4 of 19 2. Mathematical Development In the context of magnetohydrodynamics (MHD) and thermal radiation, this section provides a thorough summary of the mathematical framework used in the numerical analy- sis of Reiner-Philippoff nanofluid flow encompasses the Cattaneo-Christov double diffusion model (CADDY). The magnetic field strength, denoted by B = B0x −1 3 , is defined as the fluid flow past a nonlinearly extending sheet in the study. This formulation takes into consideration both chemical interactions and viscous dissipation. The study also includes diffusion from thermophagies, denoted by DT , and Browning diffusion, represented by DB. This study considers activation energy since it is important for nanofluid flow and has a significant effect on the rate of thermal and chemical processes, which in turn impacts the fluid’s overall behavior and performance in many applications. Characterizing the flow is a system of PDEs: the continuity equation, the momentum equation adjusted for magnetohydrodynamic phenomena, the energy equation incorporating viscous dissipation, the CCDM, the species concentration equation with activation energy phenomenon, and the chemical reactions involved. This study considers the impact of heat radiation on the sheet with the measurement and evaluation of Ohmic heating. The schematic configu- ration diagram illustrates the surface velocity, represented by the equation: uw = ax 1 3 , where a is a positive constant. Figure 1 offers a more comprehensive depiction of the flow pattern. Figure 1. Modeling geometry flow Furthermore, during the flow motion, it is posited that the pertinent temperatures remain constant. Throughout the operation, the ambient temperature T∞, indicative of the temperature distant from the sheet, and the surface temperature Tw, denoting the temperature at the surface of the stretching sheet, are both held at constant values. For M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 5 of 19 better understanding, the assumptions made for this problem are summarized below in a list for easier comprehension. i. Flow Geometry: The fluid flow is modeled as two-dimensional, with a constant velocity, and exhibits laminar behavior over the stretching surface. ii. Governing Forces: The fluid flow is affected by the presence of a magnetic field perpendicular to the flow direction, heat transfer due to radiation, and energy dissipation caused by fluid viscosity. iii. Diffusion Mechanism: The species concentration dynamics are modeled using the Cattaneo-Christov double diffusion approach and incorporate activation energy effects. iv. Similarity Transformation: Similarity transformations are used to reduce the PDEs to a system of ODEs. v. Numerical Methodology: The governing equations are solved by employing the Mohand transform in con- junction with the Adomian decomposition method. The Modified Decomposition Method is utilized to guarantee the convergence of the series solution. Considering these constraints, the governing equations for the modeled system are derived as follows [14]: ∂u ∂x + ∂v ∂y = 0, (1) ∂u ∂y = τ( µ0−µ∞ 1+ ( τ τs )2 + µ∞ ) , (2) u ∂u ∂x + v ∂u ∂y + σ ρ B2u− 1 ρ ∂τ ∂y = 0, (3) u ∂T ∂x + v ∂T ∂y − κ ρcp ( 1 + 16σ∗T 3 ∞ 3κk∗ )( ∂2T ∂y2 ) − µ ρcp ( τ ∂u ∂y ) − Ω [ DB ∂C ∂y ∂T ∂y + DT T∞ ( ∂T ∂y )2 ] − σB2 ρcp u2 + Λt [( u ∂u ∂x + v ∂u ∂y ) ∂T ∂x + ( u ∂v ∂x + v ∂v ∂y ) ∂T ∂y + u2 ∂2T ∂x2 + v2 ∂2T ∂y2 + 2uv ∂2T ∂x∂y ] = 0, (4) u ∂C ∂x + v ∂C ∂y +Λc [( u ∂u ∂x + v ∂u ∂y ) ∂C ∂x + ( u ∂v ∂x + v ∂v ∂y ) ∂C ∂y + u2 ∂2C ∂x2 + v2 ∂2C ∂y2 + 2uv ∂2C ∂x∂y ] − DB ∂2C ∂y2 − DT T∞ ∂2T ∂y2 +Kr(C − C∞) ( T T∞ )n e −Ea κT = 0. (5) M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 6 of 19 It is important to recognize that the system under study is two-dimensional. Therefore, it is thought that there are two components to the flow velocity: u in the x−axis and v in the y−axis. τ denotes shear stress, whereas τs denotes the reference shear stress. Λt represents the relaxation time for heat flux, Ω represents the effective heat capacity ratio, and Λc represents the relaxation time for mass flux. ρ denotes fluid density, µ∞ signifies ambient viscosity, ν indicates kinematic viscosity, µ0 refers to dynamic viscosity at zero shear, and σ symbolizes electrical conductivity. The concentration of a component in the fluid is denoted by C, signifying its quantity or distribution within the fluid medium, while C∞ denotes the ambient concentration. The thermal condition and total thermal energy of the nanofluid are indicated by its temperature, T . This value is essential because it sheds light on the thermal state of the nanofluid, affecting its behavior and thermal characteristics in a variety of applications. Particular boundary conditions apply to the regulation equations (2) through (5). By specifying the restrictions that are applied to the equations at the system’s edges, these conditions also define the behavior of the system at its boundaries: u = uw(x) = ax 1 3 , v = −vw, T = Tw, C = Cw, at y = 0, (6) u→ 0, C → C∞, T → T∞, at y → ∞. (7) 2.1. Dimensionless model We reformulated the governing equations and boundary conditions in a dimensionless format to enhance the efficacy of numerical analysis. By employing appropriate dimension- less variables, this transformation simplifies the equations, facilitating the identification of the primary variables influencing the system’s behavior. This approach simplifies the analysis, highlighting critical aspects and improving our comprehension of the system’s dynamics overall. Using this method simplifies the equations while highlighting the sig- nificance of the dimensionless parameters. These include the radiation parameter, the thermophoresis parameter, the chemical reaction parameter, the solutal relaxation time parameter, the Brownian motion parameter, the Eckert number, the thermal relaxation parameter, and the Prandtl number [3]: ϕ(η) = C − C∞ Cw − C∞ , θ(η) = T − T∞ Tw − T∞ , (8) τ = ρ √ νa3g(η), ψ = √ νax2/3f(η), η = y √ a ν x−1/3. (9) The continuity equation is readily fulfilled by inserting (8) and (9) into equations (2)-(5). The linked nonlinear differential equations governing motion, concentration, and energy can then be derived via the subsequent techniques, yielding the following outcomes: g = ( λγ2 + g2 γ2 + g2 ) f ′′, (10) g′ = 1 3 f ′2 +Mf ′ − 2 3 ff ′′, (11) M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 7 of 19 1 Pr ( (R+ 1)θ′′ ) +M Ecf ′2 +Ec g f ′′ + Zb ϕ ′θ′ +Υt ( η θ′′ f2 + f θ′ f ′ ) + Zt θ ′2 + 2 3 fθ′ = 0, (12) ϕ′′ + Zt Zb θ′′ − Scδr (1 + Γ θ)n e( −E 1+Γ θ )ϕ+Υc Sc ( η ϕ′′ f2 + f ϕ′ f ′ ) + 2 3 Sc fϕ ′ = 0. (13) In addition, the relevant boundary conditions are modified to meet the predefined criteria listed below. This modification guarantees the fulfillment of all requirements, upholding the procedure’s accuracy and uniformity. The following are the specific requirements: f(η) = β, f ′(η) = 1, θ(η) = 1, ϕ(η) = 1, at η = 0, (14) f ′ → 0, θ → 0, ϕ→ 0, as η → ∞. (15) Now, the model under investigation has been converted to (10)-(15), the definitions of the regulating factors are as follows: The thermal relaxation parameter is represented by Υt = Λtuw, the Brownian motion parameter by Zb = Ω(Cw−C∞)DB ν , the magnetic parameter can be expressed as M = σ ρaB 2, the thermophoresis parameter by Zt = Ω(Tw−T∞)DT νT∞ , the solutal relaxation time parameter is denoted by Υc = Λcuw, the thermal radiation parameter shown by R = 16σ∗T 3 ∞ 3κk∗ , the dimensionless activation energy variable by E = Ea κT∞ , the Reiner-Philippoff fluid parameter denoted by λ = µ0 µ∞ , the Schmidt number by Sc = ν DB , the temperature relative parameter represented by Γ = Tw−T∞ T∞ , the Prandtl number by Pr = µcp κ , the Bingham number by γ = τs ρ √ a3ν , the chemical reaction parameter by δr = Kr a and the Eckert number by Ec = u2 w cp(Tw−T∞) . 2.2. Important applicable quantities Crucial and pertinent physical parameters of practical and technological importance across various domains include the local Sherwood number Shx, the local Nusselt number Nux, and the local skin friction coefficient Cfx in the proposed physical model. The subsequent summary will help to clarify these requirements: CfxRe 1 2 = −g(0), NuxRe −1 2 = − (1 +R) θ′(0), ShxRe −1 2 = −ϕ′(0), where Re = uwx ν is the local Reynolds number. 3. Procedure Solution 3.1. Fundamental principles of the Mohand transform Definition 1: For a function f(t), the Mohand transformation indicated by M(.) is defined as [17]: M{f(t)} = F (s) = s2 ∫ ∞ 0 f(t)e−stdt, k1 ≤ s ≤ k2. M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 8 of 19 If the Mohand transform of a function f(t) is F (s) then f(t) is known as the inverse of F (s) which can be described by: M−1{F (s)} = f(t), M−1 is the inverse Mohand operator. The MT of the derivatives of the function f(t): If M{f(t)} = F (s) then we have M { f (n)(t) } = sn F (s)− sn+1f(0)− snf ′(0)− . . .− s2 f (n−1)(0), n = 1, 2, ... . (16) The MT for the power functions: M{tn} = { n! sn−1 , n ∈ N ; Γ(n+1) sn−1 , n > −1. 3.2. Implementation of modified decomposition method This part succinctly outlines the procedure of the newly adopted modified technique. To execute the MDM for addressing the proposed system (10)-(13), we will reformulate it in the subsequent operator form: f ′′(η) = N1(f, g) = ( 1 λγ2 )[ g ( γ2 + g2 ) − g2f ′′ ] , (17) g′(η) = N2(f) = 1 3 f ′2 − 2 3 ff ′′ +M f ′, (18) θ′′(η) = N3(f, g, θ, φ) = ( −Pr 1 +R )[ M Ecf ′2 + Ec g f ′′ + Zb ϕ ′θ′ +Υt ( η θ′′ f2 + f θ′ f ′ ) + Zt θ ′2 + 2 3 fθ′ ] , (19) φ′′(η) = N4(f, θ, φ) =− Zt Zb θ′′ + Scδr (1 + Γ θ)n ϕExp [ −E 1 + Γ θ ] −Υc Sc ( η ϕ′′ f2 + f ϕ′ f ′ ) − 2 3 Sc fϕ ′. (20) Take the Mohand transform of this system (17)-(20) as follows: s2F (s)− s3f(0)− s2f ′(0) = M [ N1(f, g) ] , sG(s)− s2g(0) = M [ N2(f) ] , s2Θ(s)− s3θ(0)− s2θ′(0) = M [ N3(f, g, θ, φ) ] , s2Φ(s)− s3φ(0)− s2φ′(0) = M [ N4(f, θ, φ) ] . (21) M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 9 of 19 By using the boundary conditions (14)-(15), we can solve the above algebraic system as follows: F (s) = 1 + 1 s2 M [ N1(f, g) ] , G(s) = ℓ1 s+ 1 s M [ N2(f) ] , Θ(s) = s+ ℓ2 + 1 s2 M [ N3(f, g, θ, φ) ] , Φ(s) = s+ ℓ3 + 1 s2 M [ N4(f, θ, φ) ] . (22) Take the inverse Mohand transform of the system (22) as follows: f(η) = η +M−1 [ 1 s2 M [ N1(f, g) ]] , g(η) = ℓ1 +M−1 [ 1 s M [ N2(f) ]] , θ(η) = 1 + ℓ2 η +M−1 [ 1 s2 M [ N3(f, g, θ, φ) ]] , φ(η) = 1 + ℓ3 η +M−1 [ 1 s2 M [ N4(f, θ, φ) ]] , (23) where ℓ1 = g(0), ℓ2 = θ′(0), ℓ3 = φ′(0). Consequently, the preliminary elements for the estimated solution of the specified problem will be derived as follows: f0(η) = η, g0(η) = ℓ1, θ0(η) = 1 + ℓ2 η, φ0(η) = 1 + ℓ3 η, (24) subsequently, the conclusive iterative strategy for the remaining terms is expressed as: fm+1(η) = M−1 [ 1 s2 M [ N1(f, g) ]] = M−1 [ 1 s2 M [ A1 m ]] , gm+1(η) = M−1 [ 1 s M [ N2(f) ]] = M−1 [ 1 s M [ A2 m ]] , θm+1(η) = M−1 [ 1 s2 M [ N3(f, g, θ, φ) ]] = M−1 [ 1 s2 M [ A3 m ]] , φm+1(η) = M−1 [ 1 s2 M [ N4(f, θ, φ) ]] = M−1 [ 1 s2 M [ A4 m ]] . (25) The nonlinear terms Np(f, g, θ, φ), p = 1, 2, 3, 4, are decomposed by using the Adomian polynomials defined as: Np(f, g, θ, φ) = ∞∑ m=0 Aq m, p = 1, 2, 3, 4, (26) M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 10 of 19 where, Ap m = 1 m! [ dm dλm [ Np ( ∞∑ i=0 λi fi, ∞∑ i=0 λi gi, ∞∑ i=0 λi θi, ∞∑ i=0 λi φi )]] λ=0 , p = 1, 2, 3, 4. (27) Considering these formulas, we can calculate the initial Adomian polynomials as follows: A1 0 = ( 1 λγ2 )[ g0 ( γ2 + g20 ) − g20f ′′ 0 ] = ( 1 λγ2 )[ ℓ1 ( γ2 + ℓ21 ) − ℓ21 ℓ4 ] , A2 0 = 1 3 f ′20 − 2 3 f0f ′′ 0 +M f ′0 = 1 3 +M, A3 0 = ( −Pr 1 +R )[ M Ecf ′20 + Ec g0 f ′′ 0 + Zb ϕ ′ 0θ ′ 0 +Υt ( η θ′′0 f 2 0 + f0 θ ′ 0 f ′ 0 ) + Zt θ ′2 0 + 2 3 f0θ ′ 0 ] = ( −Pr 1 +R )[ M Ec+ Ec ℓ1 ℓ4 + Zbℓ2 ℓ3 + Ztℓ 2 2 ] , A4 0 = −Zt Zb θ′′0 + Scδr (1 + Γ θ0) n ϕ0Exp [ −E 1 + Γ θ0 ] −Υc Sc ( η ϕ′′0 f 2 0 + f0 ϕ ′ 0 f ′ 0 ) − 2 3 Sc f0ϕ ′ 0 = −Zt Zb ℓ5 + Scδr (1 + Γ)n Exp [ −E 1 + Γ ] . (28) Considering the iteration formulas (25), we can calculate the first components of the approximate solution as follows: f1(η) = M−1 [ 1 s2 M [ A1 0 ]] = ( 1 2λγ2 ) η2 [ ℓ1 ( γ2 + ℓ21 ) − ℓ21 ℓ4 ] , g1(η) = M−1 [ 1 s M [ A2 0 ]] = η [ 1 3 +M ] , θ1(η) = M−1 [ 1 s2 M [ A3 0 ]] = 1 2 η2 [( −Pr 1 +R )[ M Ec+ Ec ℓ1 ℓ4 + Zbℓ2 ℓ3 + Ztℓ 2 2 ]] , φ1(η) = M−1 [ 1 s2 M [ A4 0 ]] = 1 2 η2 [ −Zt Zb ℓ5 + Scδr (1 + Γ)n Exp [ −E 1 + Γ ]] , ... . (29) Consequently, the approximate solution is derived by aggregatingm of the estimated terms as follows: f(η) = m−1∑ k=0 fk(η), g(η) = m−1∑ k=0 gk(η), θ(η) = m−1∑ k=0 θk(η), φ(η) = m−1∑ k=0 φk(η). (30) The series form solution converges to the exact solution as m approaches infinity. The values of the quantities ℓk, k = 1, 2, 3 can be determined by applying certain boundary conditions (14)-(15). M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 11 of 19 4. Verification of Adomian Decomposition Approach Through a comparison with the results from the body of previous research, particularly the work by Sajid et al. [16]. Table 1 confirms for the current findings. Validating the results acquired by the Mohand transform and the Adomian decomposition approach is the goal of this comparison. The Prandtl number Pr represents the numerical values of the following comparison with R = Zb =M = Zt = 0. The precision and dependability of the results of the current investigation are demonstrated by this comparison. It is evident from the comparison that the existing research approach and its outcomes are credible. Table 1. −θ′(0) values in varying Pr with R = Zb =M = Zt = 0 Pr Sajid et al. [16] Present work 1.0 0.556065 0.556064892 1.5 0.727928 0.727927745 2.0 0.873992 0.873991029 2.5 1.012056 1.012055496 5. Results and Discussion This section aims to demonstrate the influence of temperature, concentration, veloc- ity, the local Sherwood number, the skin friction coefficient, and the local Nusselt number as affected by the following parameters: chemical reaction parameter, Brownian motion parameter, magnetic parameter, thermophoresis parameter, Bingham number, Reiner- Philippoff fluid constraint, solutal relaxation time parameter, activation energy parameter, Eckert number, Prandtl number, and temperature. The Adomian decomposition method, grounded in the Mohand transform, offers a numerical characterization of the system governing the model under certain physical conditions. A variation in the magnetic pa- rameter M can influence the flow behavior, as illustrated in Figure 2, which demonstrates the impact of the magnetic field parameter on temperature θ(η), concentration ϕ(η), and velocity f ′(η). Graphing the data indicates that increasing the magnetic parameter M reduces the velocity gradient, enhances the temperature of the nanofluid, and slightly in- creases concentration. The fluid’s velocity diminishes due to elevated magnetic parameter values generating a Lorentz force that counteracts fluid motion. M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 12 of 19 Figure 2 (a) θ(η) and f ′(η) for assorted M (b) ϕ(η) for assorted M Reiner-Philippoff nanofluid flow behavior in terms of temperature θ(η), concentration ϕ(η), and velocity f ′(η) is illustrated in Figure 3 as a function of the Bingham number γ. First, it’s crucial to understand that a Newtonian fluid is indicated by a Bingham number of zero (γ = 0). Stated differently, a fluid that exhibits γ = 0 shows no yield stress and, as a result, follows the standard Newtonian fluid behavior in which the viscosity is independent of the applied shear rate. The graph demonstrates that while the fluid’s concentration and velocity climb with increasing Bingham number, the thermal gradient falls. This happens because a fluid is more resistant to deformation when its Bingham number is larger, which denotes a bigger yield stress. Because of the increased internal friction, this increased resistance decreases heat transfer. In addition, by promoting more consistent flow and improved particle distribution, the increased yield stress allows for higher fluid velocities and concentrations. Figure 3 (a) θ(η) and f ′(η) for assorted γ (b) ϕ(η) for assorted γ The Reiner-Philippoff parameter’s λ behavior about the temperature θ(η), velocity f ′(η), and concentration ϕ(η) fields is depicted in Figure 4. Firstly, we must remember that the fluid also exhibits Newtonian behavior at λ = 1, with a constant viscosity that remains constant at different shear rates. On the other hand, when λ is less than 1, the fluid is considered dilatant, indicating that its viscosity increases as the shear rate increases, M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 13 of 19 making it more resistant to flow under larger stresses. In contrast, the fluid is considered pseudo-plastic when λ > 1, meaning that when the shear stress increases, it will flow more readily due to its decreasing viscosity with increasing shear rate. The graph indicates that the pseudo-plastic fluid’s concentration and velocity exceed those of the dilatant and Newtonian fluids when the Reiner-Philippoff parameter improves. By contrast, an inverse correlation exists between the temperature fields and the Reiner-Philippoff parameter. Figure 4 (a) θ(η) and f ′(η) for assorted λ (b) ϕ(η) for assorted λ Figure 5 shows the impact of thermal relaxation time parameter Υt and solutal relax- ation time parameter Υc on concentration and thermal fields. As the thermal relaxation time parameter increases, the thermal gradient increases and the concentration field de- creases. A steeper temperature gradient is the result of heat diffusing over a longer period, which may be described physically by the longer thermal relaxation time. Yet, the con- centration field decreases when heat is dispersed over a greater area because the thermal gradients that drive the concentration distribution are less strong. Moreover, the concen- tration gradient rises, and the associated thermal field falls with a boost in the solutal relaxation time parameter. This is physically accounted for by the fact that the ther- mal field drops as a result of the modified concentration dynamics, but the concentration gradient is enhanced by the longer solutal relaxation time, which allows the solute to accumulate more effectively. Figure 5 (a) θ(η) and ϕ(η) for assorted Υt (b) θ(η) and ϕ(η) for assorted Υc M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 14 of 19 Figure 6 shows the impact of gradients in temperature θ(η) and concentration ϕ(η) for increasing radiation parameter R and Eckert number Ec values. The graph shows that while concentration distribution falls, thermal distribution is enhanced by rising values of the radiation parameter or Eckert number. The reason for this is that higher radiation promotes heat transfer, which results in a more even distribution of heat, and higher Eckert numbers enhance viscous dissipation and heat creation. Since temperature-driven solute transport is reduced as a result of the higher thermal effects, the concentration gradient is lowered. Figure 6 (a) θ(η) and ϕ(η) for assorted Ec (b) θ(η) and ϕ(η) for assorted R Figure 7 depicts the influence of the Brownian motion parameter Zb and the ther- mophoresis parameter Zt on the thermal θ(η) and concentration ϕ(η) fields. The plotted figure demonstrates the increase in temperature distribution and the decrease in con- centration distribution as the Brownian motion parameter escalates. Furthermore, an improvement in the temperature and concentration fields is brought about by the ther- mophoresis parameter’s expanding values. The improvement in thermal and concentration distributions can be attributed to the enhanced thermophoresis, which propels particles from warmer to colder areas with greater efficiency. Figure 7 (a) θ(η) and ϕ(η) for assorted Zb (b) θ(η) and ϕ(η) for assorted Zt M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 15 of 19 Figure 8 displays the concentration profile change for various values of the chemical reaction parameter δr or the activation energy parameter E. In this case, the mass dis- tribution is enhanced by greater activation energy parameter values while it is weakened by higher chemical reaction parameter values. This can be explained physically as faster chemical reaction rates use more reactants, which reduces mass distribution, but increases activation energy makes it simpler for particles to overcome energy barriers, enhancing mass distribution. Figure 8 (a) ϕ(η) for assorted δr (b) ϕ(η) for assorted E Skin friction (CfxRe 1 2 ), Sherwood number (ShxRe −1 2 ), and Nusselt number (NuxRe −1 2 ) fluctuations in response to various controlling parameters are shown in Table 2. These parameters affect the mass transfer rate, heat transfer rate, and surface friction, respec- tively. The data show how these modifications affect these parameters. In the domains of fluid dynamics, heat transfer, and mass transfer, in particular, these values are cru- cial. Systems in engineering applications like heat management, chemical processing, and aerospace require them to be carefully designed and optimized. Richer skin friction coeffi- cient and poorer local Nusselt number are caused by rising amounts of the magnetic field parameter, Bingham number, and Reiner-Philippoff fluid. M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 16 of 19 Table 2. ShxRe −1 2 , CfxRe 1 2 , and NuxRe −1 2 as function of some of governing parameters with β = 0.5, Sc = 2.0, n = 0.1, P r = 3.0, E = 1.0 and Γ = 0.3 M γ λ Υt Υc Ec R Zb Zt δr CfxRe 1 2 NuxRe −1 2 ShxRe −1 2 0.0 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.2 0.881327 0.538817 1.18721 1.0 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.394642 0.358658 1.16321 2.0 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.750413 0.232276 1.15214 0.5 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.170692 0.464874 1.02306 0.5 0.3 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.200923 0.435463 1.17507 0.5 0.8 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.270380 0.405517 1.33093 0.5 0.1 0.5 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.150791 0.469372 1.02313 0.5 0.1 2.0 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.178753 0.436178 1.17256 0.5 0.1 6.0 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.226634 0.407493 1.27121 0.5 0.1 1.5 0.0 0.1 0.1 0.5 0.8 0.2 0.2 1.170692 0.514926 1.16189 0.5 0.1 1.5 0.2 0.1 0.1 0.5 0.8 0.2 0.2 1.170692 0.378464 1.18108 0.5 0.1 1.5 0.3 0.1 0.1 0.5 0.8 0.2 0.2 1.170692 0.340872 1.18647 0.5 0.1 1.5 0.1 0.0 0.1 0.5 0.8 0.2 0.2 1.170692 0.395878 1.26859 0.5 0.1 1.5 0.1 0.2 0.1 0.5 0.8 0.2 0.2 1.170692 0.479547 1.08722 0.5 0.1 1.5 0.1 0.3 0.1 0.5 0.8 0.2 0.2 1.170692 0.515606 1.01995 0.5 0.1 1.5 0.1 0.1 0.0 0.5 0.8 0.2 0.2 1.170692 0.609283 1.09733 0.5 0.1 1.5 0.1 0.1 0.2 0.5 0.8 0.2 0.2 1.170692 0.277313 1.19814 0.5 0.1 1.5 0.1 0.1 0.4 0.5 0.8 0.2 0.2 1.170692 0.047912 1.29726 0.5 0.1 1.5 0.1 0.1 0.1 0.0 0.8 0.2 0.2 1.170692 0.270952 0.27095 0.5 0.1 1.5 0.1 0.1 0.1 1.0 0.8 0.2 0.2 1.170692 0.579297 1.17214 0.5 0.1 1.5 0.1 0.1 0.1 2.0 0.8 0.2 0.2 1.170692 0.813973 1.13532 0.5 0.1 1.5 0.1 0.1 0.1 0.5 0.5 0.2 0.2 1.170692 0.665106 1.10042 0.5 0.1 1.5 0.1 0.1 0.1 0.5 1.0 0.2 0.2 1.170692 0.321876 1.19234 0.5 0.1 1.5 0.1 0.1 0.1 0.5 1.5 0.2 0.2 1.170692 0.127043 1.21107 0.5 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.0 0.2 1.170692 0.509556 1.19799 0.5 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.2 1.170692 0.437905 1.17259 0.5 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.5 0.2 1.170692 0.348344 1.16616 0.5 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.0 1.170692 0.467644 1.02194 0.5 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.2 0.5 1.170692 0.408185 1.35451 0.5 0.1 1.5 0.1 0.1 0.1 0.5 0.8 0.2 1.0 1.170692 0.377344 1.59455 Also, it is demonstrated that mass transfer is reduced by the solutal relaxation time parameter and enhanced by the chemical reaction parameter. This demonstrates their opposing effects on the mass transfer process, showing that a greater chemical reaction parameter speeds mass transfer while an increase in solutal relaxation time slows it down. Additionally, the same table shows that a rise in the Eckert number increases mass trans- mission while decreasing heat transfer. This is because greater viscous dissipation increases M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5700 17 of 19 species mixing, which enhances mass transfer, but also increases the amount of heat ab- sorbed by the fluid, decreasing heat transfer efficiency. A little increase in the mass transfer rate and a decrease in the heat transfer rate are also observed when the Brownian motion parameter is raised. Additionally, since enhanced thermophoresis generates higher particle movement due to temperature gradients, it disrupts the uniform distribution of heat and mass and reduces transfer efficiency. As a result, raising the thermophoresis parameter values often lowers both mass and heat transfer rates. 6. Conclusions Under consideration are activation energy, thermal radiation, Ohmic heating, and vis- cous dissipation in the flow of a hydromagnetic non-Newtonian Reiner-Philippoff nanofluid. The theory of Cattaneo-Christov double diffusions is used in place of the traditional Fick’s and Fourier’s laws during the modeling procedure. By adding relaxation durations into the diffusion equations, this method takes into account the finite speed of thermal and mass diffusions, resulting in a more accurate description of the actual processes. Mohand transform combined with the Adomian decomposition approach yields the numerical so- lution for the converted flow governing the model. The numerical findings’ generated flow fields are shown while being influenced by several factors. Tabular data and graphics are utilized to leverage the limits on temperature concentration and velocity. The concen- tration profiles are enhanced by a surge in the Bingham number, the Reiner-Philippoff fluid parameter, and the solutal relaxation time parameter. The velocity scale grows as the Bingham number rises, but it decreases when the magnetic field parameter increases. In contrast to the chemical reaction parameter, concentration distribution improves with increasing the activation energy parameter. Higher chemical reaction parameter values cause the concentration layer to get more compact, whilst higher activation energy pa- rameter values cause it to expand. A greater estimate of the magnetic field parameter, the Bingham number, and the Reiner-Philippoff fluid enriches the skin friction coefficient and impoverishes the local Nusselt number. About the solutal relaxation time parameter, mass transfer is shown to be reduced, whilst the chemical reaction parameter exhibits enhancing behavior. 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